# Topological representations of motion groups and mapping class groups – a unified functorial construction

Martin Palmer, Arthur Soulié

► To cite this version:

Martin Palmer, Arthur Soulié. Topological representations of motion groups and mapping class groups – a unified functorial construction. Annales Henri Lebesgue, 2024, 7, pp.409-519. 10.5802/ahl.204. hal-05123144

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Submitted on 20 Jun 2025

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# Topological representations of motion groups and mapping class groups – a unified functorial construction

Martin Palmer and Arthur Soulié

11th January 2024

## Abstract

For groups of a topological origin, such as braid groups and mapping class groups, an important source of interesting and highly non-trivial representations is given by their actions on the twisted homology of associated spaces; these are known as *homological representations*. Representations of this kind have proved themselves especially important for the question of *linearity*, a key example being the family of topologically-defined representations introduced by Lawrence and Bigelow, and used by Bigelow and Krammer to prove that braid groups are linear. In this paper, we give a unified foundation for the construction of homological representations using a functorial approach. Namely, we introduce *homological representation functors* encoding a large class of homological representations, defined on categories containing all *mapping class groups* and *motion groups* in a fixed dimension. These source categories are defined using a topological enrichment of the Quillen bracket construction applied to categories of decorated manifolds. This approach unifies many previously-known constructions, including those of Lawrence-Bigelow, and yields many new representations.

## Introduction

For a smooth manifold $M$ with non-empty boundary and a closed submanifold $Y$ of its interior, the mapping class group $MCG(M, Y)$ is the group of isotopy classes of diffeomorphisms of $M$ fixing its boundary pointwise and fixing $Y$ setwise. The corresponding *motion group* $Mot_Y(M)$ is the fundamental group of the space of embeddings of $Y$ into the interior of $M$ up to diffeomorphisms of $Y$. (When $Y$ is orientable, there is also an oriented variant $Mot_Y^+(M)$ of the motion group, for which we require diffeomorphisms of $Y$ to be orientation-preserving.) For example, if $\mathbb{D}^2$ denotes the closed unit 2-disc and $\overline{n}$ a set of $n \ge 1$ distinct points in its interior, the classical braid group on $n$ strands $B_n$ is isomorphic both to the mapping class group $MCG(\mathbb{D}^2, \overline{n})$ and to the motion group $Mot_{\overline{n}}(\mathbb{D}^2)$.

The representation theory of *mapping class groups* and of *motion groups* is very rich, and the subject of much active research – see Birman and Brendle's survey [BB05, §4] for braid groups or Margalit's expository paper [Mar19] for more general mapping class groups of surfaces. A reason for this is that representation theory transforms abstract group theory problems into linear algebra questions, which are generally more accessible. In particular, a group is said to be *linear* if it acts faithfully on a finite-dimensional vector space. Whether or not a group satisfies this property is a fundamental question: if it does, the group is isomorphic to a subgroup of a general linear group over a field and falls into the class of *matrix groups*, and thus automatically satisfies many remarkable properties for free; see for instance [Sup76]. Classical braid groups provide a famous positive answer to this question by Bigelow and Krammer [Big01; Kra02], as well as the mapping class group of a torus which is classically known to be isomorphic to the symplectic group $Sp_2(\mathbb{Z})$ (see for example [FM12, Th. 2.5]), or the mapping class groups of a genus two surface thanks to the work of Bigelow and Budney [BB01]. In contrast, there are also families with a negative answer, such as automorphism groups of free groups — which may be viewed as mapping class groups of certain 3-manifolds — by Formanek and Procesi [FP92]. It is known by [But16] (see also [KLS19,

2020 Mathematics Subject Classification: Primary: 20C12, 20F36, 57K20; Secondary: 18B40, 20C07, 20J05, 55R80, 57M07, 57M10.

Key words and phrases: Homological representations, mapping class groups, surface braid groups, loop braid groups, motion groups, Lawrence-Bigelow representations.
Cor. 1.4(b)]) that any faithful, finite-dimensional representation of mapping class groups of genus at least 3 must be in characteristic zero. However, in general, the question of linearity remains a wide open fundamental problem for the vast majority of mapping class groups and motion groups; see [BH01] or [Mar19, §1].

**Homological representations.** In order to understand the representation theory of mapping class groups and motion groups, we view them not just as abstract groups, but use their geometric and topological structures. Combining this philosophy (of studying these groups via their *topology*) with the philosophy of representation theory itself (of studying groups via *linear algebra*), one is led naturally to homology. More precisely, the idea consists in constructing representations of these groups via their actions on the homology of some topological spaces naturally associated to them. Such representations are said to be *homological*.

As an archetypal example, for each braid group $B_n$, Lawrence [Law90] and Bigelow [Big01] constructed a by now well-known representation $\mathfrak{L}\mathfrak{B}_k(n)$ for each $k \ge 1$, called the $k$-th Lawrence-Bigelow representation, defined as follows. Let $\mathbb{D}_n$ be the $n$-punctured disc and $C_k(\mathbb{D}_n)$ the configuration space of $k$ unordered points in $\mathbb{D}_n$. Lawrence [Law90, §2] and Bigelow [Big01, §1.2] define via a geometrical method regular covering spaces $C_k(\mathbb{D}_n)^{\phi_k}$ of $C_k(\mathbb{D}_n)$, with deck transformation group $Q_k = \mathbb{Z}$ if $k=1$ and $\mathbb{Z}^2$ if $k \ge 2$; see §3.2.1.1 for more details. The representation $\mathfrak{L}\mathfrak{B}_k(n)$ is then defined by the natural action of $B_n$ on the homology group $H_k(C_k(\mathbb{D}_n)^{\phi_k}; \mathbb{Z})$. In particular, $\mathfrak{L}\mathfrak{B}_1(n)$ is the reduced Burau representation originally introduced by Burau [Bur35], while $\mathfrak{L}\mathfrak{B}_2(n)$ corresponds to the Lawrence-Krammer-Bigelow representation, which Bigelow [Big01] and Krammer [Kra02] independently proved to be faithful. Moreover, the representations used to prove the linearity of the mapping class groups of a torus and of a genus two surface are also homological; see [FM12, Th. 2.5] and [BB01, §3]. In particular, the representations that have thus far provided positive answers for the linearity question have all been homological representations. This suggests that a systematic treatment of constructing homological representations that works for all motion groups and all mapping class groups is an important and natural avenue of investigation related to the linearity question for these families of groups.

**A functorial approach.** In parallel, another point of view adopted in this paper is to treat mapping class groups and motion groups simultaneously in families. A family of groups means a collection of groups $G_n$ indexed by the natural numbers $\mathbb{N}$ equipped with morphisms $G_n \to G_{n+1}$. Typically, we consider a set of geometrically-consistent manifolds $\{M_n\}_{n \in \mathbb{N}}$ in order to construct a family of mapping class groups $\text{MCG}(M_n) \to \text{MCG}(M_{n+1})$, or a set of geometrically-consistent submanifolds $\{Y_n\}_{n \in \mathbb{N}}$ of a fixed manifold $M$ to construct a family of motion groups $\text{Mot}_{Y_n}(M) \to \text{Mot}_{Y_{n+1}}(M)$. For example, we consider braid groups as a family thanks to the inclusions $B_n \hookrightarrow B_{n+1}$ induced by adding a strand on the left. The purpose of considering groups in families is to require representations to respect the coherences that naturally arise between the different groups in the family. More precisely, for a family of groups $\{g_n: G_n \to G_{n+1}\}_{n \in \mathbb{N}}$ and an associative, unital ring $R$, a collection of representations $\{\rho_n: G_n \to \text{Aut}_R(V_n)\}_{n \in \mathbb{N}}$ is *coherent* if it comes equipped with module homomorphisms $v_n: V_n \to V_{n+1}$ such that $v_n$ is *equivariant* with respect to $g_n$, meaning that for all $g \in G_n$ and $x \in V_n$ we have $v_n(g.x) = g_n(g).v_n(x)$.

Furthermore, we consider a slightly larger notion of representation than the classical one of a group homomorphism $\rho_n: G_n \to \text{Aut}_R(V_n)$: we generally allow $\rho_n$ to take values in the larger group $\text{Aut}_Z(V_n) \supset \text{Aut}_R(V_n)$ and assume that it comes equipped with an action $r_n: G_n \to \text{Aut}_{\text{Rings}}(R)$ encoding its failure to respect the $R$-module structure of $V_n$; see Definition 2.3. The representation $\rho_n$ is then said to be *twisted*; if $r_n$ is trivial, then we recover the classical notion, and $\rho_n$ is said to be *genuine* or *untwisted*.

Now, the notion of coherent (twisted or genuine) representations of a family of groups can be encoded in a functorial way as follows. We denote by $\text{Mod}_R$ the category of right $R$-modules. We introduce the larger category $\text{Mod}_R \subseteq \text{Mod}_R^{\text{tw}}$ in which morphisms are permitted to act on the underlying ring $R$; we call this the category of *twisted right* $R$-modules; see Definition 2.3. Let $\mathbf{G}$ be the groupoid with objects indexed by non-negative integers, with the groups $\{G_n\}_{n \in \mathbb{N}}$ as automorphism groups and with no morphisms between distinct objects. Let us tentatively assume that there exists a category $\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$ containing $\mathbf{G}$ as its underlying groupoid and with a preferred morphism $\iota_n: n \to n+1$ for each object $n$, satisfying $\iota_n \circ g = g_n(g) \circ \iota_n$ for each $g \in G_n$. Then,
functors

$$ \langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle \to \mathrm{Mod}_{R}^{\mathrm{tw}} \quad \text{and} \quad \langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle \to \mathrm{Mod}_{R} \qquad (0.1) $$

give us coherent representations of {$G_n$}$_{n \in \mathbb{N}}$, which are respectively twisted and untwisted over $R$. It is an important question whether or not our representations take values in the untwisted subcategory $\mathrm{Mod}_R \subseteq \mathrm{Mod}_R^{\mathrm{tw}}$. Indeed, if this is not the case, the encoded representations are not genuine representations of the family of groups, since the groups also act on the ground ring $R$. Although one may always find a subring $R' \subset R$ over which the representations become genuine (i.e. untwisted), the representations typically also become infinite-dimensional as $R'$-modules. See [BPS21; PS23] for examples of such situations, where the representations are finitely generated as $R$-modules with $R = \mathbb{Z}[Q]$ for $Q$ an infinite group, while $R' = \mathbb{Z}$. We introduce a general criterion for the representations that we construct to be untwisted in §2.2.3.

The category $\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle$ is defined by the Quillen bracket construction $\langle -, - \rangle$, introduced in [Gra76, p.219], applied to a monoidal groupoid $\mathcal{G}_{\circ}$ and a $\mathcal{G}_{\circ}$-module groupoid $\mathcal{M}_{\circ}$. That there exist such groupoids $\mathcal{G}_{\circ}$ and $\mathcal{M}_{\circ}$ such that the category $\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle$ contains the groupoid $\mathbf{G}$ as its underlying groupoid is a mild assumption, naturally satisfied in all of our examples. For instance, we consider the braid groupoid $\beta$ (see [Mac98, Chap. XI, §4]) and take $\mathcal{G}_{\circ} = \mathcal{M}_{\circ} = \mathbf{G} = \beta$ to deal with the braid groups; see §3.1.2. We provisionally defer the treatment of this question until after Theorem B. For now, we assume that this category exists and mention that, for the purpose of constructing homological representations, we first introduce a topologically-enriched semicategory $\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle^{\mathrm{t}}$, and define $\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle := \pi_0(\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle^{\mathrm{t}})$ by taking path-components of all morphism spaces (this is a priori just a semicategory, but turns out to be a category). From now on, we consider a family $\{g_n: G_n \to G_{n+1}\}_{n \in \mathbb{N}}$ of mapping class groups or motion groups in a fixed dimension $d$, contained in a category $\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle = \pi_0(\langle \mathcal{G}_{\circ}, \mathcal{M}_{\circ} \rangle^{\mathrm{t}})$ as above.

In addition, thanks to our functorial point of view, notions of polynomiality may be introduced for functors of the form (0.1). We refer the reader to [RW17, §4.1] or [PS23, §4.1] for a detailed introduction to this notion. In particular, a significant application of polynomiality is homological stability with twisted coefficients for several families of mapping class groups and motion groups: see the work of Randal-Williams and Wahl [RW17, Theorems D, 5.26 and I]; this was also proven earlier by Ivanov [Iva93] and Boldsen [Bo12] for mapping class groups of orientable surfaces. Also, certain notions of polynomiality offer a classifying tool for functors (see [DV19, §1.3] for instance), while the representation theory of mapping class groups and motion groups is known to be wild (for example, this is proven by Erdmann and Nakano [EN02] for braid groups on $n \ge 6$ strands). This is therefore another argument supporting our functorial approach to representations. The study of polynomiality for homological representation functors of certain mapping class groups and motion groups is the central topic of the sequel paper [PS23].

In summary, our goal is to develop functorial constructions of homological representations that:

* apply naturally to all mapping class groups and motion groups in a fixed dimension (globality),

* respect the natural coherences between these groups (functoriality),

* produce a wide range of new, interesting and highly non-trivial representations (richness), for the purpose of studying questions of linearity and of polynomiality.

**Functorial constructions of homological representations.** Our first main result is Theorem A below, which introduces functorial constructions of coherent representations for the family of groups {$g_n: G_n \to G_{n+1}$}$_{n \in \mathbb{N}}$. They depend on the following parameters:

* the dimension $d$ of the family of mapping class groups or motion groups, which we assume from now on to be different from 4. The reason for this restriction is related to establishing the existence of a certain decomposition of embedding spaces; see Lemma 1.53 and Remark 1.54;

* a closed submanifold $Z \subset \mathbb{R}^d$;

* an open subgroup $\mathbf{G}$ of the group $\mathrm{Diff}(Z)$ of diffeomorphisms of $Z$;

* an integer $\ell \ge 1$, indexing the $\ell$-th lower central series functor $\Gamma_{\ell}: \mathrm{Grp} \to \mathrm{Grp}$ defined by sending a group $\mathbf{G}$ to $\Gamma_{\ell}(\mathbf{G})$, see Example 2.23;

* a non-negative integer $i \ge 0$, indexing the degree of the twisted homology functor $H_i$, see Proposition 2.14.

**Theorem A** (Theorems 2.19, 2.28 and 2.62) For a fixed set of parameters {$i, Z, G, \ell$} as described above, the action of the family of groups {$g_n: G_n \to G_{n+1}$}$_{n \in \mathbb{N}}$ on the twisted homology of certain
$\pi_0$: $\langle \mathcal{G}_o, \mathcal{M}_o \rangle^t \to \langle \mathcal{G}_o, \mathcal{M}_o \rangle$ given by taking path-components of morphism spaces; see Lemma 2.16. The desired output is the functor $L_i(F_{(Z,G,\ell)}): \langle \mathcal{G}_o, \mathcal{M}_o \rangle \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}]}^{\mathrm{tw}}$; this is a coherent family of (possibly twisted) representations of $\{G_n\}_{n \in \mathbb{N}}$. A more elaborate version of diagram (0.5), allowing one to take fibrewise tensor products with different coefficient systems $V$, is described in diagram (2.12) and Definition 2.20, but the underlying ideas are the same. We refer the reader to §2.1.5 for the detailed procedure defining diagram (0.5) and its elaboration (2.12).

**Categorical framework.** Let us now tackle the question of defining the topologically-enriched semicategory $\langle \mathcal{G}_o, \mathcal{M}_o \rangle^t$, which is a topological lift of the Quillen bracket category $\langle \mathcal{G}_o, \mathcal{M}_o \rangle$. This is essentially solved by introducing a certain topologically-enriched semicategory $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$ as follows.

For each fixed dimension $d \ge 2$, we construct a semi-monoidal topologically-enriched groupoid $(\mathcal{D}_d, \mathfrak{h})$, called the *decorated $d$-manifolds groupoid*; see Definition 1.16. The objects of $\mathcal{D}_d$ are pairs $(M, A)$ with $M$ a smooth $d$-manifold and $A$ a closed submanifold embedded in the interior of $M$ (together with some auxiliary data needed to construct boundary connected sums); see Definition 1.11. Its morphism spaces consist of *decorated diffeomorphisms*, namely diffeomorphisms of pairs of manifolds compatible with the auxiliary data; see Definitions 1.12 and 1.14. The semi-monoidal structure $\mathfrak{h}$ is given by the boundary connected sum of manifolds; see Proposition 1.22. See also Remark 1.23 for an explanation of why $\mathfrak{h}$ is only a *semi-monoidal structure*; this is also why $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$ is only a *semicategory*. Furthermore, we introduce in §1.4.1 a topological enrichment of the Quillen bracket construction $\langle -, - \rangle$ [Gra76, p.219]; see Proposition 1.60. The topologically-enriched semicategory $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$ is then constructed by applying the topologically-enriched Quillen bracket construction to the semi-monoidal topologically-enriched groupoid $(\mathcal{D}_d, \mathfrak{h})$.

Our second main result establishes crucial properties of the semicategory $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$, by giving a geometric description of its morphism spaces and by describing its interaction with the path-component functor $\pi_0$:

**Theorem B (Corollary 1.80)** For $d \ne 4$, the morphism spaces $\langle \mathcal{D}_d, \mathcal{D}_d \rangle(M, N)$ of the topologically-enriched semicategory $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$ may be identified with the spaces $\mathrm{Emb}_{\mathrm{dec}}(M, N)$ of decorated embeddings introduced in Definition 1.49.

Applying the path-component functor $\pi_0$ to all morphism spaces, there is an isomorphism of categories:

$$ \pi_0(\langle \mathcal{D}_d, \mathcal{D}_d \rangle) \cong \langle \pi_0(\mathcal{D}_d), \pi_0(\mathcal{D}_d) \rangle. \qquad (0.6) $$

Moreover, $\pi_0(\mathcal{D}_d)$ is the underlying groupoid of $\pi_0(\langle \mathcal{D}_d, \mathcal{D}_d \rangle)$.

The above description of the morphism spaces of $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$ as decorated embedding spaces (see Proposition 1.76), as well as the isomorphism (0.6) (see Proposition 1.66), crucially require a technical Serre fibration result for decorated diffeomorphism groups given in Theorem 1.55.

The topologically-enriched groupoid $\mathcal{D}_d$ contains, by construction, all diffeomorphism groups of $d$-manifolds equipped with configurations of submanifolds. It therefore follows that its groupoid of path-components $\pi_0(\mathcal{D}_d)$, and hence also the category $\pi_0(\langle \mathcal{D}_d, \mathcal{D}_d \rangle)$ of which $\pi_0(\mathcal{D}_d)$ is the underlying groupoid, contains all mapping class groups of $d$-manifolds, as well as all $d$-dimensional motion groups as normal subgroups; see Remark 1.43. We may thus realise the aforementioned topologically-enriched semicategory $\langle \mathcal{G}_o, \mathcal{M}_o \rangle^t$ as a certain subsemicategory $\langle \mathcal{G}, \mathcal{M} \rangle$ of $\langle \mathcal{D}_d, \mathcal{D}_d \rangle$; see Construction 3.1. In brief, we consider subgroupoids $\mathcal{G}$ and $\mathcal{M}$ of $\mathcal{D}_d$, to which we apply the topologically-enriched Quillen bracket construction of §1.4.1, and Theorem B repeats verbatim for $\langle \mathcal{G}, \mathcal{M} \rangle \subseteq \langle \mathcal{D}_d, \mathcal{D}_d \rangle$; see Corollary 1.80. The category $\langle \mathcal{G}_o, \mathcal{M}_o \rangle$ is then defined to be $\pi_0(\langle \mathcal{G}, \mathcal{M} \rangle) \cong \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$. (More precisely, it is a certain skeleton of this category; see §1.4.4.) In particular, our mild assumption on each family of groups $\{g_n: G_n \to G_{n+1}\}_{n \in \mathbb{N}}$ is to require that the groups $\{G_n\}_{n \in \mathbb{N}}$ are the automorphism groups of a category of the form $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$, which also contains the maps $\{g_n\}_{n \in \mathbb{N}}$. See Construction 3.1 for more details and §3.1 for concrete examples.

**Applications.** Finally, we survey the representations generated by the constructions of Theorem A for some illustrative examples introduced in §3.1.1–§3.1.3, which represent the archetypal examples of mapping class groups and motion groups. First, let us indicate which parameters $(Z, G, \ell)$ we use to apply Theorem A in these examples. From now on, we fix an integer $k \ge 1$ and a partition $\lambda \vdash k$. We generically denote by $\underline{k} \subset \mathbb{R}^2$ a closed submanifold consisting of $k$ distinct points, and by $\underline{k}\mathbb{S}^1 \subset \mathbb{R}^3$ a closed submanifold consisting of a collection of disjoint circles forming a
<table><caption>Table 1 A summary of Theorems C-F, describing the special cases of our construction that recover previously-known homological representations.</caption><thead><tr><th rowspan="2">Thm.</th><th rowspan="2">Family of groups</th><th colspan="5">Parameters</th><th rowspan="2">Family of representations</th><th rowspan="2">Reference</th></tr><tr><th>d</th><th>Z</th><th>G</th><th>ℓ</th><th>i</th></tr></thead><tbody><tr><td>C</td><td>B<sub>n</sub></td><td>2</td><td>k</td><td>S<sub>k</sub></td><td>2</td><td>k</td><td>Lawrence-Bigelow</td><td>[Law90; Big01]</td></tr><tr><td>D</td><td>B<sub>n</sub>(Σ<sub>g,1</sub>) (g fixed)</td><td>2</td><td>k</td><td>S<sub>k</sub></td><td>3</td><td>k</td><td>An-Ko</td><td>[AK10]</td></tr><tr><td>E</td><td>LB'<sub>n</sub> and LB<sub>n</sub></td><td>3</td><td>1</td><td>{id}</td><td>2</td><td>1</td><td>Loop Burau</td><td>[PS22a]</td></tr><tr><td>F</td><td>Γ<sub>g,1</sub></td><td>2</td><td>k</td><td>{id}</td><td>1</td><td>k</td><td>Moriyama</td><td>[Mor07]</td></tr></tbody></table>

trivial link of $k$ components. We delineate our results for each family of groups in the paragraphs below; they are also summarised in Table 1.

**Classical braid groups:** the relevant functors of Theorem A to consider for the groups $B_n = \text{MCG}(\mathbb{D}^2, \underline{n}) = \text{Mot}_n(\mathbb{D}^2)$ are those of the form $L_i(\mathcal{F}_{(Z,G,\ell)})$ with $d=2$, $Z := k$, $G := S_\lambda$, $\ell \ge 1$ and $i \ge 0$; see (3.2).

**Theorem C** (Theorem 3.5, §3.2.1.1) For $\lambda = \{k\}$ the trivial partition, $\ell = 2$ and $i = k$, the functor $L_k(\mathcal{F}_{(k,S_\lambda,2)})$ recovers the aforementioned family $\mathfrak{L}\mathfrak{B}_k(n)$ of Lawrence-Bigelow representations [Law90; Big01] of the braid groups.

Otherwise, the $B_n$-representations encoded by the functors $L_i(\mathcal{F}_{(k,S_\lambda,\ell)})$ appear to be new.

**Surface braid groups:** we denote by $\Sigma_{g,1}$ (respectively $\mathcal{N}_{h,1}$) a compact, connected, smooth, orientable (resp. non-orientable) surface of genus $g$ (resp. $h$) and with one boundary component. The relevant functors of Theorem A to consider for the surface braid groups $B_n(\Sigma_{g,1}) = \text{Mot}_n(\Sigma_{g,1})$ and $B_n(\mathcal{N}_{h,1}) = \text{Mot}_n(\mathcal{N}_{h,1})$ are those of the form $L_i(\mathcal{F}_{(Z,G,\ell)})$ with $d=2$, $Z := k$, $G := S_\lambda$, $\ell \ge 1$ and $i \ge 0$; see (3.3).

**Theorem D** (Example 3.6, §3.2.1.2) For $\lambda = \{k\}$ the trivial partition, $\ell = 3$ and $i = k$, the variant using Borel-Moore homology of the functor $L_k(\mathcal{F}_{(k,S_\lambda,3)})$ recovers the family of An-Ko representations [AK10] of the surface braid groups $B_n(\Sigma_{g,1})$.

Otherwise, the representations of the groups $B_n(\Sigma_{g,1})$ and $B_n(\mathcal{N}_{h,1})$ encoded by the functors $L_i(\mathcal{F}_{(k,S_\lambda,\ell)})$ appear to be new.

**Loop braid groups:** for the extended and non-extended loop braid groups $LB'_n$ and $LB_n$ (see §3.1.3 for their definitions), the relevant functors of Theorem A to consider are those of the form $L_i(\mathcal{F}_{(Z,G,\ell)})$ with $d=3$, $\ell \ge 1$ and $i \ge 0$, and:

* either $Z := k$ together with $G := S_\lambda$, see (3.4);

* or else $Z := k\mathbb{S}^1$ together with $G := \text{Diff}^+(\mathcal{A}\mathbb{S}^1)$ or $\text{Diff}(\mathcal{A}\mathbb{S}^1)$, see (3.5) and (3.6).

**Theorem E** (Example 3.7, §3.2.1.3) For $k=1$ and $\lambda = \{1\}$ its trivial partition, $\ell=2$ and $i=1$, the functor $L_1(\mathcal{F}_{(1,0,2)})$ recovers the family of loop Burau representations [PS22a] of the extended and non-extended loop braid groups.

Otherwise, the representations of the groups $LB'_n$ and $LB_n$ defined by the functors $L_i(\mathcal{F}_{(k,S_\lambda,\ell)})$, $L_i(\mathcal{F}_{(k\mathbb{S}^1,\text{Diff}^+(\mathcal{A}\mathbb{S}^1),\ell)})$ and $L_i(\mathcal{F}_{(k\mathbb{S}^1,\text{Diff}(\mathcal{A}\mathbb{S}^1),\ell)})$ appear to be new.

**Mapping class groups of surfaces:** for the groups $\Gamma_{g,1} = \text{MCG}(\Sigma_{g,1})$ and $\mathcal{N}_{h,1} = \text{MCG}(\mathcal{N}_{h,1})$, we consider the homological representation functors $L_i(\mathcal{F}_{(Z,G,\ell)})$ and $L_i(\mathcal{F}_{(Z,G,\ell)})$ of Theorem A with $d=2$, $Z := k$, $G := S_\lambda$, $\ell \ge 1$ and $i \ge 0$; see (3.7) and (3.8).

**Theorem F** (Proposition 3.9, §3.2.2) For $\lambda = \{1^k\}$ the discrete partition of $k$, $\ell=1$ and $i=k$, the variant using Borel-Moore homology of the functor $L_k(\mathcal{F}_{(k,0,1)})$ encodes the family of Moriyama representations [Mor07] of the mapping class groups $\Gamma_{g,1}$.

Otherwise, the representations of the groups $\Gamma_{g,1}$ and $\mathcal{N}_{h,1}$ encoded by the functors $L_i(\mathcal{F}_{(k,S_\lambda,\ell)})$ and $L_i(\mathcal{F}_{(k,S_\lambda,\ell)})$ appear to be new.
**Unified foundations.** As detailed above, several examples of the representations arising from the construction of Theorem A (see Theorems C-F and Table 1) have previously been defined and studied in the literature, at least at the level of individual groups, i.e. when restricted to the automorphism groups of $\langle G_o, M_o \rangle$. One purpose of setting up the general procedure for constructing homological representations in this paper is to give a unified foundation that encompasses all known homological representations, amongst many others, and that inspires the discovery of new representations by applying this unified context to novel settings. Regarding this last point, we mention that the examples described in Theorems C-F represent only a small fragment of the potential of the general construction of Theorem A.

**Perspectives on the construction.** Although we do not discuss it in more detail in this paper, one may equally well consider examples analogous to those of Theorems C-F, given by the homological representation functors of Theorem A on subcategories of $\mathcal{UD}_2$ and $\mathcal{UD}_3$ (or $\mathcal{UD}_d$ for $d \ge 5$) relevant for automorphism groups of free groups, Torelli groups, handlebody mapping class groups, pure braid groups, as well as higher-dimensional mapping class groups and motion groups. In addition, there are several natural variations of our constructions of Theorem A. First, one may change the “flavour” of (twisted) homology that we use: notably Borel-Moore homology rather than ordinary homology (see Theorem 2.19). Second, we may naturally package together the homological representation functors (0.3) by fixing the parameters $(Z, G)$, and by considering simultaneously all the lower central series parameters $\{\ell \ge 1\}$, which control the ground ring over which the representation is defined (for each $\ell$, it is the group-ring of a group of nilpotency class at most $\ell-1$). More precisely, these functors fit together into a tower that may be thought of as a single pro-nilpotent (functorial) representation, which may be truncated to recover the representation corresponding to each level $\ell$; see [PS22b, §1]. In particular, there are many interesting cases where we obtain in this way an infinite tower of distinct representations as $\ell \to \infty$; see [PS22b, §4–§6].

**Outline.** The aim of §1 is to set up the source categories for homological representation functors. Namely, we define the topologically-enriched groupoids $\mathcal{D}_d$ of decorated manifolds in §1.1. Then, we prove some fundamental properties of decorated diffeomorphisms and embeddings of manifolds: in §1.2 we exhibit several split short exact sequences associated to these topological groups, which are crucial for the construction of homological representation functors in §2, and then we prove in §1.3 that certain quotients of decorated diffeomorphism groups are Serre fibrations (see Theorem 1.55); these are decisive in the proof of Theorem B. Finally, we introduce the topological enrichment of the Quillen bracket construction in §1.4.1 and then in §1.4.2 we apply it to categories of decorated manifolds and study its main properties, in particular proving Theorem B.

In §2, we introduce the various tools and topological constructions to define homological representation functors, which between them imply Theorem A. In more detail, we present in §2.1 the different ingredients and steps of the construction illustrated in diagram (0.5), considering an abstract continuous semifunctor of the form (0.4); see Theorem 2.19. We then construct the two continuous semifunctors $\mathcal{F}_{(Z,G,\ell)}$ and $\mathcal{F}_{(Z,G,\ell)}$ of (0.4) in §2.2 and §2.3 respectively (see Theorems 2.28 and 2.62), thus completing the proof of Theorem A.

We finally apply our constructions of homological representation functors to the four different families of groups of Theorems C-F in §3. Namely, we first describe (specialising §1 to each case) the appropriate categorical framework encoding these families of groups in §3.1. We deal with the applications for the families of motion groups in §3.2.1, proving Theorem C in §3.2.1.1, Theorem D in §3.2.1.2 and Theorem E in §3.2.1.3, and we treat the case of the mapping class groups in §3.2.2, proving Theorem F.

**General notation.** For a category $C$, we use the abbreviation $ob(C)$ to denote its class of objects. We recall that *semicategories* and *semifunctors* between them are defined in the exact same way as (genuine) categories and functors by relaxing all structures or conditions that involve identity morphisms of objects. Throughout the paper, the word *ring* always refers to an *associative, unital ring* and we assume that ring homomorphisms preserve units. For a ring $R$, we denote by $\text{Mod}_R$ the category of (right) $R$-modules. For an $R$-module $M$, we denote by $\text{Aut}_R(M)$ the group of $R$-module automorphisms of $M$. When $R = \mathbb{Z}$, we omit it from the notation as long as there is no
ambiguity.

For a manifold $X$, we denote by $\mathrm{Diff}(X)$ its group of diffeomorphisms. If $X$ is orientable, we denote by $\mathrm{Diff}^+(X)$ the subgroup of orientation-preserving diffeomorphisms. If $X$ has non-empty boundary, $\dot{X}$ denotes its interior. For each $d \ge 1$, we denote by $\mathfrak{S}^d$ the $d$-sphere.

We denote by $\mathfrak{S}_n$ the symmetric group on a set of $n$ elements. For an integer $n \ge 1$, an *ordered partition of $n$* means an ordered $r$-tuple $\mathbf{n} = \{n_1, \dots, n_r\}$ of integers $n_i \ge 1$ (for some $r \ge 1$ called the *length of $\mathbf{n}$*) such that $n = \sum_{1 \le i \le r} n_i$ (and without the condition $n_i \ge n_{i+1}$). We recall that the *lower central series* of a group $G$ is the descending chain of subgroups $\{\Gamma_\ell(G)\}_{\ell \ge 1}$ defined by $\Gamma_1(G) := G$ and $\Gamma_{\ell+1}(G) := [G, \Gamma_\ell(G)]$, the subgroup of $G$ generated by the commutators $[g, h]$ for $g \in G$ and $h \in \Gamma_\ell(G)$. For the sake of simplicity, each quotient $G/\Gamma_\ell(G)$ is denoted by $G/\Gamma_\ell$.

**Acknowledgements.** The authors would like to thank Paolo Bellingeri, Tara Brendle, Oscar Randal-Williams, Antoine Touzé, Christine Vespa and Emmanuel Wagner for illuminating discussions and questions. In particular, they thank Antoine Touzé for pointing out the reference [EN02]. They would also like to thank Oscar Randal-Williams for inviting the first author to the University of Cambridge in November 2019, where the authors were able to make significant progress on the present article. The authors would also like to thank the anonymous referee for very helpful comments and suggestions on earlier versions of this work.

The first author was partially supported by a grant of the Romanian Ministry of Education and Research, CNCS - UEFISCDI, project number PN-III-P4-ID-PCE-2020-2798, within PNCDI III. The second author was partially supported by a Rankin-Sneddon Research Fellowship of the University of Glasgow, by the Institute for Basic Science IBS-R003-D1 and by the ANR Projects ChroK ANR-16-CE40-0003 and AlMaRe ANR-19-CE40-0001-01.

## Contents

<table><tr><td>§1. Categories of decorated manifolds and their embeddings</td><td>8</td></tr><tr><td>§1.1 Decorated manifolds and their diffeomorphisms</td><td>9</td></tr><tr><td>§1.2 Split short exact sequences</td><td>15</td></tr><tr><td>§1.3 A Serre fibration of decorated diffeomorphism groups</td><td>22</td></tr><tr><td>§1.4 The topologically-enriched Quillen bracket construction</td><td>28</td></tr><tr><td>§2. Construction of homological representation functors</td><td>37</td></tr><tr><td>§2.1 Elements of the construction</td><td>37</td></tr><tr><td>§2.2 First constructions of homological representation functors</td><td>47</td></tr><tr><td>§2.3 Further constructions of homological representation functors</td><td>59</td></tr><tr><td>§3. Applications for mapping class groups and motion groups</td><td>61</td></tr><tr><td>§3.1 Categories for mapping class groups and motion groups</td><td>61</td></tr><tr><td>§3.2 Examples of homological representation functors</td><td>65</td></tr><tr><td>§4. Appendix: computations of transformation groups</td><td>70</td></tr><tr><td>§4.1 Presentations of surface braid groups</td><td>71</td></tr><tr><td>§4.2 Properties of the transformation groups</td><td>71</td></tr><tr><td>References</td><td>75</td></tr></table>

# 1. Categories of decorated manifolds and their embeddings

In this section, we define the categories that will serve as the domain of the homological representation functors that we will construct in §2. They are obtained from certain monoidal groupoids by the *Quillen bracket construction*, an operation that enlarges a given monoidal groupoid to a category having the original monoidal groupoid as its underlying groupoid.

More precisely, we will start with certain *topologically-enriched* monoidal groupoids, so we describe, in §1.4.1, a topological enrichment of the Quillen bracket construction and show – subject to a *Serre fibration condition* – that it behaves well with respect to the functor $\pi_0$ that replaces all morphism spaces with their sets of path-components. In §1.1, we define the topologically-enriched monoidal groupoids that we wish to consider, and prove in §1.3 that they satisfy this Serre fibration condition.

Informally, the idea is that the domain category, for a given dimension $d \ge 2$, will be a
topologically-enriched category $\mathcal{U}\mathcal{D}_d$ having the property that the automorphism groups of $\pi_0(\mathcal{U}\mathcal{D}_d)$ contain all mapping class groups and motion groups in dimension $d$. To construct this, we define in §1.1 a topologically-enriched groupoid $\mathcal{D}_d$ whose automorphism groups are the diffeomorphism groups of all $d$-dimensional *decorated manifolds*. The topologically-enriched Quillen bracket construction $\mathfrak{U}$ of §1.4.1 then gives us a topologically-enriched category $\mathfrak{U}\mathcal{D}_d$ such that $\pi_0(\mathfrak{U}\mathcal{D}_d) \cong \mathfrak{U}(\pi_0(\mathcal{D}_d))$, where the latter category is defined by applying the construction $\mathfrak{U}$ to the path-component category $\pi_0(\mathcal{D}_d)$ of $\mathcal{D}_d$. The underlying groupoid of $\pi_0(\mathfrak{U}\mathcal{D}_d)$ is therefore $\pi_0(\mathcal{D}_d)$, consisting of all mapping class groups of $d$-dimensional *decorated manifolds*, which contain all $d$-dimensional motion groups as normal subgroups.

In §1.4.2, we then give an explicit description of morphism spaces of *Quillen bracket categories* of manifolds in terms of embedding spaces, which is a crucial ingredient in the construction of §2.

**Outline.** To summarise, the main points of this section are the following.

* We construct the topologically-enriched groupoid $\mathcal{D}_d$ of decorated manifolds in §1.1.

* We obtain the topologically-enriched category $\mathfrak{U}\mathcal{D}_d$ using the topologically-enriched Quillen bracket construction introduced in §1.4.1.

* We identify the morphism spaces of $\mathfrak{U}\mathcal{D}_d$ with certain embedding spaces in Proposition 1.76 (in §1.4.2). This crucially uses a Serre fibration result for decorated diffeomorphism groups proved in Theorem 1.55 (in §1.3).

* The categories that we shall use, in §3, to encode families of mapping class groups and motion groups, are defined by certain topologically-enriched Quillen bracket constructions $\langle \mathcal{G}, \mathcal{M} \rangle$, which come with natural inclusions $\langle \mathcal{G}, \mathcal{M} \rangle \hookrightarrow \mathfrak{U}\mathcal{D}_d$. The homological representation functors (defined on $\mathfrak{U}\mathcal{D}_d$) constructed in §2 therefore restrict naturally along these inclusions. In §3.1, we describe in detail the relevant categories $\langle \mathcal{G}, \mathcal{M} \rangle$ for mapping class groups and motion groups.

**Preliminaries on categorical tools.** We refer to [Mac98, Chap. VII] for a complete introduction to the notions of monoidal categories and modules over them. We generically denote a monoidal category by $(\mathcal{C}, \sharp, \mathbf{0})$, where $\mathcal{C}$ is a category, $\sharp$ is the monoidal product and $\mathbf{0}$ is the monoidal unit. A left-module $(\mathcal{M}, \sharp)$ over $(\mathcal{C}, \sharp, \mathbf{0})$ is a category $\mathcal{M}$ with a functor $\sharp: \mathcal{C} \times \mathcal{M} \to \mathcal{M}$ that is unital and associative. Similarly, semi-monoidal categories and left-modules over them are defined in the exact same way as in the genuine monoidal case by relaxing all the structures or conditions involving (left or right) units; in other words they only involve a binary operation admitting an associator that satisfies the pentagon condition. Note that a (semi-)monoidal category $(\mathcal{C}, \sharp)$ is equipped with a left-module structure over itself, induced by its monoidal product. Each left-module structure $\sharp$ in this paper is defined from some underlying (semi-)monoidal structure $\sharp$ (see §1.1 and §3.1), so we abuse notation by using the same symbol $\sharp$ for $\sharp$. The reason for introducing this *semi-monoidal* setting is that the boundary connected sum of manifolds naturally equips the topologically-enriched groupoid $\mathcal{D}_d$ with a non-unital such structure; see Remark 1.23 for more details.

We recall that, by *topologically-enriched category*, we mean a category enriched over the symmetric monoidal category of topological spaces with its Cartesian product. The functor $\pi_0$ replaces each morphism space with its set of path-components, and so defines a functor from (small) topologically-enriched categories to (small) categories. For any topologically-enriched category $\mathcal{C}$, there is a natural functor $\mathcal{C} \to \pi_0(\mathcal{C})$, also denoted by $\pi_0$, sending each point of a morphism space to the path-component that it lies in.

**Remark 1.1** Although most of the (semi-)monoidal categories and their modules that we consider in this paper are *symmetric* or *braided*, we will not dwell on these considerations, as these properties are unnecessary for our work, in particular to define the topologically-enriched bracket construction of §1.4 or the homological representation functors in §2.

## 1.1. Decorated manifolds and their diffeomorphisms

In this section, we introduce the topologically-enriched groupoids $\mathcal{D}_d$ of *decorated manifolds* and their semi-monoidal structures induced by boundary connected sum. We fix an integer $d \ge 2$.
**Convention 1.2** All of the (topological) categories that we shall consider are essentially small, i.e. they are each equivalent to a small (topological) category. One standard way to see this is as follows; see [Gal21, Rem. 1.2.7]. In each case, the objects of the categories that we shall consider consist of manifolds, submanifolds and some additional data, such as collar neighbourhoods, etc. Fixing a set $\Omega$, we may consider the full subcategory whose objects are only those where the underlying set of the manifold is a subset of $\Omega$. This is a small category, and if the cardinality of $\Omega$ is at least $|\mathbb{R}|$, its inclusion into the whole (large) category is essentially surjective on objects; thus it is an equivalence. (We assume that manifolds are second-countable, which implies that their cardinality is no larger than $|\mathbb{R}|$.) In the following, we implicitly fix a sufficiently large set $\Omega$ and we shall consider the corresponding small subcategories, whenever we need our categories to be small.

**Open path-components.** For a topological space $X$, if we give its set of path-components $\pi_0(X)$ the discrete topology, the natural map $X \to \pi_0(X)$ sending each point to the path-component that it lies in is continuous if and only if all path-components of $X$ are open. Similarly, for a topologically-enriched category $C$, the natural functor $C \to \pi_0(C)$ mentioned above is continuous if and only if all path-components of all morphisms spaces of $C$ are open. It will be important for us to ensure that this property holds, so we collect here some facts about this property that we shall make use of. We first record some basic facts:

**Lemma 1.3** If $X$ has the property that its path-components are open, then the quotient space $X/\sim$ also has this property, for any equivalence relation $\sim$ on $X$. If a collection $X_i$ of spaces all have this property, then so does their disjoint union $\bigsqcup_i X_i$.

There is a canonical way to refine the topology on an arbitrary space in order to force this property to hold.

**Construction 1.4** For a topological space $X$, denote by $o(X)$ the topological space with the same underlying set as $X$, equipped with the topology generated by the base consisting of $C \cap U$ for $U$ an open subset of $X$ and $C$ a path-component of $X$.

One may easily verify the following properties of this construction.

**Lemma 1.5** The path-components of $o(X)$ are open, and they are the same as the path-components of $X$. The canonical map $o(X) \to X$ defined by the identity map of the underlying sets is continuous, and becomes a homeomorphism when restricted to any path-component. Consequently, it is also a weak equivalence. In addition, the functor $o(-)$ preserves products.

**Remark 1.6** In fact, the operation $o(-)$ is a right adjoint to the inclusion of the full subcategory of topological spaces having the property that their path-components are open; this subcategory is therefore coreflective.

Finally, we will need the following lemma, which explains how the operation $o(-)$ interacts with quotient maps that are Serre fibrations.

**Lemma 1.7** Let $X$ be a topological space and $\sim$ an equivalence relation on $X$. If the quotient map $X \to X/\sim$ is a Serre fibration, then so is the quotient map $o(X) \to o(X)/\sim$.

*Proof.* The continuous bijection $o(X) \to X$ induces a continuous bijection $o(X)/\sim \to X/\sim$, fitting into a commutative square with the two quotient maps. Let us consider the lifting problem:

$$ \begin{tikzcd} [0,1]^{d-1} \arrow[r] \arrow[d, "?"'] & o(X) \arrow[r] \arrow[d] & X \arrow[d] \\ [0,1]^d \arrow[r, "o(X)/\sim"] & X/\sim \arrow[r, "X/\sim"] & X \arrow[d] \end{tikzcd} \quad (1.1) $$

Since $X \to X/\sim$ is a Serre fibration by hypothesis, there is a diagonal map $[0,1]^d \to X$ making the two outer triangles commute. Since the cube $[0,1]^d$ is path-connected, its image in $X$ lies in a single path-component. Restricted to this path-component, the continuous bijection $o(X) \to X$ is a homeomorphism by Lemma 1.5, so we may lift the diagonal map $[0,1]^d \to X$ to a diagonal
<figure><img src="image_2.png" /><figcaption>Figure 1.1 An illustration of the notation for the solid cylinder $\mathbb{B}_t^d$ from Notation 1.9 for $d = 3$. Its lower boundary $\partial_t \mathbb{B}_t^3$ is coloured yellow, its base $b\mathbb{B}_t^3$ is yellow-green and the codimension-2 stratum $\partial(b\mathbb{B}_t^3) = \partial\mathbb{D}^2 \times \{0\}$ is blue.</figcaption></figure>

map $[0, 1]^d \to o(X)$ for the left-hand square, as desired. The two triangles into which this splits the left-hand square of (1.1) commute, since the triangles that the outer rectangle was split into commute and the two right-hand horizontal maps in (1.1) are bijective. □

**Remark 1.8** Lemma 1.7 holds more generally for any class of fibrations defined by the right lifting property with respect to a collection of maps whose codomains are path-connected. Moreover, it also holds more generally for any two topologies on the same underlying set that have the same path-components and that agree on each path-component.

**Categories of decorated manifolds.** First, we define the notion of *decorated manifolds* of dimension $d$, their morphisms and associated categories. A motivation to work with this type of manifolds is that the groups $\pi_0(\text{Diff}(-))$ of decorated manifolds will contain as normal subgroups all of the mapping class groups and motion groups that we are interested in; see Remark 1.43. Also, the notion of decorated manifolds is formed so that their associated categories have a well-defined semi-monoidal structure induced by the boundary connected sum, which would not be the case if we just considered manifolds without decorations; see Remark 1.20. The terminology “decorated” originates from [BT01]; see also [Bas23].

**Notation 1.9** (Solid cylinders.) We denote by $\mathbb{D}^{d-1}$ the closed unit $(d-1)$-dimensional disc in $\mathbb{R}^{d-1}$ in the usual $L^2$ metric. For a real number $t > 0$, we write $\mathbb{B}_t^d = \mathbb{D}^{d-1} \times [0, t]$ for the solid $d$-dimensional cylinder of height $t$. We also write

$$ \partial_t \mathbb{B}_t^d = (\partial \mathbb{D}^{d-1} \times [0, t]) \cup (\mathbb{D}^{d-1} \times \{0\}) $$

and call this the *lower boundary* of $\mathbb{B}_t^d$, as well as $b\mathbb{B}_t^d = \mathbb{D}^{d-1} \times \{0\}$ and call this the *base* of $\mathbb{B}_t^d$. This is illustrated in Figure 1.1.

**Definition 1.10** (Boundary-cylinders.) Let $M$ be a smooth $d$-manifold with non-empty boundary. A boundary-cylinder for $M$ is a topological embedding $e: \mathbb{B}_1^d \hookrightarrow M$ such that $e^{-1}(\partial M) = \partial_t \mathbb{B}_1^d$ and $e$ is a smooth embedding on $\mathbb{B}_1^d \setminus \partial(b\mathbb{B}_1^d)$. Two boundary-cylinders $e, e'$ are equivalent if they are equal when restricted to $\mathbb{B}_\epsilon^d \subseteq \mathbb{B}_1^d$ for some $\epsilon > 0$. An equivalence class of boundary-cylinders is called a *boundary-cylinder germ*.

**Definition 1.11** (Decorated manifolds.) A *decorated manifold* is a smooth $d$-manifold $M$ with non-empty boundary, equipped with a closed submanifold $A \subset \dot{M}$ and a pair $(e_1, e_2)$ of boundary-cylinder germs for $M \setminus A$ such that $e_1(b\mathbb{B}_1^d)$ and $e_2(b\mathbb{B}_1^d)$ are disjoint. See Figure 1.2 for a schematic picture.

**Definition 1.12** (Morphisms of decorated manifolds) A morphism of decorated manifolds from $(M, A, e_1, e_2)$ to $(M', A', e'_1, e'_2)$ is a smooth, proper (preimages of compact subspaces are compact) map $\varphi: M \to M'$ such that $\varphi(A) \subseteq A'$ and such that, for some $\epsilon > 0$ and for each $i \in \{1, 2\}$, we have $\varphi(e_i(\mathbb{B}_\epsilon^d)) = e'_i(\mathbb{B}_\epsilon^d)$ and the composition $(e'_i)^{-1} \circ \varphi \circ e_i: \mathbb{B}_\epsilon^d \to \mathbb{B}_\epsilon^d$ is the identity. Write $C_{\text{dec}}^\infty(M, M')$ for the set of such morphisms, where by abuse of notation we are abbreviating $(M, A, e_1, e_2)$ to $M$ and $(M', A', e'_1, e'_2)$ to $M'$.

For decorated manifolds $M$ and $M'$, we equip the set $C_{\text{dec}}^\infty(M, M')$ with a topology induced by a colimit of Whitney topologies, as follows.
<figure><img src="image_3.png" /><figcaption>Figure 1.2 Two decorated manifolds and their boundary connected sum.</figcaption></figure>

• Choose representative boundary cylinders for the boundary-cylinder germs $e_i$ and $e'_i$. This ensures that the condition in Definition 1.12 makes sense for a fixed $\epsilon \in (0,1)$, not just for an unspecified $\epsilon \in (0,1)$ that is quantified over.

• Fix $\epsilon \in (0,1)$ and write $C_{\text{dec},\epsilon}^{\infty}(M, M')$ for the subset of $C_{\text{dec}}^{\infty}(M, M')$ where the condition in Definition 1.12 holds for this fixed $\epsilon$. Equip each subset $C_{\text{dec},\epsilon}^{\infty}(M, M')$ with the subspace topology induced from the smooth Whitney topology on the set $C^{\infty}(M, M')$ of all smooth maps from $M$ to $M'$; for details of the Whitney topology, see for example [Hir76, Chap. 2]. As a set, note that $C_{\text{dec}}^{\infty}(M, M')$ is the union of $C_{\text{dec},\epsilon}^{\infty}(M, M')$ over all choices of $\epsilon \in (0,1)$.

**Lemma 1.13** *The colimit topology induced by the increasing filtration $\{C_{\text{dec},\epsilon}^{\infty}(M, M')\}_{\epsilon \in (0,1)}$ does not depend on the choices of representative boundary-cylinders for the boundary-cylinder germs $e_i$ and $e'_i$.*

*Proof.* Different choices of representative boundary-cylinders for the boundary-cylinder germs $e_i$ and $e'_i$ result in different filtrations, which are cofinal in each other, so the induced colimit topologies are equal. $\square$

**Definition 1.14** (*Morphism spaces.*) The morphism set $C_{\text{dec}}^{\infty}(M, M')$ is topologised as follows: we first consider the colimit topology induced by the increasing filtration $\{C_{\text{dec},\epsilon}^{\infty}(M, M')\}_{\epsilon \in (0,1)}$ and then we apply the functor $o(-)$ of Construction 1.4 to ensure that all path-components are open. Thus we may write $C_{\text{dec}}^{\infty}(M, M') := o(\text{colim}_{\epsilon \to 0}(C_{\text{dec},\epsilon}^{\infty}(M, M')))$.

We note that composition is continuous in this topology since the functor $o(-)$ preserves products (by Lemma 1.5) and because composition of smooth, proper maps is continuous in the Whitney topology (see [Mat69, §2, Prop. 1]).

**Remark 1.15** Depending on $M$ and $M'$, the topology of Definition 1.14 may differ from the subspace topology inherited directly from the Whitney topology on $C^{\infty}(M, M')$: the colimit topology is in general finer than the directly inherited Whitney topology, and the operation $o(-)$ refines the topology further. However, these three topologies on $C_{\text{dec}}^{\infty}(M, M')$ are all weakly equivalent (see Lemma 1.5). In particular, they have the same $\pi_0$.

We can now introduce the topologically-enriched categories associated to decorated manifolds:

**Definition 1.16** (*Decorated manifold categories.*) Let $\mathcal{Dec}_d$ be the topologically-enriched category defined as follows. Its objects are all *decorated manifolds* $(M, A, e_1, e_2)$ of dimension $d$, as in Definition 1.11. The space of morphisms from $M = (M, A, e_1, e_2)$ to $M' = (M', A', e'_1, e'_2)$ is the space $C_{\text{dec}}^{\infty}(M, M')$ whose underlying set is specified in Definition 1.12 and which is topologised in Definition 1.14.

Let $\mathcal{D}_d$ be the underlying topologically-enriched groupoid of $\mathcal{Dec}_d$. In other words, its objects are all decorated manifolds of dimension $d$ and its morphisms are those morphisms $(M, A, e_1, e_2) \to (M', A', e'_1, e'_2)$ of decorated manifolds whose underlying smooth map $\varphi: M \to M'$ is a diffeomorphism and $\varphi(A) = A'$.
Let $\mathcal{D}_d^+$ denote the topologically-enriched groupoid whose objects are decorated $d$-manifolds $(M, A, e_1, e_2)$ together with an orientation of $A \subset \tilde{M}$ (called an *orientedly* decorated $d$-manifold), and whose morphisms are maps $\varphi$ as in Definition 1.12 that are diffeomorphisms and such that the restriction $\varphi|_A: A \to A'$ is an orientation-preserving diffeomorphism.

The morphisms of the groupoid $\mathcal{D}_d$, i.e. the invertible morphisms of $\mathcal{Dec}_d$, are called *decorated diffeomorphisms*, for which we introduce the following notation.

**Notation 1.17** Let us write $\text{Diff}_{\text{dec}}(M, N)$ for the space of decorated diffeomorphisms $M \to N$ and abbreviate $\text{Diff}_{\text{dec}}(M) = \text{Diff}_{\text{dec}}(M, M)$. We similarly define $\text{Diff}_{\text{dec}}^{+}(M) = \text{Diff}_{\text{dec}}^{+}(M, M)$ for an *orientedly* decorated $d$-manifold $M$, where the superscript $+$ means that diffeomorphisms are orientation-preserving on the submanifold $A \subset M$.

The space $\text{Diff}_{\text{dec}}(M, N)$ is topologised as a subspace of $C_{\text{dec}}^{\infty}(M, N)$, which is topologised as in Definition 1.14.

**Semi-monoidal structure.** The topologically-enriched categories $\mathcal{Dec}_d$, $\mathcal{D}_d$ and $\mathcal{D}_d^+$ are naturally endowed with semi-monoidal structures defined from the boundary connected sum, which we detail here. Let $(M, A, e_1, e_2)$ and $(M', A', e'_1, e'_2)$ be two decorated $d$-manifolds. Define

$$M \dashv M' = (M \sqcup M')/\sim,$$

where $\sim$ is the equivalence relation generated by $e_2(x, 0) \sim e'_1(x, 0)$ for all $(x, 0) \in b\mathbb{B}_1^d$. We give this a smooth structure as follows. There are obvious topological embeddings

$$M \ltimes e_2(b\mathbb{B}_1^d) \to M \dashv M' \quad \text{and} \quad M' \ltimes e'_1(b\mathbb{B}_1^d) \to M \dashv M', \qquad (1.2)$$

and another topological embedding

$$\mathbb{D}^{d-1} \times [-1, 1] \to M \dashv M' \qquad (1.3)$$

given by $(x, t) \mapsto e_2(x, -t)$ for $t \le 0$ and $(x, t) \mapsto e'_1(x, t)$ for $t \ge 0$, where we have implicitly chosen representative boundary-cylinders for the boundary-cylinder germs $e_2$ and $e'_1$.

**Lemma 1.18** *Declaring that the topological embeddings (1.2) and (1.3) are both smooth embeddings defines a smooth structure on $M \dashv M'$.*

*Proof.* First, note that the smooth structures induced by the topological embeddings (1.2) and (1.3) are compatible on intersections, since boundary-cylinders are smooth embeddings away from $\partial b\mathbb{B}_1^d$. Then the smooth structure of $M \dashv M'$ is determined, except on $e_2(b\mathbb{B}_1^d) = e'_1(b\mathbb{B}_1^d)$, by the smooth structures of $M$ and $M'$. The embedding of $\mathbb{D}^{d-1} \times [-1, 1]$ induced by (boundary-cylinders representing the boundary-cylinder germs) $e_2$ and $e'_1$ is therefore only required to extend this smooth structure to $e_2(b\mathbb{B}_1^d) = e'_1(b\mathbb{B}_1^d)$. As a result, the smooth structure does not depend on the choice of representative boundary-cylinders, but only their germs. Hence the smooth structure is well-defined. $\square$

**Definition 1.19** (*Boundary connected sum.*) We define the boundary connected sum of decorated $d$-manifolds by

$$(M, A, e_1, e_2) \dashv (M', A', e'_1, e'_2) = (M \dashv M', A \sqcup A', e_1, e_2),$$

where $M \dashv M'$ is defined by Lemma 1.18 and $A \sqcup A'$ is the disjoint union of $A$ and $A'$. For orientedly decorated $d$-manifolds, the boundary connected sum is defined in the same way, with the orientation for $A \sqcup A'$ being induced from those of $A$ and $A'$. See Figure 1.2 for a schematic illustration.

**Remark 1.20** The usual definition of boundary connected sum of two smooth manifolds $M, M'$ depends on a choice of embedded disc in the boundary of each manifold, and a method of “straightening corners” after gluing these discs together. Up to diffeomorphism, the resulting smooth manifold $M \dashv M'$ depends only on the choice of a boundary component of $M$ and of $M'$, and orientations of these if they are orientable (this is a result of Palais’ *Disc theorem* [Pal60a, Th. B and Cor. 1] and the existence of collar neighbourhoods [Bro62]). However, in order for $\dashv$ to induce a well-defined monoidal structure on some category of manifolds with boundary (which we will do just below), it must be well-defined on the nose, not just up to diffeomorphism (since objects are manifolds, not diffeomorphism classes of manifolds). Here, these additional choices are built in for decorated manifolds, and no additional choices are required in Definition 1.19 above.
**Remark 1.21** The boundary connected sum of Definition 1.19 is a fragment of a richer structure: a non-unital cyclic operad (see [GK95]) in the category of small categories. This consists in a contravariant functor $P$ from the category of finite sets and bijections to the category of small categories, together with structure maps (functors)

$$ x \circ_{y} : P(X) \times P(Y) \to P((X \setminus \{x\}) \sqcup (Y \setminus \{y\})) \qquad (1.4) $$

for each pair $(x, y) \in X \times Y$, satisfying certain associativity and equivariance axioms. In our setting, the category $P(X)$ is defined just as $\mathcal{Dec}_d$, except that each manifold is equipped with $|X|$ boundary-cylinder germs, labelled by the elements of $X$, each bijection $f: X \to X'$ induces a functor $P(X) \to P(X')$ by applying $f$ to the labels and the functor (1.4) acts by gluing together the boundary-cylinder germs labelled by $x$ and $y$. We will only use the fragment of this structure given by $\mathcal{Dec}_d = P(\{1, 2\})$ and (1.4) for $X = Y = \{1, 2\}$ and $(x = 2, y = 1)$, which is the operation described in Definition 1.19.

The boundary connected sum of Definition 1.19 induces a semi-monoidal structure on $\mathcal{Dec}_d$ as follows. Let $\varphi: (L, A, e_1, e_2) \to (L', A', e'_1, e'_2)$ and $\psi: (M, B, f_1, f_2) \to (M', B', f'_1, f'_2)$ be morphisms in $\mathcal{Dec}_d$. By definition, these are smooth, proper maps $L \to L'$ and $M \to M'$ that take $A$ and $B$ into $A'$ and $B'$ respectively, and are compatible with the given boundary-cylinder germs. This compatibility implies that they glue to a well-defined, smooth, proper map $L \natural M \to L' \natural M'$ such that the image of $A \sqcup B$ is contained in $A' \sqcup B'$ and satisfying the boundary-cylinder germ conditions of Definition 1.12: this is thus a morphism

$$ (L, A, e_1, e_2) \natural (M, B, f_1, f_2) \to (L', A', e'_1, e'_2) \natural (M', B', f'_1, f'_2) $$

of $\mathcal{Dec}_d$ that we denote by $\varphi \natural \psi$. This extends to morphisms of $\mathcal{D}_d^+$ since the restricted diffeomorphisms $A \to A'$ and $B \to B'$ are orientation-preserving by assumption and $(\varphi \natural \psi)|_{A \sqcup B} = \varphi|_A \sqcup \psi|_B$.

That the operation $\natural$ preserves composition of morphisms is a straightforward consequence of its assignments. Also, it immediately follows from the definitions that the boundary connected sum of Definition 1.19 is associative, and that the isomorphisms defined from this associativity are natural with respect to the morphisms of $\mathcal{Dec}_d$. Hence, we equip $\natural$ with the associator defined by these natural isomorphisms, for which one easily checks the pentagon diagram. Hence we prove:

**Proposition 1.22** (Semi-monoidal structure.) There is a semifunctor

$$ \natural: \mathcal{Dec}_d \times \mathcal{Dec}_d \to \mathcal{Dec}_d $$

defined by assigning the boundary connected sum of Definition 1.19 on objects and $\varphi \natural \psi$ for two morphisms $\varphi$ and $\psi$ in $\mathcal{Dec}_d$. Moreover, the semifunctor $\natural$ induces a semi-monoidal structure on the topologically-enriched categories $\mathcal{Dec}_d$, $\mathcal{D}_d$ and $\mathcal{D}_d^+$.

**Remark 1.23** (Monoidal and semi-monoidal structures.) The semi-monoidal structure on $\mathcal{Dec}_d$ defined above does not have a unit, since there is no natural way of identifying $M \natural \mathbb{B}_1^d$ with $M$ for all $M$ (although they are clearly non-naturally diffeomorphic). If decorated manifolds had been defined to be equipped with boundary cylinders (not just germs of boundary-cylinders), then $\mathcal{Dec}_d$ would have had an obvious genuine monoidal structure. However, the proof of Theorem 1.55 below, which tells us that the Serre fibration hypothesis of Proposition 1.66 is satisfied for subgroupoids of $\mathcal{D}_d$ (see Lemma 1.71), depends crucially on the fact that morphisms of decorated manifolds are only required to preserve germs of boundary-cylinders, rather than entire boundary-cylinders. Thus we are forced either to formally adjoin a unit to $\mathcal{Dec}_d$ (which is unnatural since $\pi_0(\mathcal{Dec}_d)$ already has a unit by Lemma 1.24), or to work directly with semi-monoidal categories, which is the choice that we make here.

**Groupoids of path-components.** Applying the functor $\pi_0$, we may consider the discrete groupoids $\pi_0(\mathcal{D}_d)$ and $\pi_0(\mathcal{D}_d^+)$. The following lemma may be checked straightforwardly from the definitions.

**Lemma 1.24** The discrete groupoids $\pi_0(\mathcal{D}_d)$ and $\pi_0(\mathcal{D}_d^+)$ inherit semi-monoidal structures from those on $\mathcal{D}_d$ and $\mathcal{D}_d^+$. Moreover, these semi-monoidal structures are monoidal, with unit object
given in each case by the solid cylinder $(\mathbb{B}_1^d, \emptyset, \text{id}, r)$, where $r: \mathbb{B}_1^d \to \mathbb{B}_1^d$ is the reflection $(x, t) \mapsto (x, 1-t)$.

More generally, if $\mathcal{G} \subseteq \mathcal{D}_d$ is any subgroupoid closed under the semi-monoidal structure and containing the solid cylinder $(\mathbb{B}_1^d, \emptyset, \text{id}, r)$, then the semi-monoidal structure inherited by $\pi_0(\mathcal{G})$ is monoidal. Furthermore, if $\mathcal{M} \subseteq \mathcal{D}_d$ is a subgroupoid closed under the operation $g \triangleright_\mathcal{M}$ for each object $g$ of $\mathcal{G}$ (thus $\mathcal{M}$ is a left module over the semi-monoidal groupoid $\mathcal{G}$), then $\pi_0(\mathcal{M})$ inherits a structure of a (genuine) left module over the monoidal groupoid $\pi_0(\mathcal{G})$. Similar results hold for analagous subgroupoids $\mathcal{G} \subseteq \mathcal{D}_d^+$ and $\mathcal{M} \subseteq \mathcal{D}_d^+$.

## 1.2. Split short exact sequences

In this subsection, we establish several split homotopy fibration sequences for embedding spaces and decorated diffeomorphism groups of manifolds, whose associated split short exact sequences on $\pi_1$ and $\pi_0$ will be used in our constructions in §2. In particular, the split short exact sequences (1.7) and (1.9) below will be used in the two versions of our general construction of homological representations in §2.2 and §2.3 respectively. In addition, the split short exact sequence (1.9) implies that any motion group, under a certain peripherality condition, is a *braided mapping class group* (see Proposition 1.42); in particular, a normal subgroup of a mapping class group.

Throughout §1.2, we fix a closed submanifold $Z \subset \mathbb{R}^d$ and an open subgroup $\mathcal{G} \subseteq \text{Diff}(Z)$. Note that open subgroups $\mathcal{G} \subseteq \text{Diff}(Z)$ correspond to subgroups of $\pi_0(\text{Diff}(Z))$. We recall our notational convention that, for $X$ a manifold with boundary, $\tilde{X}$ denotes its interior.

### 1.2.1. The first short exact sequence

The first split short exact sequence (1.7) that we construct deals with fundamental groups of certain spaces of embeddings, which we now define.

**Definition 1.25** For smooth manifolds $X$ and $Y$, let us write $\mathcal{E}(X, Y) = \text{Emb}(X, Y)/\text{Diff}(X)$. For a subgroup $H \subseteq \text{Diff}(X)$, we also write $\mathcal{E}_H(X, Y) = \text{Emb}(X, Y)/H$. In the special case when $H = \text{Diff}^+(X)$ (when $X$ is orientable), we also write $\mathcal{E}^+(X, Y) = \text{Emb}(X, Y)/\text{Diff}^+(X)$.

We specify once and for all an identification of $\mathbb{R}^d$ with the interior of the solid cylinder $\mathbb{B}_1^d$. The choice of closed submanifold $Z \subset \mathbb{R}^d$ above therefore determines a decorated manifold $(\mathbb{B}_1^d, Z) = (\mathbb{B}_1^d, Z, [\text{id}], [r])$ whose boundary-cylinder germs are represented, respectively, by the identity and by the reflection $r$ of the solid cylinder $\mathbb{B}_1^d = \mathbb{D}^{d-1} \times [0, 1]$ in its second coordinate. To justify that this is a decorated manifold, we need to verify that there are boundary-cylinders $e_1$ and $e_2$, having the same germs as $\text{id}$ and $r$, whose images are disjoint from $Z$. We will explain this for $e_1$, the case of $e_2$ being similar. Since $Z$ is a compact subset of the interior of $\mathbb{D}^{d-1} \times [0, 1]$, it is disjoint from $\mathbb{D}^{d-1} \times [0, \epsilon]$ for some $\epsilon > 0$. Choose a diffeomorphism $[0, 1] \cong [0, \epsilon]$ that agrees with the identity on $[0, \delta]$ for some $0 < \delta < \epsilon$. The product of this with the identity of $\mathbb{D}^{d-1}$ is then a boundary-cylinder $e_1$ for $\mathbb{B}_1^d$ whose image is disjoint from $Z$ and whose germ is the same as the identity.

For any other decorated manifold $(M, A) \in \mathcal{D}_d$, we may therefore consider the boundary connected sum $(M, A) \sharp(\mathbb{B}_1^d, Z)$. If we fix an orientation of $Z$, we may do the same for any orientedly decorated manifold $(M, A) \in \mathcal{D}_d^+$.

**Notation 1.26** To shorten notation, we write $\bar{M} = M \sharp \mathbb{B}_1^d$ and we denote the interior of $\bar{M}$ by $\tilde{M}$. Notice that $\tilde{M}$ contains (disjointly) both $\tilde{M}$ and $\mathbb{R}^d$ under its fixed identification with the interior of $\mathbb{B}_1^d$.

**Construction 1.27** It will be useful at several points (see for instance the proofs of Propositions 1.29 and 1.33) to have a specified self-embedding $\Theta$ of $\bar{M}$ whose image is $M \subset \bar{M}$, together with a path of embeddings from $\Theta$ to the identity. Let us first choose a representative for the boundary-cylinder germ $e_2$ associated to $M$; this is an embedding $\mathbb{B}_1^d = \mathbb{D}^{d-1} \times [0, 1] \hookrightarrow M$, which we also denote by $e_2$. Let us moreover assume that its image $e_2(\mathbb{B}_1^d)$ is disjoint from the submanifold $A \subset \tilde{M}$. The decorated manifold $\bar{M}$ may then be viewed as the union of $M$ and $\mathbb{D}^{d-1} \times [-1, 0]$ along $\mathbb{D}^{d-1} \times \{0\}$. Using this viewpoint, we construct a path of self-embeddings

$$ \Theta_t: \bar{M} \hookrightarrow \bar{M} $$
such that $\Theta_0 = \text{id}$ and $\Theta_1(\bar{M}) = M$. To do this, we first specify a path of embeddings of intervals $\theta_t: [-1, 1] \hookrightarrow [-1, 1]$ where $\theta_0 = \text{id}$, each $\theta_t$ agrees with the identity near 1 and $\theta_t([-1, 1]) = [t-1, 1]$. Taking the product of $\theta_t$ with the identity on $\mathbb{D}^{d-1}$ and extending by the identity on the complement $\bar{M} \setminus (\mathbb{D}^{d-1} \times [-1, 1])$, we obtain a self-embedding $\bar{M} \hookrightarrow \bar{M}$, which we define to be $\Theta_t$. In particular, $\Theta = \Theta_1$ is the advertised self-embedding of $\bar{M}$.

**Remark 1.28** We emphasise that, by construction, each self-embedding $\Theta_t$ of Construction 1.27 agrees with the identity on $M \setminus e_2(\mathbb{B}_1^d)$; in particular it agrees with the identity on $A \subset \dot{M}$.

**Proposition 1.29** For any decorated manifold $(M, A) \in \mathcal{D}_d$ and an open subgroup $G \leq \text{Diff}(Z)$, there is a homotopy fibration sequence

$$ \mathcal{E}_G(Z, \check{M} \setminus A) \xrightarrow{\kappa} \mathcal{E}_{\text{Diff}(A) \times G}(A \sqcup Z, \check{M}) \xrightarrow{\kappa} \mathcal{E}(A, \check{M}), \quad (1.5) $$

in which the second map admits a section up to homotopy, as pictured. The same holds for objects of $\mathcal{D}_d^+$, with $\text{Diff}(A)$ replaced by $\text{Diff}^+(A)$ in the middle term of (1.5).

*Proof.* The map $\mathcal{E}_{\text{Diff}(A) \times G}(A \sqcup Z, \check{M}) \to \mathcal{E}(A, \check{M})$ that forgets the embedding of $Z$ is equivariant with respect to the left action of the topological group $\text{Diff}_c(\check{M})$ of compactly-supported diffeomorphisms of $\check{M}$. By [Pal21, Prop. 4.15], the action of $\text{Diff}_c(\check{M})$ on $\mathcal{E}(A, \check{M})$ is locally retractile, i.e. it admits local sections. Thus, by [Pal60b, Th. A], the map

$$ \mathcal{E}_{\text{Diff}(A) \times G}(A \sqcup Z, \check{M}) \to \mathcal{E}(A, \check{M}) \quad (1.6) $$

is a fibre bundle, in particular a Serre fibration. Write incl. for the inclusion of $A$ into $\check{M}$ and [incl.] for its $\text{Diff}(A)$-orbit; this is a natural basepoint for $\mathcal{E}(A, \check{M})$. The point-set fibre of (1.6) over [incl.] $\in \mathcal{E}(A, \check{M})$ is clearly equal to $\mathcal{E}_G(Z, \check{M} \setminus A)$, so (1.5) is a homotopy fibration sequence.

A section up to homotopy for (1.6) is defined by sending $[\varphi] \in \mathcal{E}(A, \check{M})$ to $[(\Theta \circ \varphi) \sqcup \iota]$, where $\iota$ is the inclusion $Z \subset \mathbb{R}^d \subset \check{M}$ and $\Theta$ is the self-embedding of Construction 1.27. $\square$

**Corollary 1.30** For any decorated manifold $(M, A) \in \mathcal{D}_d$ there is a split short exact sequence

$$ 1 \to \pi_1(\mathcal{E}_G(Z, \check{M} \setminus A)) \xrightarrow{\kappa} \pi_1(\mathcal{E}_{\text{Diff}(A) \times G}(A \sqcup Z, \check{M})) \xrightarrow{\kappa} \pi_1(\mathcal{E}(A, \check{M})) \to 1, \quad (1.7) $$

where the basepoint of each space of embeddings modulo diffeomorphisms is given by the inclusion. The same holds for objects of $\mathcal{D}_d^+$, with $\text{Diff}(A)$ replaced by $\text{Diff}^+(A)$ in the middle term of (1.7).

We will sometimes consider spaces of embeddings of $Z$ into $\bar{M}$, rather than its interior $\check{M}$. We record here the following fact, for later reference.

**Proposition 1.31** The inclusion $\mathcal{E}_G(Z, \check{M} \setminus A) \subset \mathcal{E}_G(Z, \bar{M} \setminus A)$ is a homotopy equivalence.

*Proof.* A deformation retraction may be defined by post-composing embeddings with an isotopy of self-embeddings of $\bar{M}$ starting from the identity and “shrinking” a collar neighbourhood of its boundary. $\square$

### 1.2.2. The second short exact sequence

We now construct a second split short exact sequence (1.9). We first need some notation. For a decorated manifold $(M, A) = (M, A, e_1, e_2) \in \mathcal{D}_d$, recall from Notation 1.17 that we write $\text{Diff}_{\text{dec}}(M, A)$ for its automorphism group in $\mathcal{D}_d$, in other words, the self-diffeomorphisms of $M$ that send $A$ onto itself and that are compatible with the boundary-cylinder germs $e_1$ and $e_2$. Similarly, we write $\text{Diff}_{\text{dec}}^+(M, A)$ for the automorphism group of an orientedly decorated manifold $(M, A) \in \mathcal{D}_d^+$, the additional condition being that the restriction to $A$ must be orientation-preserving.

**Definition 1.32** For a decorated manifold $(M, A) \in \mathcal{D}_d$ such that $A$ decomposes as $A_1 \sqcup \dots \sqcup A_k$, we define $\text{Diff}_{\text{dec}}(M, A_1, \dots, A_k)$ to be the subgroup of $\text{Diff}_{\text{dec}}(M, A)$ of those diffeomorphisms that preserve this decomposition, in other words, that send each $A_i$ onto itself for each $1 \leq i \leq k$. In the setting of $\mathcal{D}_d^+$, we analogously define the subgroup $\text{Diff}_{\text{dec}}^+(M, A_1, \dots, A_k) \subset \text{Diff}_{\text{dec}}^+(M, A)$.
In the following, we continue to use the Notation 1.26 from §1.2.1.

**Proposition 1.33** For any decorated manifold $(M, A) \in \mathcal{D}_d$ and an open subgroup $G \leq \mathrm{Diff}(Z)$, there is a homotopy fibration sequence

$$ \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G) \xrightarrow{\sim} \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A) \longrightarrow \mathcal{E}_G(Z, \check{M} \setminus A), \quad (1.8) $$

in which the first map admits a section up to homotopy, as pictured, and where $\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$ denotes the subgroup of $\varphi \in \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z)$ such that $\varphi|_Z \in G \leq \mathrm{Diff}(Z)$.

The analogous statement also holds for objects of $\mathcal{D}_d^+$, with $\mathrm{Diff}_{\mathrm{dec}}$ replaced by $\mathrm{Diff}_{\mathrm{dec}}^+$.

*Proof.* The right-hand map above is equivariant with respect to the left action of $\mathrm{Diff}_c(\check{M} \setminus A)$. By [Pal21, Prop. 4.15], since $G \leq \mathrm{Diff}(Z)$ is an open subgroup, its action on $\mathcal{E}_G(Z, \check{M} \setminus A)$ is locally retractile, so by [Pal60b, Th. A], the right-hand map above is a fibre bundle. Its point-set fibre over the basepoint [incl.] is clearly equal to $\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$.

We now define a section up to homotopy for the inclusion $\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G) \subset \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)$, using the path of self-embeddings $\Theta_t$ for $t \in [0, 1]$ of Construction 1.27. Let us define a homotopy $H: \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A) \times [0, 1] \to \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)$ by $H(\varphi, t)(x) = \Theta_t(\varphi(\Theta_t^{-1}(x)))$ if $x \in \Theta_t(\bar{M})$ and by $H(\varphi, t)(x) = x$ if $x \notin \Theta_t(\bar{M})$. To see that this defines a section up to homotopy for the inclusion, we must check (i) that $H(\varphi, t)$ is indeed a diffeomorphism of $\bar{M}$, (ii) that $H$ is continuous, (iii) that $H(-, t)$ is a group homomorphism for each $t \in [0, 1]$, (iv) that $H(-, 0)$ is the identity and (v) that $H(\varphi, 1) \in \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$ for each $\varphi \in \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)$.

By construction, the conjugate $\Theta_t \circ \varphi \circ \Theta_t^{-1}$ is a diffeomorphism of $\Theta_t(\bar{M})$. Since $\varphi$ is a decorated diffeomorphism of $\bar{M}$, it agrees with the identity on $\mathbb{D}^{d-1} \times [-1, -1 + \delta]$ for some $\delta > 0$ and so $\Theta_t \circ \varphi \circ \Theta_t^{-1}$ agrees with the identity on $\mathbb{D}^{d-1} \times [t - 1, t - 1 + \delta']$ for some $\delta' > 0$. The result of extending this by the identity on $\bar{M} \setminus \Theta_t(\bar{M}) = \mathbb{D}^{d-1} \times [-1, t - 1)$ is therefore also a diffeomorphism. This is precisely $H(\varphi, t)$, so we have verified point (i).

Composition of proper maps (such as diffeomorphisms or closed embeddings like $\Theta_t$) is continuous in the Whitney topology (see [Mat69, §2, Prop. 1]), as is extending diffeomorphisms by the identity. Since the topology on decorated diffeomorphism groups is derived from the Whitney topology (see Definitions 1.14 and 1.16 for precise details), it follows that $H(-, t)$ is continuous for each $t \in [0, 1]$. Continuity in $t$ follows from the fact that $t \mapsto \Theta_t$ was constructed to be a continuous path of self-embeddings; see Construction 1.27. This verifies point (ii).

Points (iii) and (iv) are obvious: conjugating by a fixed self-embedding and extending by the identity is clearly a group homomorphism, and $H(-, 0)$ is the identity since $\Theta_0$ is the identity. Finally, $H(\varphi, 1)$ agrees with the identity on $\mathbb{B}_1^d \subset \bar{M}$, by construction, so in particular it acts by the identity on $Z \subset \mathbb{B}_1^d$; hence it lies in $\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$. This verifies point (v).

Thus $H(-, 1): \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A) \to \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$ is a section up to homotopy for the inclusion, witnessed by the homotopy $H$. $\square$

**Corollary 1.34** For any decorated manifold $(M, A) \in \mathcal{D}_d$ there is a split short exact sequence

$$ 1 \to \pi_1(\mathcal{E}_G(Z, \check{M} \setminus A)) \xrightarrow{\sim} \pi_0(\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)) \xrightarrow{\sim} \pi_0(\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)) \to 1, \quad (1.9) $$

where the basepoint of the space $\mathcal{E}_G(Z, \check{M} \setminus A)$ is the orbit of the inclusion. The analogous statement also holds for objects of $\mathcal{D}_d^+$, with $\mathrm{Diff}_{\mathrm{dec}}$ replaced by $\mathrm{Diff}_{\mathrm{dec}}^+$.

*Proof.* In contrast to Corollary 1.30, where it was immediate that (1.5) induces (1.7) by considering the long exact sequence of homotopy groups, this corollary requires a little more explanation, since it involves also $\pi_0$, while we aim to prove that we obtain a short exact sequence of groups and homomorphisms: more precisely, it is a priori unclear that the connecting homomorphism $\pi_1(\mathcal{E}_G(Z, \check{M} \setminus A)) \to \pi_0(\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G))$ of the homotopy long exact sequence is a group homomorphism. Let us write $\Gamma = \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)$, $H = \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$ and $B = \mathcal{E}_G(Z, \check{M} \setminus A)$. Since we are not interested in $\pi_0(B)$, we may implicitly replace $B$ with the image of the fibre bundle $\pi: \Gamma \to B$. This has the effect of possibly throwing away some path-components of $B$ (not containing its basepoint) and ensuring that $\pi$ is a surjective fibre bundle. It is also $\Gamma$-equivariant,
where in each case $\Gamma$ acts on the left by post-composition. Lemma 1.35 below thus implies that $\pi: \Gamma \to B$ is isomorphic (in the category of maps out of $\Gamma$) to the quotient map $\Gamma \to \Gamma/H$, which is therefore a fibre bundle. We shall prove in a moment that it is in fact a principal $H$-bundle. This implies that the homotopy fibration sequence $H \hookrightarrow \Gamma \to \Gamma/H$ continues two steps to the right with $\Gamma/H \to BH \to B\Gamma$ (where $BH$ and $B\Gamma$ denote the classifying spaces of $H$ and $\Gamma$ respectively). Moreover, the map $BH \to B\Gamma$ in this sequence, which is induced by the inclusion, admits a section up to homotopy, induced by the section up to homotopy in (1.8). Considering the long exact sequence of homotopy groups associated to the split fibration sequence $\Gamma/H \to BH \to B\Gamma$, we obtain the split short exact sequence (1.9), as claimed.

It remains to prove our assertion that $\Gamma \to \Gamma/H$ is a principal $H$-bundle. For this, we will use a refinement of [Pal60b, Th. A], which is remarked on page 307 of [Pal60b] and written down explicitly in [Pal21, Prop. 4.8]. Namely, let $X$ be a space with a left $\Gamma_0$-action and a right $H$-action that commute, and such that the induced left $\Gamma_0$-action on $X/H$ is locally retractile. Assume that the $H$-action on $X$ is free and that the action map $x.-$: $H \to X$ is a topological embedding for each $x \in H$. Then $X \to X/H$ is a principal $H$-bundle. In our setting we have $X = \Gamma = \text{Diff}_{\text{dec}}(\bar{M}, A)$ and we take $\Gamma_0 = \text{Diff}_c(\check{M} \setminus A)$ acting on $\Gamma$ by extending compactly-supported diffeomorphisms of $\check{M} \setminus A$ to $\bar{M}$ by the identity and post-composing. This obviously commutes with the right action of $H$ by pre-composition. The induced left action of $\Gamma_0$ on $\Gamma/H$ is the same as its left action on $\mathcal{E}_G(Z, \check{M} \setminus A)$ by post-composition under the homeomorphism $\Gamma/H \cong B = \mathcal{E}_G(Z, \check{M} \setminus A)$, which is locally retractile by [Pal21, Prop. 4.15], as mentioned already in the proof of Proposition 1.33. The right $H$-action on $\Gamma$ is simply right multiplication ($H$ is a subgroup of $\Gamma$), so it is obviously free. It is also clear that the action map $g.-$: $H \to \Gamma$ is a topological embedding for each $g \in \Gamma$, since it is just the inclusion $H \subset \Gamma$ (which is a topological embedding) followed by the homeomorphism $\Gamma \cong \Gamma$ given by left multiplication by $g$. It therefore follows from [Pal21, Prop. 4.8] that $\Gamma \to \Gamma/H$ is a principal $H$-bundle, as claimed. $\square$

**Lemma 1.35** Let $\Gamma$ be a topological group acting on the left on a space $B$ and let $\pi: \Gamma \to B$ be a surjective, $\Gamma$-equivariant fibre bundle. Denote by $H$ the stabiliser of the basepoint $\pi(1_\Gamma) \in B$. Then there is a homeomorphism $B \cong \Gamma/H$ commuting with $\pi$ and the projection $\Gamma \to \Gamma/H$ that quotients by the right action of $H$ on $\Gamma$ by multiplication.

*Proof.* Surjective fibre bundles are quotient maps, so in order to show that the two quotient maps $\Gamma \to \Gamma/H$ and $\pi: \Gamma \to B$ are isomorphic, it suffices to check that their fibres give the same partition of the underlying set of $\Gamma$. The fibres of $\Gamma \to \Gamma/H$ are by definition the right $H$-cosets in $\Gamma$. Let $b \in B$ and choose $g \in \Gamma$ so that $\pi(g) = b$, which is possible since $\pi$ is surjective. The fibre $\pi^{-1}(b)$ consists of all $g' \in \Gamma$ such that $g'\pi(1_\Gamma) = \pi(g') = b = \pi(g) = g\pi(1_\Gamma)$, where we have used $\Gamma$-equivariance of $\pi$. This condition is equivalent to $g^{-1}g' \in \text{Stab}(\pi(1_\Gamma)) = H$, in other words $g' \in gH$. Hence the fibres of $\pi$ are also the right $H$-cosets in $\Gamma$. $\square$

### 1.2.3. Connecting the short exact sequences

We now describe the relationship between the split short exact sequences (1.7) and (1.9).

**Proposition 1.36** Let $(M, A) \in \mathcal{D}_d$ be a decorated manifold, let $Z \subset \mathbb{R}^d$ be a closed submanifold and let $G \leq \text{Diff}(Z)$ be an open subgroup. Then there is a commutative diagram of the form:

$$
\begin{tikzcd}
1 \arrow[r] \arrow[d, "\text{id}"] & \pi_1(\mathcal{E}_G(Z, \check{M} \setminus A)) \arrow[r] \arrow[d] & \pi_1(\mathcal{E}_{\text{Diff}(A) \times G}(A \sqcup Z, \check{M})) \arrow[r, "\leftarrow--"] \arrow[d] & \pi_1(\mathcal{E}(A, \check{M})) \arrow[r] \arrow[d] & 1 \\
1 \arrow[r] \arrow[d] & \pi_1(\mathcal{E}_G(Z, \check{M} \setminus A)) \arrow[r] \arrow[d] & \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A, Z|G)) \arrow[r, "\leftarrow--"] \arrow[d] & \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A)) \arrow[r] \arrow[d] & 1 \quad (1.10) \\
1 \arrow[r] \arrow[d] & 1 \arrow[r] \arrow[d] & \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, \emptyset)) \arrow[r, "\text{id}"] \arrow[d] & \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, \emptyset)) \arrow[r] \arrow[d] & 1
\end{tikzcd}
$$

where the top and middle rows are (1.7) and (1.9) respectively and the columns are also exact. The top-right square also commutes when the solid horizontal arrows are replaced by the dotted arrows.
Proof. Let us consider the following diagram:

$$
\begin{tikzcd}
\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G) \arrow[r, "\kappa"] \arrow[d, "\sim"] & \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A) \arrow[r, "\sim"] \arrow[d, "\sim"] & \mathcal{E}_G(Z, \check{M} \setminus A) \arrow[r, "\sim"] \arrow[d, "\sim"] & \mathcal{E}_G(Z, \check{M} \setminus A) \\
\mathcal{E}_G(Z, \check{M} \setminus A) \arrow[r, "\sim"] & \mathcal{E}_{\mathrm{Diff}(A) \times G}(A \sqcup Z, \check{M}) \arrow[r, "\kappa"] & \mathcal{E}_G(Z, \check{M}) \arrow[r, "\sim"] & \mathcal{E}_G(Z, \check{M})
\end{tikzcd}
\qquad (1.11)
$$

The top row is (1.8) and the bottom row is (1.5). The bottom vertical maps are the restriction maps to $A \sqcup Z$ and to $A$ respectively. They are fibre bundles by the same argument as in the proof of Proposition 1.33, and the top two vertical maps are the corresponding inclusions of point-set fibres. Both squares evidently commute, since restricting a diffeomorphism to $A \sqcup Z$ and then to $A$ gives the same result as restricting directly to $A$. Taking the associated long exact sequences of homotopy groups of these homotopy fibration sequences horizontally and vertically, we obtain the diagram (1.10), where for the two columns we apply the same argument as in the proof of Corollary 1.34 (except without the section up to homotopy). One may check directly that the top-right square of (1.10) commutes when the solid horizontal arrows are replaced by the dotted arrows by considering the diagram (1.11) and using the definition of the connecting homomorphisms in the vertical long exact sequences. $\square$

**Remark 1.37** We shall prove in §1.4.3 below that the diagram (1.11) is functorial with respect to morphisms of the category $\mathcal{U}\mathcal{D}_d$ defined in §1.4 below; in particular it is functorial with respect to decorated diffeomorphisms of $(M, A)$, i.e. with respect to the morphisms of the groupoid $\mathcal{D}_d$.

### 1.2.4. Motion groups and braided mapping class groups

The goal of this section is to introduce *braided diffeomorphism groups* (see Definition 1.38) and *motion groups* (see Definition 1.39), and to show (see Proposition 1.42) that the groups of path-components of braided diffeomorphism groups (*braided mapping class groups*) are naturally identified with the corresponding motion groups when $A \subset M$ is *peripheral* in the sense of Definition 1.40 below.

**Definition 1.38** The *braided diffeomorphism group* $\mathrm{Diff}^{\mathrm{br}}_{\mathrm{dec}}(M, A)$ of a decorated manifold $(M, A)$ is the kernel of the natural homomorphism

$$ \mathrm{Diff}_{\mathrm{dec}}(M, A) \to \pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, \emptyset)), $$

in other words, it is the subgroup of diffeomorphisms of $(M, A)$ that become isotopic to the identity after forgetting $A$. We also define $\mathrm{Diff}^{\mathrm{br},+}_{\mathrm{dec}}(M, A) = \mathrm{Diff}^{\mathrm{br}}_{\mathrm{dec}}(M, A) \cap \mathrm{Diff}^{+}_{\mathrm{dec}}(M, A)$.

**Definition 1.39** Given a closed submanifold $Y \subset \check{M}$, the corresponding *motion group* $\mathrm{Mot}_Y(M)$ is the fundamental group $\pi_1(\mathcal{E}(Y, \check{M}))$. When $Y$ is orientable, the corresponding *oriented motion group* is $\mathrm{Mot}_Y^+(M) = \pi_1(\mathcal{E}^+(Y, \check{M}))$.

**Definition 1.40** Let $(M, A)$ be a decorated manifold. The submanifold $A \subset M$ is called *peripheral* if it is contained in a codimension-zero ball in $M$.

**Remark 1.41** If $A \subset M$ is peripheral, the columns of the commutative diagram (1.10) are moreover split short exact. Indeed, in this case, one may construct sections up to homotopy for the top two vertical inclusions in (1.11) (or alternatively consider the two columns as instances of (1.8) with $Z$ replaced by $A$ or $A \sqcup Z$), which implies that the columns in (1.10) are split short exact.

When $A \subset M$ is peripheral (see Definition 1.40), there is a canonical identification of (oriented) motion groups with $\pi_0$ of (oriented) braided diffeomorphism groups. Thus – in this case – it follows that motion groups are braided mapping class groups (see Remark 1.43 below).

**Proposition 1.42** Let $(M, A)$ be a decorated manifold. There is a canonical surjection

$$ \mathrm{Mot}_A(M) = \pi_1(\mathcal{E}(A, \check{M})) \longrightarrow \pi_0(\mathrm{Diff}^{\mathrm{br}}_{\mathrm{dec}}(M, A)) \quad (1.12) $$
that is an isomorphism if $A \subset M$ is peripheral. The analogous statement also holds in the oriented setting, adding a superscript + to all groups and embedding spaces involved.

*Proof.* The restriction map $\text{Diff}_{\text{dec}}(M, \emptyset) \to \mathcal{E}(A, \mathring{M})$ is a fibre bundle by the same reasoning as in the proof of Proposition 1.33, so (using the same argument as in the proof of Corollary 1.34) it induces a long exact sequence of the form

$$ \dots \to \pi_1(\text{Diff}_{\text{dec}}(M, \emptyset)) \to \text{Mot}_A(M) \to \pi_0(\text{Diff}_{\text{dec}}(M, A)) \to \pi_0(\text{Diff}_{\text{dec}}(M, \emptyset)). $$

By exactness, the image of the middle map is precisely $\pi_0(\text{Diff}_{\text{dec}}^{\text{br}}(M, A))$, so we obtain the surjection (1.12). When $A \subset M$ is peripheral, we may choose a codimension-zero ball in $M$ containing $A$ and an embedded path from a point on its boundary to a point $p \in \partial M$ that lies in one of the boundary cylinders of $M$. Since all diffeomorphisms in $\text{Diff}_{\text{dec}}(M, \emptyset)$ are the identity in a neighbourhood of $p$, we may deform them continuously until they are the identity on the chosen ball and path; in particular on $A$. This deformation provides a section up to homotopy of the inclusion $\text{Diff}_{\text{dec}}(M, A) \hookrightarrow \text{Diff}_{\text{dec}}(M, \emptyset)$. Hence this inclusion induces a split-surjection on homotopy groups; by exactness, this implies that the middle map in the long exact sequence above is injective, and hence (1.12) is an isomorphism. $\square$

**Remark 1.43 (Globality.)** In particular, Proposition 1.42 implies: each motion group $\text{Mot}_A(M)$, where $A \subset M$ is peripheral, is canonically isomorphic to a normal subgroup of the corresponding decorated mapping class group $\pi_0(\text{Diff}_{\text{dec}}(M, A))$. All of the examples of motion groups discussed in this paper are of this form; see §3.1.

Although motion groups and braided mapping class groups are in general different without the peripherality condition, there is always the canonical surjection (1.12). Thus we may conclude: a representation of the discrete groupoid $\pi_0(D_d)$ canonically induces representations of all motion groups and all mapping class groups of d-dimensional manifolds.

### 1.2.5. Actions of motion groups

To conclude this subsection, we prove a result identifying the actions coming from the two split short exact sequences (1.7) and (1.9). We first need the following general result about connecting homomorphisms in long exact sequences associated to quotients of topological groups.

**Proposition 1.44** Let $\Gamma$ be a topological group and $H \leqslant \Gamma$ a subgroup such that the quotient map $\pi: \Gamma \to \Gamma/H$ is a fibre bundle. There is a left $H$-action

* on $\pi_1(\Gamma/H)$ induced by the obvious left $H$-action on $\Gamma/H$,

* on $\pi_0(H)$ by conjugation.

With respect to these actions, the connecting homomorphism

$$ \delta: \pi_1(\Gamma/H) \to \pi_0(H) \qquad (1.13) $$

of the long exact sequence associated to $\pi$ is $H$-equivariant.

**Remark 1.45** Note that the obvious left action of the whole group $\Gamma$ on $\Gamma/H$ does not induce an action on $\pi_1(\Gamma/H)$, since it does not preserve the basepoint. Also, Proposition 1.44 may seem counterintuitive, since the action on the domain of (1.13) is induced by multiplication in $\Gamma$ whereas the action on its codomain is induced by conjugation in $H$. The reason for this is explained in Remark 1.46 after the proof.

*Proof of Proposition 1.44.* Let $[\alpha] \in \pi_1(\Gamma/H)$ represented by a loop $\alpha: [0, 1] \to \Gamma/H$ and let $h \in H$. Our goal is to show that $h.\delta([\alpha]) = \delta(h.[\alpha])$. We first compute the left-hand side of this expression. Choose a path $\tilde{\alpha}: [0, 1] \to \Gamma$ such that $\tilde{\alpha}(0) = 1_\Gamma$ and $\pi \circ \tilde{\alpha} = \alpha$ (this is possible since $\pi$ is a fibre bundle, in particular a Serre fibration). By definition of the connecting homomorphism, we have $\delta([\alpha]) = [\tilde{\alpha}(1)]$. By definition of the left-action of $H$ on $\pi_0(H)$ by conjugation, we thus have

$$ h.\delta([\alpha]) = [h\tilde{\alpha}(1)h^{-1}]. \qquad (1.14) $$

We now compute the right-hand side, namely $\delta(h.[\alpha])$. The element $h.[\alpha] \in \pi_1(\Gamma/H)$ is represented by the loop $h.\alpha: [0, 1] \to \Gamma/H$ given by applying pointwise the left action of $H$ on $\Gamma/H$, i.e. it
(ii) Precomposing with the canonical map (see Proposition 1.42)

$$ \pi_1(\mathcal{E}(A, \mathring{M})) \longrightarrow \pi_0(\mathrm{Diff}_{\mathrm{dec}}^{\mathrm{br}}(M, A)) \leq \pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A)), \quad (1.20) $$

the action (1.19) agrees with the action induced by the split short exact sequence (1.7), pre-composed with the isomorphism $\pi_1(\mathcal{E}(A, \mathring{M})) \cong \pi_1(\mathcal{E}(A, \mathring{M}))$ induced by the inclusion $\mathring{M} \subset \mathring{M}$.

Both statements also hold in the oriented setting, adding a superscript + to all groups and all embedding spaces (except those already with a subscript G) in the statement.

*Proof.* As explained in the proof of Corollary 1.34, the fibre bundle $\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A) \to \mathcal{E}_G(Z, \mathring{M} \setminus A)$ from (1.8) is isomorphic (in the category of maps out of $\Gamma$) to the quotient map $\Gamma \to \Gamma/H$, where $\Gamma = \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)$ and $H = \mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G)$ and we have possibly thrown away some non-basepoint path-components of $\mathcal{E}_G(Z, \mathring{M} \setminus A)$, which does not matter since we are only interested in its fundamental group. We are therefore in the setting of Corollary 1.47, with (1.18) = (1.9). It then follows from Corollary 1.47 that the action of $\pi_0(\Gamma)$ on $\pi_1(\mathcal{E}_G(Z, \mathring{M} \setminus A))$ induced by the split short exact sequence (1.9) is equal to the section $\pi_0(\Gamma) \to \pi_0(H)$ followed by the action of $\pi_0(H) = \pi_0(\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A, Z|G))$ on $\pi_1(\mathcal{E}_G(Z, \mathring{M} \setminus A))$ by post-composition. Pre-composing the latter action by the isomorphism $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A)) \cong \pi_0(\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)) = \pi_0(\Gamma)$, we obtain exactly the action (1.19), because extending by the identity on $\bar{M} \setminus M$, conjugating by the isomorphism $\Theta: \bar{M} \to M$ (see Construction 1.27) and extending by the identity again is isotopic to simply extending by the identity once. This establishes part (i) of the proposition.

Part (ii) of the proposition then follows from part (i), together with diagram (1.10) of Proposition 1.36 (in particular the fact that its top-right square commutes also when the solid horizontal arrows are replaced by the dotted arrows) and the observation that the top-right vertical map of that diagram coincides with the canonical map (1.20) under the two isomorphisms

$$ \pi_1(\mathcal{E}(A, \mathring{M})) \cong \pi_1(\mathcal{E}(A, \mathring{M})) \quad \text{and} \quad \pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A)) \cong \pi_0(\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, A)). \quad (1.21) $$

This last observation follows because (1.20) is the connecting homomorphism of the restriction fibre bundle $\mathrm{Diff}_{\mathrm{dec}}(M, \emptyset) \to \mathcal{E}(A, \mathring{M})$ (see the proof of Proposition 1.42), the top-right vertical map of (1.10) is the connecting homomorphism of the restriction fibre bundle $\mathrm{Diff}_{\mathrm{dec}}(\bar{M}, \emptyset) \to \mathcal{E}(A, \mathring{M})$ (see diagram (1.11)) and the isomorphisms (1.21) are induced by the map of fibre bundles induced by extending decorated diffeomorphisms of $M$ to $\bar{M}$ by the identity on $\bar{M} \setminus M$ and post-composing embeddings into $\mathring{M}$ with the inclusion $\mathring{M} \subset \mathring{M}$. $\square$

## 1.3. A Serre fibration of decorated diffeomorphism groups

This section is devoted to the proof of a technical result (see Theorem 1.55) concerning decorated diffeomorphism groups, which will play an important role in §1.4 to ensure that the topologically-enriched Quillen bracket construction behaves well with respect to the $\pi_0$ functor on the topologically-enriched groupoids of decorated manifolds that we shall consider; see Lemma 1.71 and Proposition 1.76. From this point onwards, we assume that the dimension $d$ is not equal to 4; see Remark 1.54 for the reason why this is necessary. We start by introducing the following notion.

**Definition 1.49 (Decorated embeddings.)** Let $M = (M, A)$ and $N = (N, B)$ be decorated manifolds. Define $\mathrm{Emb}_{\mathrm{dec}}(M, N)$ to be the set of smooth, proper embeddings $\varphi: M \hookrightarrow N$, equipped with a germ of an extension $\varphi'$ to an embedding $\mathbb{B}_1^d \hookrightarrow M \hookrightarrow N$, such that:

* $\varphi(A) \subseteq B$;

* for some $\epsilon > 0$ we have $\varphi(e_2(\mathbb{B}_\epsilon^d)) = e'_2(\mathbb{B}_\epsilon^d)$ and $(e'_2)^{-1} \circ \varphi \circ e_2$ is the identity map $\mathbb{B}_\epsilon^d \to \mathbb{B}_\epsilon^d$, where $e_1, e_2$ are the boundary-cylinder germs of $(\bar{M}, A)$ and $e'_1, e'_2$ are those of $(N, B)$;

* there is a decorated manifold $M'$ and diffeomorphism of decorated manifolds $\bar{\varphi}: M' \hookrightarrow M \to N$ such that $\varphi = \bar{\varphi} \circ \iota_{M,M'}$, where $\iota_{M,M'}$ denotes the canonical embedding of $M$ into $M' \hookrightarrow M$. This extension of $\varphi$ to $\bar{\varphi}$ should be compatible with the given germ of an extension $\varphi'$ of $\varphi$.

In other words – modulo decorated diffeomorphisms of the codomain – a decorated embedding is the canonical inclusion of (a germ of a neighbourhood of) the right-hand factor of a boundary connected sum. As an illustration, Figure 1.3 depicts the inclusion of a neighbourhood $M_t$ of $M$ into $L \hookrightarrow M$, representing a decorated embedding of $M$ into $L \hookrightarrow M$.
We now describe the topology on this set. Recall that $\bar{M} = \mathbb{B}_1^d[M] = (\mathbb{D}^{d-1} \times [-1, 0]) \cup M$ and write $M_t = (\mathbb{D}^{d-1} \times [-t, 0]) \cup M$ for $t \in (0, 1)$, as depicted in Figure 1.3. For $\epsilon \in (0, 1)$, write $\text{Emb}_{\text{dec},\epsilon}(M_t, N)$ for the set of embeddings defined as above except that $\epsilon$ is fixed in the second bullet point and the embedding is defined on $M_t$ rather than on $M$ plus a germ of a neighbourhood. Topologise $\text{Emb}_{\text{dec},\epsilon}(M_t, N)$ as a subspace of $C^\infty(M_t, N)$ equipped with the Whitney topology. We may take the colimit as $\epsilon \to 0$ (along inclusion maps) and $t \to 0$ (along restriction maps), and we define $\text{Emb}_{\text{dec}}(M, N) := o(\text{colim}_{\epsilon,t \to 0}(\text{Emb}_{\text{dec},\epsilon}(M_t, N)))$, where $o(-)$ is the functor of Construction 1.4.

**Notation 1.50** For decorated manifolds $L, M, N$, we denote by $\text{Emb}_{\text{dec}}(M, N)_L$ the subspace of $\text{Emb}_{\text{dec}}(M, N)$ of those embeddings for which we may take $M' = L$ in the third point above. Additionally, if $A$ and $B$ are oriented, we denote by $\text{Emb}_{\text{dec}}^+(M, N)$ the subspace of $\text{Emb}_{\text{dec}}(M, N)$ of those decorated embeddings $\varphi: M \hookrightarrow N$ whose restriction $\varphi|_A: A \hookrightarrow B$ is orientation-preserving.

The composition of decorated embeddings is defined as follows.

**Definition 1.51** Let $\varphi \in \text{Emb}_{\text{dec}}(M, N)$ and $\psi \in \text{Emb}_{\text{dec}}(N, P)$ for decorated manifolds $M, N$ and $P$. Choose decorated diffeomorphisms $\bar{\varphi}: M'\natural M \to N$ and $\bar{\psi}: N'\natural N \to P$ extending $\varphi$ and $\psi$ respectively; these exist by definition. We may then form the composite decorated diffeomorphism $\bar{\psi} \circ (\text{id}_{N'\natural}\bar{\varphi}) : N'\natural M'\natural M \to P$, which restricts to the composite embedding $\psi \circ \varphi$ on $M \subset N'\natural M'\natural M$. Moreover, its restriction to an $\epsilon$-neighbourhood of $M \subset N'\natural M'\natural M$ provides a germ of an extension of $\psi \circ \varphi$ to $\mathbb{B}_1^d[M]$. The first two points of Definition 1.49 clearly hold for the composite embedding $\psi \circ \varphi$, so this embedding together with the germ of an extension is an element of $\text{Emb}_{\text{dec}}(M, P)$, which we define to be the composition of the decorated embeddings $\varphi$ and $\psi$.

**Lemma 1.52** *The composition of decorated embeddings given in Definition 1.51 defines a continuous, associative operation* $\text{Emb}_{\text{dec}}(M, N) \times \text{Emb}_{\text{dec}}(N, P) \to \text{Emb}_{\text{dec}}(M, P)$.

*Proof.* We must first check that this operation is a well-defined map, i.e. that the construction of Definition 1.51 does not depend on the choice of decorated diffeomorphisms $\bar{\varphi}$ and $\bar{\psi}$ extending $\varphi$ and $\psi$. The only part of the construction that depends on these choices of extensions is the germ of an extension of $\psi \circ \varphi$ to an embedding $\mathbb{B}_1^d[M] \hookrightarrow P$. The fact that this germ is independent of the choice of decorated diffeomorphisms extending the decorated embeddings follows from the fact that decorated diffeomorphisms are required to be compatible with boundary-cylinder germs. Continuity of the operation follows from the fact that composition of proper, smooth maps is continuous in the Whitney topology; see [Mat69, §2, Prop. 1].

Finally, let us check associativity of this operation. For $\varphi \in \text{Emb}_{\text{dec}}(M, N)$, $\psi \in \text{Emb}_{\text{dec}}(N, P)$ and $\chi \in \text{Emb}_{\text{dec}}(P, Q)$, the two iterated compositions $(\chi \circ \psi) \circ \varphi$ and $\chi \circ (\psi \circ \varphi) \in \text{Emb}_{\text{dec}}(M, Q)$ each consist of a smooth embedding $M \hookrightarrow Q$ and a germ of an extension to $\mathbb{B}_1^d[M] \hookrightarrow Q$. The two smooth embeddings $M \hookrightarrow Q$ are evidently equal, by associativity of composition of maps. Moreover, the fact that the two germs of extensions to embeddings $\mathbb{B}_1^d[M] \hookrightarrow Q$ are equal follows from the fact that they are restrictions, to an $\epsilon$-neighbourhood of $M \subset P'\natural N'\natural M'\natural M$, of the decorated diffeomorphisms $(\bar{\chi} \circ (\text{id}_{P'\natural}\bar{\psi})) \circ (\text{id}_{P'\natural N'\natural}\bar{\varphi})$ and $\bar{\chi} \circ ((\text{id}_{P'\natural}\bar{\psi}) \circ (\text{id}_{P'\natural N'\natural}\bar{\varphi}))$, which are equal by associativity of composition of maps. $\square$

**Lemma 1.53** *The embedding space* $\text{Emb}_{\text{dec}}(M, N)$ *decomposes as a topological disjoint union*

$$ \text{Emb}_{\text{dec}}(M, N) \cong \bigsqcup_L \text{Emb}_{\text{dec}}(M, N)_L, \qquad (1.22) $$

where the disjoint union runs over representatives of isomorphism classes of decorated manifolds.

*Proof.* Let us denote by $(-)^c$: $\text{Emb}_{\text{dec}}(M, N) \to \{\text{isomorphism classes of decorated manifolds}\}$ the function that assigns to each decorated embedding the isomorphism class of the decorated manifold given by the complement of its image. We must show that this function is locally constant. Let $\varphi \in \text{Emb}_{\text{dec}}(M, N)$ be any decorated embedding; we must show that it admits a neighbourhood $\mathcal{U}$ such that $\bar{\psi}^c = \varphi^c$ for each $\psi \in \mathcal{U}$.

Since $\varphi$ is a decorated embedding, we may fix an identification of $N$ with $L\natural M$, for some decorated manifold $L$, such that $\varphi$ is (the germ of an extension to a collar neighbourhood of) the canonical inclusion of $M$ into $L\natural M$.
We first note that there exists an open neighbourhood U of $\varphi$ such that, for each $\psi \in U$, the interface $\psi(M) \cap ((L\sharp M)) \setminus \psi(M))$ between the image of $\psi$ and its complement is a smoothly embedded $(d-1)$-dimensional disc $D(\psi)$ contained in the central solid cylinder $D^{d-1} \times [-1, 1]$ along which the boundary connected sum of $L\sharp M$ is formed; see Figure 1.3. Moreover, $D(\psi)$ intersects the boundary of $D^{d-1} \times [-1, 1]$ precisely in $\partial D(\psi)$. Such an open neighbourhood U exists in the compact-open topology, and hence also in the topology that we are considering on $Emb_{dec}(M, N)$ (a refinement of a colimit of Whitney topologies), which is finer than the compact-open topology.

It remains to show, for a given $\psi \in U$, that $\psi^c = \varphi^c$. The embedded codimension-1 disc $D(\psi)$ cuts the solid cylinder $D^{d-1} \times [-1, 1]$ into two pieces; let us denote by $A(\psi)$ the piece that contains $D^{d-1} \times \{-1\}$. We claim that $A(\psi)$ is diffeomorphic to $D^{d-1} \times [-1, 0]$ by a diffeomorphism $\Xi$ that acts by the identity on a neighbourhood of $D^{d-1} \times \{-1\}$. This claim will immediately imply that $\psi^c = \varphi^c$, as desired, by extending $\Xi$ by the identity on $L \setminus (D^{d-1} \times [-1, 0])$.

Smoothing corners, we may rephrase the claim in the previous paragraph as follows. Consider the standard $d$-ball $D^d \subset \mathbb{R}^d$, denote by $D_+^d$ the closed upper half-ball and fix a small $(d-1)$-dimensional subdisc $D_0$ of the boundary $\partial D^d$ centred at the north pole. Suppose we are given a smoothly embedded $(d-1)$-dimensional disc $D \subset D^d$ with $D \cap \partial D^d = \partial D$ disjoint from $D_0$ and denote by $A$ the closure of the component of $D^d \setminus D$ containing $D_0$. The claim is that there exists a diffeomorphism $\Xi: A \cong D_+^d$ that acts by the identity on a neighbourhood of $D_0$. As explained above, this claim will complete the proof.

To prove this claim, we first consider the double $2(D^d)$ of $D^d$, which is a smooth $d$-sphere. Smoothing corners again, the boundary of $A \subset D^d \subset 2(D^d)$ is a smoothly embedded $(d-1)$-sphere. The smooth Schoenflies theorem (see [Sch06] for $d=2$, [Maz59; Mor60; Bro60] for $d=3$ and [Mil65, Prop. D, §9] for $d \ge 5$) then implies that $A$ is diffeomorphic to a $d$-ball. Smoothing corners yet again, this gives a diffeomorphism $\Xi': A \cong D_+^d$. However, this does not necessarily act by the identity on a neighbourhood of $D_0$.

To ensure this second condition, let $e: D^d \to D^d$ be an orientation-preserving embedding so that $e(D^d)$ is a neighbourhood of $D_0$ and so that $e(D^d)$ is contained in $A \cap D_+^d$. Then $e$ and $\Xi' \circ e$ are two embeddings of $D^d$ into $D_+^d$. By composing $\Xi'$ with a reflection if necessary, $\Xi' \circ e$ is also orientation-preserving. Palais' disc embedding theorem [Pal60a, Thm. B] implies that there is a diffeomorphism $\Xi'': D_+^d \cong D_+^d$ such that $\Xi'' \circ e = \Xi' \circ e$. The composition $\Xi := (\Xi'')^{-1} \circ \Xi': A \cong D_+^d$ is then a diffeomorphism that acts by the identity on the neighbourhood $e(D^d)$ of $D_0$, as desired. $\square$

**Remark 1.54** The proof above uses the smooth Schoenflies theorem, which is an open question (rather than a theorem) in dimension $d=4$; this is the reason for the restriction $d \ne 4$ that we make in the present subsection §1.3, as well as consequently in §1.4.2–§1.4.3, §2.2 and §2.3.

Now, for the remainder of §1.3, we consider two decorated manifolds $L = (L, A, e_1, e_2)$ and $M = (M, B, e'_1, e'_2)$. There is a continuous right action of $Diff_{dec}(L)$ on $Diff_{dec}(L\sharp M)$ given by $\varphi \cdot \psi = \varphi \circ (\psi\sharp \text{id}_M)$, and hence a quotient map

$$ \Psi: \mathrm{Diff}_{\mathrm{dec}}(L\sharp M) \to \mathrm{Diff}_{\mathrm{dec}}(L\sharp M)/\mathrm{Diff}_{\mathrm{dec}}(L). \qquad (1.23) $$

**Theorem 1.55** The quotient map (1.23) is a Serre fibration. There is a homeomorphism between its codomain and $Emb_{dec}(M, L\sharp M)_L$, induced by the restriction map.

If the submanifolds $A \subset \tilde{L}$ and $B \subset \tilde{M}$ are oriented and we consider the subgroups $Diff_{dec}^+(\cdot)$ of $Diff_{dec}(\cdot)$, then the analogue of the quotient map (1.23) is also a Serre fibration and there is a homeomorphism from its codomain to $Emb_{dec}^+(M, L\sharp M)_L$, induced by the restriction map.

**Remark 1.56** This is related to results of Cerf [Cer61, p. 294, §II.2.2.2, Cor. 2], Palais [Pal60b, Th. B] and Lima [Lim63], but we were not able to find an instance of their results that covers exactly the setting that we need here. We therefore give a complete proof of Theorem 1.55 below, using as an input two results of Cerf and Palais, namely Lemme II.2.1.2 (page 291) of [Cer61] and [Pal60b, Th. A].

*Proof of Theorem 1.55.* In the proof below, we use the topology on the spaces of decorated diffeomorphisms and of decorated embeddings given by a colimit of Whitney topologies, as described in Definitions 1.14 and 1.49, thus proving that (1.23) is a Serre fibration for this topology. This result automatically implies that the same result holds when using the finer topology making all
<figure><img src="image_4.png" /><figcaption>Figure 1.3 The boundary connected sum $L\sharp M$ from the proof of Theorem 1.55.</figcaption></figure>

path-components open; see Lemma 1.7. Hence we shall make no further mention of this finer topology and work directly with the colimit of Whitney topologies.

Moreover, we detail here the proof for the first (unoriented) case of Theorem 1.55. All of the arguments below repeat mutatis mutandis in the setting where the submanifolds A and B are oriented and the morphisms of decorated manifolds preserve these orientations.

The decorated manifolds $L = (L, A, e_1, e_2)$ and $M = (M, B, e'_1, e'_2)$ come equipped with germs $e_1, e_2, e'_1, e'_2$ of boundary cylinders; let us once and for all choose representative boundary cylinders for these germs, and denote them by the same symbols, by abuse of notation.

For $\epsilon \in (0,1)$, let $\text{Diff}_{\text{dec},\epsilon}(L\sharp M)$ denote the group of self-diffeomorphisms of $L\sharp M$ sending $A \sqcup B$ onto itself and restricting to the identity on $e_1(\mathbb{B}_\epsilon^d)$ and on $e'_2(\mathbb{B}_\epsilon^d)$. If we give this the Whitney topology, then we have

$$ \text{Diff}_{\text{dec}}(L\sharp M) \cong \text{colim}_{\epsilon \to 0} (\text{Diff}_{\text{dec},\epsilon}(L\sharp M)), \quad (1.24) $$

by Definitions 1.12 and 1.14 (recalling that, after the first paragraph of the present proof, we are ignoring the refinement $o(-)$ of the topology). Similarly, for each $\epsilon, t \in (0,1)$, let $\text{Diff}_{\text{dec},\epsilon,t}(L)$ denote the subgroup of $\text{Diff}_{\text{dec},\epsilon}(L\sharp M)$ consisting of diffeomorphisms that restrict to the identity on the submanifold $M_t = M \cup e_2(\mathbb{B}_\epsilon^d)$ of $L\sharp M$ pictured in Figure 1.3. We have a quotient map

$$ \Psi_{\epsilon,t}: \text{Diff}_{\text{dec},\epsilon}(L\sharp M) \to \text{Diff}_{\text{dec},\epsilon}(L\sharp M)/\text{Diff}_{\text{dec},\epsilon,t}(L). $$

For any $\epsilon, \epsilon', t, t' \in (0,1)$ with $\epsilon' \le \epsilon$ and $t' \le t$ there are natural maps

$$ \text{Diff}_{\text{dec},\epsilon}(L\sharp M)/\text{Diff}_{\text{dec},\epsilon,t}(L) \to \text{Diff}_{\text{dec},\epsilon'}(L\sharp M)/\text{Diff}_{\text{dec},\epsilon',t'}(L), $$

so we may take the directed colimit of the maps $\Psi_{\epsilon,t}$ to obtain

$$ \text{colim}_{\epsilon,t \to 0} (\Psi_{\epsilon,t}): \text{Diff}_{\text{dec}}(L\sharp M) \to \text{colim}_{\epsilon,t \to 0} (\text{Diff}_{\text{dec},\epsilon}(L\sharp M)/\text{Diff}_{\text{dec},\epsilon,t}(L)), $$

where we have used the identification (1.24) in the domain. Since each $\Psi_{\epsilon,t}$ is a quotient map, using the characterisation of quotient maps as coequalisers and the fact that colimits commute, we deduce that $\text{colim}_{\epsilon,t \to 0}(\Psi_{\epsilon,t})$ is also a quotient map. The map

$$ \Psi: \text{Diff}_{\text{dec}}(L\sharp M) \to \text{Diff}_{\text{dec}}(L\sharp M)/\text{Diff}_{\text{dec}}(L), $$

i.e. the map (1.23) that we would like to show is a Serre fibration, is also a quotient map, with the same domain. Since $M_t$ is a cofinal family of neighbourhoods of $M$ in $L\sharp M$, two diffeomorphisms of $\text{Diff}_{\text{dec}}(L\sharp M)$ have the same image under $\Psi$ if and only if they have the same image under $\text{colim}_{\epsilon,t \to 0}(\Psi_{\epsilon,t})$. As they are quotient maps with the same domain, it follows that $\Psi \cong \text{colim}_{\epsilon,t \to 0}(\Psi_{\epsilon,t})$.

We will prove below that each $\Psi_{\epsilon,t}$ is a fibre bundle (and hence a Serre fibration), and then deduce that $\Psi$ is also a Serre fibration using the following general fact.

(*) Any filtered colimit of based Serre fibrations between compactly-generated weak-Hausdorff spaces is again a Serre fibration.

For a reference for this fact, see [TV08, Prop. 1.2.3.5(1)], which states that a filtered colimit of fibrations is a fibration in any compactly generated model category. The classical model category of based compactly-generated weak-Hausdorff spaces, with its Quillen model structure in which the fibrations are the Serre fibrations, is compactly generated; see for example [MMSS01, Prop. 6.3].
To apply (*) in our situation, first note that we are taking a directed colimit, which is in particular a filtered colimit. We then need to check that the diffeomorphism groups $Diff_{\mathrm{dec},\epsilon}(L\sharp M)$ and their quotients are compactly-generated weak-Hausdorff spaces. Recall that, when restricting to subspaces of proper maps, the Whitney topology on spaces of smooth maps coincides with the weak topology, which is second-countable and hence also first-countable; see [Hir76, Chap. 2.1]. Also, diffeomorphism groups of smooth manifolds are Hausdorff in the compact-open topology, and thus also in the Whitney topology, since the latter is a finer topology. Hence diffeomorphism groups of smooth manifolds, in the Whitney topology, are always first-countable and Hausdorff, and thus compactly-generated and weak-Hausdorff. Moreover, the property of being compactly-generated is preserved when taking quotients. The property of being (weak) Hausdorff is not preserved when taking quotients; however, in the process of proving that each $\Psi_{\epsilon,t}$ is a fibre bundle below, we will also show that its target space $\mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)/\mathrm{Diff}_{\mathrm{dec},\epsilon,t}(L)$ is Hausdorff.

It therefore remains to show that each $\Psi_{\epsilon,t}$ is a fibre bundle and that its target space is Hausdorff. Write

$$ \mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L $$

for the space of smooth, proper embeddings $\varphi: M_t \to L\sharp M$ such that $\varphi(B) \subseteq A \sqcup B$, the restriction of $\varphi$ to $e'_2(\mathbb{B}_\epsilon^d)$ is the identity and there exists $\bar{\varphi} \in \mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)$ such that $\varphi = \bar{\varphi} \circ \iota$, where $\iota$ is the inclusion of $M_t$ into $L\sharp M$. Then we have

$$ \mathrm{Emb}_{\mathrm{dec}}(M, L\sharp M)_L \cong \operatorname{colim}_{\epsilon,t \to 0} (\mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L), \quad (1.25) $$

see Definition 1.49, Notation 1.50 and Lemma 1.53. There is a restriction map

$$ \Phi_{\epsilon,t}: \mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M) \to \mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L, $$

which is equivariant with respect to the left action of $\mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)$ by post-composition. This factors through the quotient map $\Psi_{\epsilon,t}$, so we have an induced map

$$ \begin{tikzcd} \mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M) \arrow[r, "\Phi_{\epsilon,t}"] \arrow[d, "\Psi_{\epsilon,t}", "\cong"] & \mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L \arrow[d, "\widehat{\Phi}_{\epsilon,t}", "\cong"'] \\ \mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)/\mathrm{Diff}_{\mathrm{dec},\epsilon,t}(L) \arrow[r, "\cong"] & \mathrm{Emb}_{\mathrm{dec}}(M, L\sharp M)_L \arrow[d, "\cong", "\cong"'] \end{tikzcd} $$

By definition of the right-hand embedding space, the map $\Phi_{\epsilon,t}$ is surjective, and so is the induced map $\widehat{\Phi}_{\epsilon,t}$. Moreover, if two diffeomorphisms of $\mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)$ have the same image under $\Phi_{\epsilon,t}$, their difference lies in $\mathrm{Diff}_{\mathrm{dec},\epsilon,t}(L)$, so the induced map $\widehat{\Phi}_{\epsilon,t}$ is also injective.

We claim that it is sufficient to prove that $\Phi_{\epsilon,t}$ is a fibre bundle. Indeed, it is then a quotient map, since surjective fibre bundles are always quotient maps. Thus the induced map $\widehat{\Phi}_{\epsilon,t}$ must be a homeomorphism. This implies:

* The map $\Psi_{\epsilon,t}$ is also a fibre bundle, hence a Serre fibration.

* Its target space is homeomorphic to the embedding space $\mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L$, which we have given the Whitney topology, so it is Hausdorff.

* We also obtain the second statement of the proposition:

$$ \begin{align*} \mathrm{Diff}_{\mathrm{dec}}(L\sharp M)/\mathrm{Diff}_{\mathrm{dec}}(L) &\cong \operatorname{colim}_{\epsilon,t \to 0} (\mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)/\mathrm{Diff}_{\mathrm{dec},\epsilon,t}(L)) \\ &\cong \operatorname{colim}_{\epsilon,t \to 0} (\mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L) \\ &\cong \mathrm{Emb}_{\mathrm{dec}}(M, L\sharp M)_L, \end{align*} $$

by combining the identification of $\Psi \cong \operatorname{colim}_{\epsilon,t \to 0} (\Psi_{\epsilon,t})$ with the colimit of $\widehat{\Phi}_{\epsilon,t}$ and (1.25).

It therefore remains just to prove that $\Phi_{\epsilon,t}$ is a fibre bundle. Since it is equivariant with respect to the left action of $\mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)$, it suffices to prove that the action of $\mathrm{Diff}_{\mathrm{dec},\epsilon}(L\sharp M)$ on $\mathrm{Emb}_{\mathrm{dec},\epsilon}(M_t, L\sharp M)_L$ is locally retractile. This is because, by [Pal60b, Th. A], any $G$-equivariant map into a $G$-locally retractile space is a fibre bundle.
Thus, we have to prove the following statement: given an embedding $e \in \text{Emb}_{\text{dec},\epsilon}(M_t, L_M)$, we may find an open neighbourhood $\mathcal{U}$ of $e$ and a continuous map $\gamma: \mathcal{U} \to \text{Diff}_{\text{dec},\epsilon}(L_M)$ such that $\gamma(e) = \text{id}$ and $\gamma(f) \circ e = f$ for any $f \in \mathcal{U}$. Note that, since $\text{Diff}_{\text{dec},\epsilon}(L_M)$ acts transitively on $\text{Emb}_{\text{dec},\epsilon}(M_t, L_M)$ (because $\Phi_{\epsilon,t}$ is both equivariant and surjective), it suffices to prove this for just one such $e$, which we take to be the inclusion $M_t \hookrightarrow L_M$.

To prove this, we apply a result of Cerf [Cer61, p. 291, Lem. II.2.1.2], which we first recall. Let $X$ be a manifold-with-corners. This means in particular that $X$ has a stratification into faces (for example, if $X$ is a connected manifold with boundary, but no higher-codimension corners, then its set of faces is $\pi_0(\partial X) \sqcup \{X\}$). Each point $x \in X$ may lie in many faces, but it has a unique smallest face (according to inclusion) in which it lies, which we denote by $C_X(x)$. Now if $Y$ is any submanifold-with-corners of $X$, we define

$$C_{\text{face}}^{\infty}(Y, X) = \{\text{smooth maps } \varphi: Y \to X \text{ such that } C_X(\varphi(x)) = C_X(x) \text{ for each } x \in Y\},$$

equipped with the Whitney topology. The Extension Lemma II.2.1.2 of [Cer61] says that, if $Y$ is closed in $X$ and $V$ is any neighbourhood of $Y$ in $X$, then the restriction map

$$C_{\text{face}}^{\infty}(X, X) \to C_{\text{face}}^{\infty}(Y, X)$$

admits a section $s$ defined on an open neighbourhood $\mathcal{V}$ of the inclusion in $C_{\text{face}}^{\infty}(Y, X)$, such that $s(\text{incl}) = \text{id}$ and $s(f)(x) = x$ for all $f \in \mathcal{V}$ and $x \in X \setminus V$.

**Step 1.** Let us write $\partial_{\bullet}L$ for the union of all boundary components of $L$ except for the one that intersects the image of $e_2$; see Figure 1.3. Note that $\partial_{\bullet}L$ may or may not intersect the image of $e_1$. Then there is a canonical identification:

$$\pi_0(\partial(L_M)) \cong \pi_0(\partial_{\bullet}L) \sqcup \pi_0(\partial M). \qquad (1.26)$$

This is necessarily asymmetric in $L$ and $M$. Each embedding $f \in \text{Emb}_{\text{dec},\epsilon}(M_t, L_M)_L$ extends to a diffeomorphism of $L_M$, so it induces an injection $f_{\partial}: \pi_0(\partial M) \to \pi_0(\partial(L_M))$. In particular, if $f$ is the inclusion, then $f_{\partial}$ is also the inclusion, under the identification (1.26). In addition, we know that $f$ sends $B$ into $A \sqcup B$, so it also induces a map $f_{\partial}: \pi_0(B) \to \pi_0(A) \sqcup \pi_0(B)$, which must be an injection since $A$ and $B$ are closed manifolds and $f$ is an embedding. The function $f \mapsto (f_{\partial}, f_{\partial})$ is locally constant, so its fibres are open. Let $\mathcal{U}'$ be the open subset of $\text{Emb}_{\text{dec},\epsilon}(M_t, L_M)_L$ consisting of all $f$ such that $f_{\partial}$ is the inclusion and $f_{\partial}(\pi_0(B)) = \pi_0(B)$. Note that the second condition implies that $f(B) = B$, since $f$ is an embedding and $B$ is a closed manifold.

**Step 2.** Write $M_{\epsilon,t} = e_1(\mathbb{B}_{\epsilon}^d) \sqcup M_t$ (pictured in Figure 1.4). Let

$$\gamma': \text{Emb}_{\text{dec},\epsilon}(M_t, L_M)_L \to C^{\infty}(M_{\epsilon,t}, L_M)$$

be the continuous map that extends a given embedding $M_t \hookrightarrow L_M$ to a smooth map $M_{\epsilon,t} \to L_M$ by defining it to be the identity on $e_1(\mathbb{B}_{\epsilon}^d)$. Note that this may fail to be injective, so it is just a smooth map, not necessarily an embedding. Also observe that, if $f$ lies in the open subset $\mathcal{U}'$ from Step 1, then $\gamma'(f)$ lies in the subspace $C_{\text{face}}^{\infty}(M_{\epsilon,t}, L_M)$, since it takes points of $(L_M^{\mathring{\partial}}) \cap M_{\epsilon,t}$ into the interior of $L_M$ and, for any boundary component $P$ of $L_M$, it takes $P \cap M_{\epsilon,t}$ into $P$ (this uses the fact that $f_{\partial} = \text{id}$). Restricting $\gamma'$ to $\mathcal{U}'$, we therefore have a continuous map

$$\gamma': \mathcal{U}' \to C_{\text{face}}^{\infty}(M_{\epsilon,t}, L_M)$$

such that $\gamma'(\text{incl}) = \text{incl}$ and $\gamma'(f)|_{M_t} = f$ for all $f \in \mathcal{U}'$.

**Step 3.** Now set $X = L_M$ and $Y = M_{\epsilon,t}$ in the Extension Lemma of Cerf above, and choose $V$ to be any open neighbourhood of $M_{\epsilon,t}$ in $L_M$ that is disjoint from the submanifold $A \subset \mathring{L}$. Composing the local section $s$ obtained from the Extension Lemma with $\gamma'$, we have a continuous map

$$\gamma'' = s \circ \gamma' : \mathcal{U}'' = (\gamma')^{-1}(\mathcal{V}) \to C_{\text{face}}^{\infty}(L_M, L_M)$$

such that $\gamma''(\text{incl}) = \text{id}$ and for any $f \in \mathcal{U}''$ we have $\gamma''(f)|_{M_t} = f$ and $\gamma''(f)(A) = A$. Moreover, by construction, we also know that $\gamma''(f)(B) = B$ and $\gamma''(f)(x) = x$ for all $x \in e_1(\mathbb{B}_{\epsilon}^d) \sqcup e_2(\mathbb{B}_{\epsilon}^d)$.

**Step 4.** Finally, note that $\text{Diff}(L_M)$ is open in $C^{\infty}(L_M, L_M)$, so

$$\mathcal{U} = (\gamma'')^{-1}(C_{\text{face}}^{\infty}(L_M, L_M) \cap \text{Diff}(L_M))$$
<figure><img src="image_5.png" /><figcaption>Figure 1.4 The submanifold $M_{\epsilon,t}$ (shaded in green) of $L_{\vdash}M$ from the proof of Theorem 1.55.</figcaption></figure>

is an open neighbourhood of the inclusion in $Emb_{dec,\epsilon}(M_t, L_{\vdash}M)_L$. For each $f \in \mathcal{U}$, the diffeomorphism $\gamma''(f)$ of $L_{\vdash}M$ fixes each point of $e_1(\mathbb{B}_{\epsilon}^d) \sqcup e'_2(\mathbb{B}_{\epsilon}^d)$ and sends $A \sqcup B$ onto itself, so it is an element of $Diff_{dec,\epsilon}(L_{\vdash}M)$. So we have a continuous map

$$ \gamma = \gamma''|_{\mathcal{U}}: \mathcal{U} \to \mathrm{Diff}_{\mathrm{dec},\epsilon}(L_{\vdash}M) $$

such that $\gamma(\mathrm{incl}) = \mathrm{id}$ and, for all $f \in \mathcal{U}$, we have $\gamma(f) \circ \mathrm{incl} = \gamma(f)|_{M_t} = f$.

**Summary.** The 4-step construction above may be summarised in the following commutative diagram:

<figure><img src="image_6.png" /></figure>

where the construction of $\gamma$ ensures that its image lies in $\mathrm{Diff}_{\mathrm{dec},\epsilon}(L_{\vdash}M) \subseteq \mathrm{Diff}(L_{\vdash}M)$. $\square$

<figure><img src="image_7.png" /></figure>

## 1.4. The topologically-enriched Quillen bracket construction

This section deals with the bracket construction due to Quillen, its generalisations (see §1.4.1) and its applications for subcategories of the groupoids of decorated manifolds (see §1.4.2). The various instances of the Quillen bracket construction will be essential for the construction of the homological representation functors in §2. First, we recall the definition of the original version of Quillen's bracket construction, which is a particular case of a more general construction described in [Gra76, p.219]; see also [RW17, §1.1].

**Definition 1.57** (Original Quillen bracket construction.) We fix a (discrete) monoidal groupoid $(\mathcal{G}_\circ, \vdash, \mathbf{0})$ and a (discrete) left module $(\mathcal{M}_\circ, \vdash)$ over $\mathcal{G}_\circ$. The Quillen bracket construction $\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$ is the category with the same objects as $\mathcal{M}_\circ$ and whose morphisms are given by the colimit

$$ \langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle(X, Y) = \operatorname{colim}_{\mathcal{G}_\circ} [\mathcal{M}_\circ(-\vdash X, Y)]. \qquad (1.27) $$

**Example 1.58** As an example, a monoidal groupoid $\mathcal{G}_\circ$ has a canonical left action on itself given by its monoidal structure, so we may always take $\mathcal{M}_\circ = \mathcal{G}_\circ$ and consider the category $\langle \mathcal{G}_\circ, \mathcal{G}_\circ \rangle$. As an abbreviation, we denote the category $\langle \mathcal{G}_\circ, \mathcal{G}_\circ \rangle$ by $\mathcal{U}\mathcal{G}_\circ$.

**Notation 1.59** By (1.27), a morphism in $\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$ from $X$ to $Y$ corresponds to the equivalence class of a morphism $\varphi \in \mathcal{M}_\circ(A\vdash X, Y)$ for an object $A$ of $\mathcal{G}_\circ$, which we denote by $[A, \varphi]$.

There is a canonical faithful functor $c_{\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle}: \mathcal{M}_\circ \hookrightarrow \langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$ defined as the identity on objects and sending each morphism $\varphi$ of $\mathcal{M}_\circ(X, Y)$ to $[0, \varphi]$. Assuming in addition that $\mathcal{M}_\circ$ is a groupoid (as for all the examples discussed in this paper, see §3.1), following mutatis mutandis [RW17, Prop. 1.7], if $(\mathcal{G}_\circ, \vdash, \mathbf{0})$ has no zero divisors – meaning that $A\vdash B \cong \mathbf{0}$ if and only if $A \cong B \cong \mathbf{0}$ for all objects $A$ and $B$ of $\mathcal{G}_\circ$ – and if $\mathrm{Aut}_{\mathcal{G}_\circ}(\mathbf{0}) = \{\mathrm{id}_0\}$, then the above canonical functor $c_{\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle}$ is an isomorphism from $\mathcal{M}_\circ$ onto the maximal subgroupoid of $\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$ (i.e. the subcategory with the same objects as $\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$ and whose morphisms are the isomorphisms of $\langle \mathcal{G}_\circ, \mathcal{M}_\circ \rangle$).
### 1.4.1. A topological enrichment of the Quillen bracket construction

We now explain how to generalise the Quillen bracket construction to semicategories and topologically-enriched semicategories.

**Proposition 1.60** Let $(\mathcal{G}, \mathfrak{h})$ be a semi-monoidal groupoid and let $\mathcal{M}$ be a category with a left action of $\mathcal{G}$ also denoted by $\mathfrak{h}$. There is a semicategory $\langle\mathcal{G}, \mathcal{M}\rangle$ with the same objects as $\mathcal{M}$, given by assigning $\langle\mathcal{G}, \mathcal{M}\rangle(X, Y)$, for objects $X, Y$ of $\mathcal{M}$, to be the quotient set

$$ \left( \bigsqcup_{\mathrm{ob}(\mathcal{G})} \mathcal{M}(-\mathfrak{h}X, Y) \right) / \sim, $$

where $\sim$ is the equivalence relation given by $(A, \varphi) \sim (A', \varphi')$ if and only if $\varphi = \varphi' \circ (\sigma\mathfrak{h}\mathrm{id}_X)$ for some $\sigma \in \mathcal{G}(A, A')$. For two morphisms $[A, \varphi]: X \to Y$ and $[B, \psi]: Y \to Z$ in $\langle\mathcal{G}, \mathcal{M}\rangle$, their composition is defined to be $[B, \psi] \circ [A, \varphi] = [B\mathfrak{h}A, \psi \circ (\mathrm{id}_{B\mathfrak{h}}\varphi)]$.

Moreover, if the categories $\mathcal{G}$ and $\mathcal{M}$ are topologically-enriched and the left-action of $\mathcal{G}$ on $\mathcal{M}$ is continuous, then performing the same constructions on spaces rather than sets produces a topologically-enriched semicategory $\langle\mathcal{G}, \mathcal{M}\rangle$.

**Proof.** It straightforwardly follows from the assignments that $\sim$ is an equivalence relation, and so $\langle\mathcal{G}, \mathcal{M}\rangle(X, Y)$ is well-defined. Hence the associativity of the composition is the only point to check in order to prove that we define a semicategory $\langle\mathcal{G}, \mathcal{M}\rangle$. Let $[A, \varphi]: X \to Y$, $[B, \psi]: Y \to Z$ and $[C, \chi]: Z \to W$ be morphisms in $\langle\mathcal{G}, \mathcal{M}\rangle$. Recall that the associator of $\mathcal{G}$ provides an isomorphism $\alpha_{C,B,A}: C\mathfrak{h}(B\mathfrak{h}A) \cong (C\mathfrak{h}B)\mathfrak{h}A$. Then the functoriality of the left action $\mathcal{G} \times \mathcal{M} \to \mathcal{M}$ implies that $\mathrm{id}_{C\mathfrak{h}}(\psi \circ (\mathrm{id}_{B\mathfrak{h}}\varphi)) = (\mathrm{id}_{C\mathfrak{h}}\psi) \circ (\mathrm{id}_{C\mathfrak{h}}(\mathrm{id}_{B\mathfrak{h}}\varphi))$. We also note from the naturality of the associator that $\mathrm{id}_{C\mathfrak{h}}(\mathrm{id}_{B\mathfrak{h}}\varphi) = (\alpha_{C,B,A}^{-1}\mathfrak{h}\mathrm{id}_X) \circ (\mathrm{id}_{C\mathfrak{h}B}\mathfrak{h}\varphi) \circ (\alpha_{C,B,A}\mathfrak{h}\mathrm{id}_X)$. Therefore, we deduce from the associativity of the composition in $\mathcal{M}$ that

$$ \chi \circ (\mathrm{id}_{C\mathfrak{h}}(\psi \circ (\mathrm{id}_{B\mathfrak{h}}\varphi))) = ((\chi \circ (\mathrm{id}_{C\mathfrak{h}}\psi)) \circ (\alpha_{C,B,A}^{-1}\mathfrak{h}\mathrm{id}_X) \circ (\mathrm{id}_{C\mathfrak{h}B}\mathfrak{h}\varphi)) \circ (\alpha_{C,B,A}\mathfrak{h}\mathrm{id}_X) $$

and so $[C\mathfrak{h}(B\mathfrak{h}A), \chi \circ (\mathrm{id}_{C\mathfrak{h}}(\psi \circ (\mathrm{id}_{B\mathfrak{h}}\varphi)))] = [(C\mathfrak{h}B)\mathfrak{h}A, (\chi \circ (\mathrm{id}_{C\mathfrak{h}}\psi)) \circ (\alpha_{C,B,A}^{-1}\mathfrak{h}\mathrm{id}_X) \circ (\mathrm{id}_{C\mathfrak{h}B}\mathfrak{h}\varphi)]$. Hence, it follows from the assignment for the composition in $\langle\mathcal{G}, \mathcal{M}\rangle$ that $[C, \chi] \circ ([B, \psi] \circ [A, \varphi]) = ([C, \chi] \circ [B, \psi]) \circ [A, \varphi]$, which proves associativity of the composition.

In the topologically-enriched setting, the definition of the hom-set $\langle\mathcal{G}, \mathcal{M}\rangle(X, Y)$ in the statement of the lemma equips it with an evident topology, namely the quotient topology induced from the topological disjoint union of the hom-spaces $\mathcal{M}(A\mathfrak{h}X, Y)$ of the topologically-enriched category $\mathcal{M}$. Continuity of the composition with respect to this topology straightforwardly follows from that of $\mathcal{M}$, since morphisms of $\langle\mathcal{G}, \mathcal{M}\rangle$ are equivalence classes of morphisms of $\mathcal{M}$. $\square$

**Remark 1.61** If the morphism spaces of $\mathcal{M}$ have all of their path-components open, then the same holds for $\langle\mathcal{G}, \mathcal{M}\rangle$. This follows from Lemma 1.3, since the morphism spaces of $\langle\mathcal{G}, \mathcal{M}\rangle$ are constructed as quotients of disjoint unions of morphism spaces of $\mathcal{M}$.

When $\mathcal{G}$ and $\mathcal{M}$ are both discrete categories with a genuine monoidal structure and left $\mathfrak{h}$-module structure respectively, then the version of the Quillen bracket construction defined by Proposition 1.60 recovers that of Definition 1.57. Just as for the non-enriched case, there is a canonical continuous faithful functor $c_{\langle\mathcal{G}, \mathcal{M}\rangle}: \mathcal{M} \hookrightarrow \langle\mathcal{G}, \mathcal{M}\rangle$ defined as the identity on objects and sending $\phi \in \mathcal{M}(X, Y)$ to $[0, \phi]$. Furthermore, Example 1.58 (i.e. taking $\mathcal{M} = \mathcal{G}$ and defining the category $\mathcal{U}\mathcal{G} := \langle\mathcal{G}, \mathcal{G}\rangle$) repeats verbatim for $\mathcal{G}$ a topologically-enriched monoidal groupoid, since the canonical left action of $\mathcal{G}$ on itself given by its monoidal structure is continuous.

**Remark 1.62** From now on, we assume that all the categories we consider are small. The point is that the morphisms in the (topologically-enriched) Quillen bracket construction quantify over all objects of the groupoid that we start with, so if that groupoid is not small, then the Quillen bracket construction will fail even to be locally small. Hence it is important that we do this before applying the Quillen bracket construction, so that the categories arising from this construction are small. In particular, following Convention 1.2, we stress here that we may assume that the decorated manifold category $\mathrm{Dec}_d$ (see Definition 1.16), and hence all of its subcategories that we consider, are small. These are the categories to which we will apply the Quillen bracket construction; see §1.4.2.
where the horizontal maps are homeomorphisms.

Proof. First, the equivalence classes $\mathcal{O}_\alpha$ provide a partition of $\mathrm{ob}(\mathcal{G})$, which defines the top horizontal homeomorphism $\bigsqcup_\alpha \Phi_\alpha \cong \Phi$ of (1.28). The equivalence relation $\sim_t$ on $\Phi$ clearly preserves the topological disjoint union $\bigsqcup_\alpha \Phi_\alpha$, so this homeomorphism descends to $\bigsqcup_\alpha (\Phi_\alpha/\sim_t) \cong \Phi/\sim_t$. Moreover, note that, for a fixed $\alpha$ and for any two objects $A, A' \in \mathcal{O}_\alpha$, we have $\varrho(\mathcal{M}(A\sharp X, Y)) = \varrho(\mathcal{M}(A'\sharp X, Y))$. Hence we have a decomposition of $\Phi/\sim_t$ as a topological disjoint union:

$$ \Phi/\sim_t \cong \bigsqcup_\alpha \varrho(\mathcal{M}(A_\alpha\sharp X, Y)). \qquad (1.29) $$

Thus we have established the analogue of the commutative diagram (1.28) where the bottom-left corner is $\bigsqcup_\alpha \varrho(\mathcal{M}(A_\alpha\sharp X, Y))$.

For each $\alpha$ and $A \in \mathcal{O}_\alpha$, we note that two elements $\varphi, \varphi' \in \mathcal{M}(A\sharp X, Y)$ have the same image under $\varrho$ if and only if they are $\sim_t$-equivalent, which is equivalent to saying that they lie in the same orbit of the $\mathrm{Aut}_G(A)$-action on $\mathcal{M}(A\sharp X, Y)$. Hence the restriction

$$ \varrho_A: \mathcal{M}(A\sharp X, Y) \to \varrho(\mathcal{M}(A\sharp X, Y)) \qquad (1.30) $$

of $\varrho$ is isomorphic to

$$ \mathcal{M}(A\sharp X, Y) \to \mathcal{M}(A\sharp X, Y)/\mathrm{Aut}_G(A), \qquad (1.31) $$

at least on underlying sets. It now just remains to prove that (1.30) and (1.31) are isomorphic also as continuous maps of spaces. Since (1.30) and (1.31) are surjective continuous maps with the same domain and the same point-fibres, and we know moreover that (1.31) is a quotient map, it suffices to prove that (1.30) is also a quotient map.

Let $U \subseteq \varrho(\mathcal{M}(A\sharp X, Y))$ be a subset such that $\varrho_A^{-1}(U)$ is open in $\mathcal{M}(A\sharp X, Y)$. We need to show that $U$ is open in $\varrho(\mathcal{M}(A\sharp X, Y))$. The fact that the equivalence relation $\sim_t$ preserves the decomposition of $\Phi$ into $\Phi_\alpha$ implies that the restriction

$$ \varrho_\alpha = \varrho|_{\Phi_\alpha} : \Phi_\alpha \to \varrho(\Phi_\alpha) = \varrho(\mathcal{M}(A\sharp X, Y)) $$

is a quotient map. So it suffices to show that $\varrho_\alpha^{-1}(U)$ is open in $\Phi_\alpha$. Now, from the definitions, we observe the following description of the subset

$$ \varrho_\alpha^{-1}(U) \subseteq \bigsqcup_{A' \in \mathcal{O}_\alpha} \mathcal{M}(A'\sharp X, Y). $$

For each object $A' \in \mathcal{O}_\alpha$, choose an isomorphism $\sigma_{A'}: A' \to A$ in $\mathcal{G}$. This induces a homeomorphism

$$ \Upsilon_{A'} = -\circ (\sigma_{A'}\sharp \mathrm{id}) : \mathcal{M}(A\sharp X, Y) \to \mathcal{M}(A'\sharp X, Y). $$

Then we have

$$ \varrho_\alpha^{-1}(U) = \bigsqcup_{A' \in \mathcal{O}_\alpha} \Upsilon_{A'}(\varrho_A^{-1}(U)). $$

Since $\varrho_A^{-1}(U)$ is open in $\mathcal{M}(A\sharp X, Y)$, it follows that $\Upsilon_{A'}(\varrho_A^{-1}(U))$ is open in $\mathcal{M}(A'\sharp X, Y)$ for each $A' \in \mathcal{O}_\alpha$. Thus $\varrho_\alpha^{-1}(U)$ is open in $\Phi_\alpha$, as required. $\square$

**Proposition 1.66** Let $(\mathcal{G}, \sharp)$ be a topologically-enriched semi-monoidal groupoid. Let $\mathcal{M}$ be a topologically-enriched category with a continuous left action of $\mathcal{G}$ also denoted by $\sharp$. Assume that, for each object $A$ of $\mathcal{G}$ and each pair of objects $X, Y$ of $\mathcal{M}$, the quotient map

$$ \mathcal{M}(A\sharp X, Y) \to \mathcal{M}(A\sharp X, Y)/\mathrm{Aut}_G(A) \qquad (1.32) $$

is a Serre fibration. Then there is a canonical isomorphism of semicategories

$$ \pi_0(\langle \mathcal{G}, \mathcal{M} \rangle) \cong \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle. \qquad (1.33) $$

Moreover, if the semi-monoidal structure induced from $(\mathcal{G}, \sharp)$ on the groupoid $\pi_0(\mathcal{G})$ admits a unit (and thus upgrades to a genuine monoidal structure) that acts by the identity on $\pi_0(\mathcal{M})$ (so that $\pi_0(\mathcal{M})$ is a genuine left $\sharp$-module over $\pi_0(\mathcal{G})$), then the isomorphism (1.33) upgrades to an isomorphism of categories.
Proof. First note that $\pi_0(\langle\mathcal{G}, \mathcal{M}\rangle)$ and $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$ have the same class of objects, by the definition of the discrete and topologically-enriched Quillen bracket constructions, and the functor $\pi_0$. Specifically, their common class of objects is $\text{ob}(\mathcal{M})$. It therefore remains to show that, for objects $X$ and $Y$ of $\mathcal{M}$, there is a natural bijection between $\pi_0(\langle\mathcal{G}, \mathcal{M}\rangle(X, Y))$ and $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle(X, Y)$. We shall use the above notation $\Phi = \bigsqcup_{A\in\text{ob}(\mathcal{G})} \mathcal{M}(A\otimes X, Y)$ and the equivalence relation $\sim_t$ from Lemma 1.65. Unravelling the definitions, what we need to prove is that there is a natural bijection

$$ \pi_0(\Phi/\sim_t) \cong \pi_0(\Phi)/\sim_h, $$

where $\sim_h$ is the equivalence relation given by $(A, [\varphi]) \sim_h (A', [\varphi'])$ if and only if there is a morphism $\sigma \in \mathcal{G}(A, A')$ such that $\varphi \simeq \varphi' \circ (\sigma\otimes\text{id}_A)$. We note that the only difference between the definitions of $\sim_t$ and $\sim_h$ is that the equality is replaced by a homotopy in the definition of $\sim_h$. As sets, these are both quotients of (the underlying set of) $\Phi$, so we just need to show that, given two elements $(A, \varphi)$ and $(A', \varphi')$ of $\Phi$, they have the same image in $\pi_0(\Phi/\sim_t)$ if and only if they have the same image in $\pi_0(\Phi)/\sim_h$.

(a) Suppose first that $(A, \varphi)$ and $(A', \varphi')$ have the same image in $\pi_0(\Phi)/\sim_h$. This means that there is a morphism $\sigma \in \mathcal{G}(A, A')$ and a path $\gamma: [0, 1] \to \mathcal{M}(A\otimes X, Y) \subseteq \Phi$ with $\gamma(0) = (A, \varphi)$ and $\gamma(1) = (A, \varphi' \circ (\sigma\otimes\text{id}_A))$. Composing with the projection $\rho: \Phi \to \Phi/\sim_t$ and writing $[-]_t$ for the equivalence classes with respect to $\sim_t$, we obtain a path in $\Phi/\sim_t$ from $[(A, \varphi)]_t$ to $[(A, \varphi' \circ (\sigma\otimes\text{id}_A))]_t = [(A', \varphi')]_t$. Hence $(A, \varphi)$ and $(A', \varphi')$ have the same image in $\pi_0(\Phi/\sim_t)$.

(b) To prove the converse, we first show that the quotient map $\rho: \Phi \to \Phi/\sim_t$ is a Serre fibration. Directly from the definition, one may straightforwardly verify the following two facts:

* $\bigsqcup_i f_i: \bigsqcup_i E_i \to B$ is a Serre fibration if and only if each $f_i: E_i \to B$ is a Serre fibration.

* $f: E \to B$ is a Serre fibration if and only if $f(E)$ is a union of path-components of $B$ and $f: E \to f(E)$ is a Serre fibration.

Hence, since we know by hypothesis that (1.32) is a Serre fibration, so is the left-hand vertical map of (1.28), and thus so is $\rho: \Phi \to \Phi/\sim_t$ by Lemma 1.65.

Now assume that $(A, \varphi)$ and $(A', \varphi')$ have the same image in $\pi_0(\Phi/\sim_t)$, so there is a path $\delta: [0, 1] \to \Phi/\sim_t$ with $\delta(0) = [(A, \varphi)]_t$ and $\delta(1) = [(A', \varphi')]_t$. Since $\rho$ is a Serre fibration, we may lift this to a path $\varepsilon: [0, 1] \to \Phi$ with $\varepsilon(0) = (A, \varphi)$ and $\varepsilon(1) \sim_t (A', \varphi')$. Its image $\varepsilon([0, 1])$ is path-connected, so it must lie in $\mathcal{M}(A\otimes X, Y) \subseteq \Phi$. Hence we have a path $\varepsilon: [0, 1] \to \mathcal{M}(A\otimes X, Y)$ with $\varepsilon(0) = (A, \varphi)$ and $\varepsilon(1) = (A, \varphi'') \sim_t (A', \varphi')$, for some $\varphi'' \in \mathcal{M}(A\otimes X, Y)$. The relation $(A, \varphi'') \sim_t (A', \varphi')$ means that there is a morphism $\sigma \in \mathcal{G}(A, A')$ such that $\varphi'' = \varphi' \circ (\sigma\otimes\text{id}_A)$. Hence $\varepsilon$ is a homotopy witnessing that $\varphi \simeq \varphi' \circ (\sigma\otimes\text{id}_A)$, so we have shown that $(A, [\varphi]) \sim_h (A', [\varphi'])$, in other words, $(A, \varphi)$ and $(A', \varphi')$ have the same image in $\pi_0(\Phi)/\sim_h$.

Finally, if the semi-monoidal structure induced from $(\mathcal{G}, \otimes)$ on $\pi_0(\mathcal{G})$ admits a unit, then $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$ is a genuine category via the original Quillen bracket construction of Definition 1.57. The isomorphism (1.33) automatically preserves identities, since the image of each identity morphism of $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$ satisfies the relations of an identity and thus defines the identity morphism of its associated object, making $\pi_0(\langle\mathcal{G}, \mathcal{M}\rangle)$ a genuine category. $\square$

### 1.4.2. Quillen bracket categories of manifolds

We now focus on applications of the Quillen bracket construction to (subgroupoids of) the groupoids of decorated manifolds $D_d$ and $D_d^+$, studying in particular their path components (see Lemma 1.71) and describing their morphism spaces in terms of embedding spaces (see Proposition 1.76). The results that we shall need for our construction of homological representation functors are summarised in Corollary 1.80. Since we make use of the results of §1.3, we resume the assumption that the dimension $d$ is not equal to 4; cf. Remark 1.54. To begin with, we introduce the notion of 0-full subcategory, which is weaker than that of full subcategory:

**Definition 1.67** An inclusion of topologically-enriched categories $C \subseteq D$ is called 0-full if, for each pair of objects $c, c'$ of $C$, the subspace $C(c, c') \subseteq D(c, c')$ is a union of path-components.

**Remark 1.68** We recall that a full inclusion of topologically-enriched categories $C \subseteq D$ is one where, for each pair of objects $c, c'$ of $C$, the inclusion $C(c, c') \subseteq D(c, c')$ is an equality. We also note that a 0-full inclusion $C \subseteq D$ induces an inclusion $\pi_0(C) \subseteq \pi_0(D)$. Indeed, subcategories of $\pi_0(D)$ correspond bijectively to 0-full subcategories of $D$.
For the remainder of this section, we place ourselves in the following standard framework that will be further used at many points in the paper:

**Hypothesis 1.69** We fix an integer $d \ge 2$. Let $\mathcal{G}$ and $\mathcal{M}$ be subgroupoids of either $\mathcal{D}_d$ or $\mathcal{D}_d^+$ (see §1.1) such that $\mathcal{G}$ is full and closed under $\mathfrak{h}$, while $\mathcal{M}$ is 0-full and closed under the (continuous) left action of $\mathcal{G}$ via $\mathfrak{h}$.

Hypothesis 1.69 implies that $\mathcal{G}$ is a semi-monoidal groupoid with a left action on $\mathcal{M}$; we may thus form the Quillen bracket construction $\langle \mathcal{G}, \mathcal{M} \rangle$, which is a topologically-enriched semicategory; see §1.4.1. It is called a *Quillen bracket category of manifolds*.

**Remark 1.70** If $\mathcal{G}$ and $\mathcal{M}$ are groupoids as in Hypothesis 1.69, then the morphism spaces of the Quillen bracket category of manifolds $\langle \mathcal{G}, \mathcal{M} \rangle$ have all of their path-components open, i.e. the projection $\langle \mathcal{G}, \mathcal{M} \rangle \to \pi_0(\langle \mathcal{G}, \mathcal{M} \rangle)$ is continuous. Indeed, we ensured that the morphism spaces of $\mathcal{D}_d$ and $\mathcal{D}_d^+$ have all of their path-components open in Definition 1.14, so it follows that the same is true for $\mathcal{M}$ since it is 0-full, and hence also for $\langle \mathcal{G}, \mathcal{M} \rangle$ by Remark 1.61.

**Path-components.** The following result shows that the topologically-enriched Quillen bracket construction commutes with the path-component functor $\pi_0$ under the above framework.

**Lemma 1.71** *The Serre fibration condition of Proposition 1.66 is satisfied for $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69, and hence we have an isomorphism of semicategories*

$$ \pi_0(\langle \mathcal{G}, \mathcal{M} \rangle) \cong \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle, $$

which upgrades to an isomorphism of categories if the semi-monoidal structure induced from $(\mathcal{G}, \mathfrak{h})$ on $\pi_0(\mathcal{G})$ admits a unit that acts by the identity on $\pi_0(\mathcal{M})$.

*Proof.* Let $X, Y$ be objects of $\mathcal{M}$ and let $A$ be an object of $\mathcal{G}$; we must show that (1.32) is a Serre fibration. If $A \mathfrak{h} X$ is not isomorphic to $Y$ in $\mathcal{M}$ then it is the map $\emptyset \to \emptyset$, which is vacuously a Serre fibration. If $A \mathfrak{h} X$ is isomorphic to $Y$ in $\mathcal{M}$ then we may use this isomorphism to replace $\mathcal{M}(A \mathfrak{h} X, Y)$ with $\mathcal{M}(A \mathfrak{h} X, A \mathfrak{h} X)$; thus we may assume that $Y = A \mathfrak{h} X$. The fact that (1.32) is a Serre fibration in this case follows directly from Theorem 1.55 if $\mathcal{M}$ is a full subgroupoid of $\mathcal{D}_d^{(+)}$. If $\mathcal{M}$ only satisfies the weaker property of being a 0-full subgroupoid of $\mathcal{D}_d^{(+)}$, then it follows from Theorem 1.55 together with Lemma 1.72 below. $\square$

**Lemma 1.72** Let $X$ be a space with a continuous right action of a topological group $G$ such that the projection $X \to X/G$ is a Serre fibration. Let $X_0 \subseteq X$ be a union of path-components such that the $G$-action sends $X_0$ into itself. Then the projection $X_0 \to X_0/G$ is also a Serre fibration.

*Proof.* More generally, by considering lifting diagrams, one may prove that, in the following commutative square:

<figure><img src="image_8.png" /></figure>

if $f$ is a Serre fibration, $a$ is an inclusion of a union of path-components, and $b$ is injective, then $g$ is also a Serre fibration. In our setting, $a$ is the inclusion $X_0 \hookrightarrow X$, which is assumed to be a union of path-components, and $b$ is the induced map $X_0/G \to X/G$, which is injective. $\square$

**Morphism spaces.** For decorated manifolds $M = (M, A)$ and $N = (N, B)$, recall from Notation 1.17 and Definition 1.49 the topological group of decorated diffeomorphisms $\text{Diff}_{\text{dec}}(M)$ and the space of decorated embeddings $\text{Emb}_{\text{dec}}(M, N)$ respectively, as well as their oriented versions $\text{Diff}_{\text{dec}}^{+}(M)$ and $\text{Emb}_{\text{dec}}^{+}(M, N)$.
**Definition 1.73** (Decorated diffeomorphisms and embeddings, revisited.) If $M$ lies in a subgroupoid $\mathcal{H} \subseteq \mathcal{D}_d$, define $\text{Diff}_{\mathcal{H}}(M) \subseteq \text{Diff}_{\text{dec}}(M)$ to be the subgroup of automorphisms of $M$ in $\mathcal{H}$.

If $M$ and $N$ lie in $\mathcal{M} \subseteq \mathcal{D}_d$, define $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N) \subseteq \text{Emb}_{\text{dec}}(M, N)$ to be the subspace where, in the third condition of Definition 1.49, the decorated manifold $M'$ lies in $\mathcal{G}$ and the decorated diffeomorphism $\bar{\varphi}$ lies in $\mathcal{M}$. For an object $L$ of $\mathcal{G}$, we also write $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)_L$ for the subspace of $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)$ where we may take $M' = L$ in Definition 1.49 (and $\bar{\varphi}$ lies in $\mathcal{M}$).

For subgroupoids of $\mathcal{D}_d^+$, we make the analogous definitions, with a superscript $+$ in all of the notation.

**Remark 1.74** It is easy to check that composition of decorated embeddings $\text{Emb}_{\text{dec}}(-, -)$, as described in Definition 1.51, restricts to a well-defined composition on $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(-, -)$, using the facts that $\text{ob}(\mathcal{G})$ is closed under $\mathfrak{h}$ and $\mathcal{M}$ is closed under composition and the action of $\mathcal{G}$ via $\mathfrak{h}$.

**Remark 1.75** We note that the inclusion $\text{Diff}_{\mathcal{H}}(M) \subseteq \text{Diff}_{\text{dec}}(M)$ when $M$ lies in a subgroupoid $\mathcal{H} \subseteq \mathcal{D}_d$ is an equality if $\mathcal{H} \subseteq \mathcal{D}_d$ is full. Also, the condition in the definition of $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)$ that the decorated diffeomorphism $\bar{\varphi}$ lies in $\mathcal{M}$ is automatic if $\mathcal{M} \subseteq \mathcal{D}_d$ is full.

Furthermore, similarly to the decomposition (1.22), the space $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)$ decomposes as a topological disjoint union:

$$ \text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N) \cong \bigsqcup_L \text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)_L, \qquad (1.34) $$

where the disjoint union runs over representatives of isomorphism classes of objects $L$ of $\mathcal{G}$. This follows directly from Lemma 1.53, since $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N) \subseteq \text{Emb}_{\text{dec}}(M, N)$ is equipped with the subspace topology and $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)_L = \text{Emb}_{\text{dec}}(M, N)_L \cap \text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N)$.

**Proposition 1.76** Let $\mathcal{G}$ and $\mathcal{M}$ be subgroupoids of $\mathcal{D}_d$ satisfying the conditions of Hypothesis 1.69. Then the morphism spaces of the Quillen bracket category of manifolds $\langle \mathcal{G}, \mathcal{M} \rangle$ may be identified as follows:

$$ \langle \mathcal{G}, \mathcal{M} \rangle(M, N) \cong \text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, N). \qquad (1.35) $$

Moreover, composition in the semicategory $\langle \mathcal{G}, \mathcal{M} \rangle$ on the left-hand side corresponds to composition of decorated embeddings on the right-hand side, as described in Definition 1.51 (see Remark 1.74). The same also holds when $\mathcal{G}$ and $\mathcal{M}$ are subgroupoids of $\mathcal{D}_d^+$.

**Remark 1.77** When $\mathcal{M} \subseteq \mathcal{D}_d$ is full, this says that morphisms in $\langle \mathcal{G}, \mathcal{M} \rangle$ are embeddings of decorated manifolds such that the complement of the image of the embedding is an object of $\mathcal{G}$. In particular, the morphism space $\langle \mathcal{G}, \mathcal{M} \rangle(M, N)$ is non-empty if and only if there exists an object $M'$ of $\mathcal{G}$ such that $M' \mathfrak{h} M$ is diffeomorphic to $N$ as decorated manifolds.

*Proof of Proposition 1.76.* The space $\langle \mathcal{G}, \mathcal{M} \rangle(M, N)$ is described in Proposition 1.60: using the notation of Lemma 1.65, it is the quotient space $\Phi/\sim_t$. Lemma 1.65 then implies that we have a homeomorphism

$$ \langle \mathcal{G}, \mathcal{M} \rangle(M, N) = \Phi/\sim_t \cong \bigsqcup_L \mathcal{M}(L \mathfrak{h} M, N)/\text{Aut}_{\mathcal{G}}(L), $$

where the disjoint union runs over representatives $L$ of isomorphism classes of objects of $\mathcal{G}$. Since $\mathcal{M}$ is a groupoid, the space $\mathcal{M}(L \mathfrak{h} M, N)/\text{Aut}_{\mathcal{G}}(L)$ is empty unless $L \mathfrak{h} M$ is isomorphic to $N$ in $\mathcal{M}$, in which case we may rewrite it as $\text{Aut}_{\mathcal{M}}(L \mathfrak{h} M)/\text{Aut}_{\mathcal{G}}(L) = \text{Diff}_{\mathcal{M}}(L \mathfrak{h} M)/\text{Diff}_{\text{dec}}(L)$, using the notation of Definition 1.73 (recall that $\mathcal{G} \subseteq \mathcal{D}_d$ is full). By Theorem 1.55, the restriction map induces a homeomorphism

$$ \text{Diff}_{\text{dec}}(L \mathfrak{h} M)/\text{Diff}_{\text{dec}}(L) \cong \text{Emb}_{\text{dec}}(M, L \mathfrak{h} M)_L, \qquad (1.36) $$

and one may easily see that this sends the subspace $\text{Diff}_{\mathcal{M}}(L \mathfrak{h} M)/\text{Diff}_{\text{dec}}(L)$ homeomorphically onto the subspace $\text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, L \mathfrak{h} M)_L$. Putting this all together, we have a homeomorphism

$$ \langle \mathcal{G}, \mathcal{M} \rangle(M, N) \cong \bigsqcup_L \text{Emb}_{\langle \mathcal{G}, \mathcal{M} \rangle}(M, L \mathfrak{h} M)_L, \qquad (1.37) $$

where the disjoint union runs over representatives $L$ of isomorphism classes of objects of $\mathcal{G}$ such that $L \mathfrak{h} M$ is isomorphic to $N$ in $\mathcal{M}$.
Next, consider the topological decomposition (1.34), where the disjoint union is indexed by representatives L of isomorphism classes of all objects of G. If L_M \not\cong N in M, the corresponding term is empty, whereas if L_M \cong N in M, we may rewrite the corresponding term by replacing N with L_M, to obtain:

$$ \mathrm{Emb}_{(G, \mathcal{M})}(M, N) \cong \bigsqcup_L \mathrm{Emb}_{(G, \mathcal{M})}(M, L_M)_{\mathrm{L}}, \qquad (1.38) $$

where the disjoint union now runs over representatives L of isomorphism classes of objects of G such that L_M is isomorphic to N in M. Combining (1.37) and (1.38), we obtain the desired identification (1.35).

To prove the second statement of the proposition, we first note that the isomorphism (1.35) is given concretely as follows. A morphism f on the left-hand side is represented by an object L of G and a decorated diffeomorphism L_M \to N in M \subseteq D_d. Restricting to an \epsilon-neighbourhood of M \subset L_M we obtain a decorated embedding of M into N. This is the image of f under (1.35). (This is ultimately because the identification (1.36) from Theorem 1.55 is given by the restriction map.) From this description, it is evident that the square formed by (two instances of) (1.35), composition in (G, M) and composition of decorated embeddings is commutative.

Finally, all of the above proof adapts identically to the setting where D_d is replaced by D_d^+. □

**Corollary 1.78** Suppose that cancellation holds for our chosen subgroupoids G and M — meaning that L_M \cong L'_M in M implies L \cong L' in G for objects L, L' of G and M of M. Then we have:

$$ \langle G, \mathcal{M} \rangle(M, N) \cong \begin{cases} \mathrm{Emb}_{\langle G, \mathcal{M} \rangle}(M, L_M)_{\mathrm{L}} & \text{if } \exists L \in G \text{ such that } L_M \cong N \text{ in } \mathcal{M}; \\ \emptyset & \text{otherwise.} \end{cases} $$

*Proof.* This follows from the fact that the disjoint union (1.37) is taken either over the empty set or a set of size one in this situation. □

**Remark 1.79** All of our examples of G and M in §3.1 satisfy cancellation. This follows from the classification of compact surfaces in all of our examples in dimension 2 (see §3.1.2 and §3.1.1). In the examples of §3.1.3, it follows from the fact that the groupoid of finite sets and bijections under disjoint union satisfies cancellation, since the objects of the groupoids in the examples of §3.1.3 are determined up to isomorphism by the number of copies of S^1 in the submanifold A \subset M.

As a conclusion of this subsection, we have the following corollary, which gives an explicit description of the semicategory \mathcal{UD}_d defined by applying the topologically-enriched Quillen bracket construction (see Proposition 1.60) to the semi-monoidal groupoid D_d (see Definition 1.16).

**Corollary 1.80** For each d \ge 2, the semicategory \mathcal{UD}_d is isomorphic to the semicategory whose objects are the decorated manifolds M = (M, A, e_1, e_2) of dimension d and whose spaces of morphisms are given by the embedding spaces Emb_{dec}(M, N) introduced in Definition 1.49 with composition as described in Definition 1.51. For subgroupoids G and M of D_d satisfying the conditions of Hypothesis 1.69, the subsemicategory (G, M) corresponds under this isomorphism to the semicategory whose objects are the objects of M and whose morphisms are given by the embedding spaces Emb_{(G, M)}(M, N) of Definition 1.73.

In addition, we have isomorphisms of discrete semicategories $\pi_0(\mathcal{UD}_d) \cong \mathcal{U}(\pi_0(\mathcal{D}_d))$ and more generally $\pi_0(\langle G, M \rangle) \cong \langle \pi_0(G), \pi_0(M) \rangle$ for G and M as above. In particular, $\pi_0(\mathcal{D}_d)$ and $\pi_0(M)$ are the underlying groupoids of $\pi_0(\mathcal{UD}_d)$ and $\pi_0(\langle G, M \rangle)$ respectively.

All of the above holds equally for \mathcal{UD}_d^+ and subgroupoids G, M \subseteq D_d^+ as in Hypothesis 1.69.

*Proof.* Let us denote by \mathcal{UD}'_d the semicategory described in the first statement of the corollary. To begin with, we note that this is a well-defined (topologically-enriched) semicategory: its composition is defined in Definition 1.51 and is verified to be continuous and associative in Lemma 1.52. An isomorphism of semicategories between \mathcal{UD}_d and \mathcal{UD}'_d is then given by the identity on objects and by Proposition 1.76 (with G = M = D_d) on morphisms. Moreover, the second statement of the corollary also follows directly from Proposition 1.76 (in its general setting). The third statement is Lemma 1.71. □
### 1.4.3. Contractor split short exact sequences

As an immediate application of the description in Corollary 1.80 of $\mathcal{MD}_d$ in terms of decorated embeddings of manifolds, we prove that the split short exact sequences (1.7) and (1.9) – and indeed the whole diagram (1.10) – constructed in §1.2 are functorial on $\mathcal{MD}_d$. Recall that we are continuing to assume that $d \neq 4$ (cf. Remark 1.54).

**Proposition 1.81** Each morphism $\varphi: (M, A) \to (N, B)$ in $\mathcal{MD}_d$ induces a map of diagrams of the form (1.11), preserving basepoints of the embedding spaces and respecting the group structure of the diffeomorphism groups. As a result, we obtain a map of diagrams of the form (1.10). In particular, we obtain maps of split short exact sequences of the form (1.7) and (1.9).

*Proof.* By Corollary 1.80, $\varphi$ is a decorated embedding $(M, A) \hookrightarrow (N, B)$, which means by definition (see Definition 1.49) that there is a diffeomorphism of decorated manifolds $\bar{\varphi}: (M', A') \dashrightarrow (M, A) \to (N, B)$, for some decorated manifold $(M', A')$, so that $\varphi$ is equal to its pre-composition with the inclusion of (an arbitrarily small neighbourhood of) $(M, A)$ into $(M', A') \dashrightarrow (M, A)$. Let us fix such a diffeomorphism $\bar{\varphi}$ and extend it by the identity on $\mathbb{B}_1^d$ to $\hat{\varphi}: (M', A') \dashrightarrow (\bar{M}, A) \to (\bar{N}, B)$, where $\bar{M} = M \dashrightarrow \mathbb{B}_1^d$ and $\bar{N} = N \dashrightarrow \mathbb{B}_1^d$ as in Notation 1.26.

On each of the diffeomorphism groups of (1.11), the induced map is defined by sending a decorated diffeomorphism $\psi$ of $\bar{M}$ to the decorated diffeomorphism $E_\varphi(\psi) = \hat{\varphi} \circ (\mathrm{id}_{M'} \dashrightarrow \psi) \circ \hat{\varphi}^{-1}$ of $\bar{N}$. It is straightforward to check that if $\psi$ sends $A$ onto itself then $E_\varphi(\psi)$ sends $B$ onto itself and that if $\psi$ sends $Z$ onto itself then $E_\varphi(\psi)$ sends $Z$ onto itself by the same diffeomorphism. It is also clear that $E_\varphi(-)$ is a group homomorphism.

On the embedding spaces of (1.11) with domain $Z$, the induced map is defined simply by post-composing with $\hat{\varphi}$. On the embedding spaces of (1.11) with domain $A$, the induced map is defined by sending an orbit of embeddings $[e]$ to $[\hat{\varphi} \circ (\mathrm{incl}_{A'} \sqcup e) \circ (\hat{\varphi}^{-1})|_B]$, where $\mathrm{incl}_{A'}$ denotes the inclusion $A' \subset \bar{M}'$. On the embedding spaces of (1.11) with domain $A \sqcup Z$, the induced map is defined separately on $Z$ and on $A$ as in the previous two sentences. These maps preserve basepoints because $\hat{\varphi}$ sends $A' \sqcup A$ onto $B$ and because it fixes $Z$ pointwise, since $Z \subset \mathbb{B}_1^d \subset \bar{M}$.

It remains to check that these induced maps commute with the maps of diagram (1.11). For the (solid) maps between two diffeomorphism groups or between two embedding spaces, this is obvious from the construction. For the three maps that go from a diffeomorphism group to an embedding space, it is also straightforward to check, the only subtlety being the observation that $E_\varphi(\psi)|_Z = \hat{\varphi} \circ \psi|_Z$, since $\hat{\varphi}$ restricts to the identity on $Z \subset \mathbb{B}_1^d$. Finally, for the dotted maps of (1.11), their commutation with the maps induced by $\varphi$ may also be checked directly, and follows essentially because the dotted maps are defined by “squashing” the second (right-hand) boundary-cylinder of $M$ or $N$ using the self-embedding $\Theta$ (see Construction 1.27), whereas $\varphi$ is defined by taking the boundary connected sum with $M'$ or $N'$ along the first (left-hand) boundary-cylinder of $M$ or $N$ and then applying the diffeomorphism $\bar{\varphi}$; the two operations thus have “disjoint support”. $\square$

### 1.4.4. Skeleta of Quillen bracket categories

It is often convenient to be able to pass to a skeleton of a category. For a plain (small) category this is easy: one simply has to pick one object from each isomorphism class (using the axiom of choice, if necessary) and consider the full subcategory on these objects. (We recall that all categories considered in this paper are small; see Convention 1.2 and Remark 1.62.) For categories with additional structure, one has to be a little more careful to preserve the structure. The authors learned of the following folklore fact from [Sch].

**Proposition 1.82 (i)** Let $C$ be a monoidal category and let $C_0$ be any skeleton of $C$. Then there is a monoidal structure on $C_0$ such that the inclusion $C_0 \subset C$ may be enhanced to a monoidal equivalence.

(ii) Let $M$ be a category equipped with a left action of a skeletal monoidal category $C_0$ and let $M_0$ be any skeleton of $M$. Then there is a left action of $C_0$ on $M_0$ such that the inclusion $M_0 \subset M$ may be enhanced to an equivalence of categories equipped with left $C_0$-actions.

*Sketch proof.* Let us first consider part (i). By [Mac98, Thm. 1, §IV.4], the inclusion $C_0 \subset C$ is part of an adjoint equivalence, namely an adjunction whose unit and counit are natural isomorphisms.
Choosing such an adjoint equivalence, one may transfer the monoidal structure of $\mathcal{C}$ along it to obtain a monoidal structure on $C_0$. The coherence isomorphisms needed to enhance the inclusion $C_0 \subset \mathcal{C}$ to a (strong) monoidal equivalence may then be constructed from the units and counits of the adjoint equivalence. The same strategy also deals with part (ii). $\square$

**Remark 1.83** In general, the monoidal structure induced on $C_0$ and the enhancement of the inclusion $C_0 \subset \mathcal{C}$ to a monoidal equivalence depend on the choice of adjoint inverse to the inclusion in the proof above. We also note that Proposition 1.82 is not true if one insists on the inclusion $C_0 \subset \mathcal{C}$ being a *strict* monoidal equivalence.

As a consequence, we may always find a skeleton of a Quillen bracket category that is itself a Quillen bracket category:

**Corollary 1.84** Let $\mathcal{C}$ be a monoidal category and let $\mathcal{M}$ be a category with a left action of $\mathcal{C}$. Then there are skeleta $C_0 \subset \mathcal{C}$ and $M_0 \subset \mathcal{M}$, together with a monoidal structure on $C_0$ and a left $C_0$-action on $M_0$, such that the inclusion $\langle C_0, M_0 \rangle \subset \langle \mathcal{C}, \mathcal{M} \rangle$ is an equivalence. Moreover, the subcategory $\langle C_0, M_0 \rangle$ is skeletal.

*Proof.* We may choose $C_0 \subset \mathcal{C}$ and $M_0 \subset \mathcal{M}$ to be any skeleta. Proposition 1.82 then provides us with a monoidal structure on $C_0$ and a left $C_0$-action on $M_0$ together with enhancements of $C_0 \subset \mathcal{C}$ to a monoidal equivalence and of $M_0 \subset \mathcal{M}$ to an equivalence of categories equipped with left $C_0$-actions. These induce an inverse equivalence for the inclusion $\langle C_0, M_0 \rangle \subset \langle \mathcal{C}, \mathcal{M} \rangle$. Finally, $\langle C_0, M_0 \rangle$ is skeletal since its underlying groupoid is $M_0$, which is skeletal. $\square$

## 2. Construction of homological representation functors

In this section, we construct homological representation functors for mapping class groups and motion groups. We consider the topologically-enriched category $\mathcal{UD}_d$ introduced in §1, which contains all motion groups and all mapping class groups of $d$-dimensional manifolds, as explained in Remark 1.43. The construction consists of two main parts:

(1) constructing continuous semifunctors $\mathcal{UD}_d \to \mathrm{Cov}_\bullet$;

(2) using the output of step (1) to construct functors $\pi_0(\langle \mathcal{G}, \mathcal{M} \rangle) \to \mathrm{Mod}_\bullet$, by restricting to some subcategory $\langle \mathcal{G}, \mathcal{M} \rangle \subset \mathcal{UD}_d$ and applying the procedure summarised in diagram (2.12).

Here, $\mathrm{Cov}_\bullet$ and $\mathrm{Mod}_\bullet$ denote categories of spaces equipped, respectively, with coverings and with right modules over rings; see Definitions 2.2 and 2.3.

In §2.1, we deal with step (2) in a general setting: for any topologically-enriched semicategory $\mathcal{C}$ and continuous semifunctor $F: \mathcal{C} \to \mathrm{Cov}_\bullet$, as well as a continuous semifunctor $V: \mathcal{C} \to \bullet \mathrm{Mod}_\bullet$ that is “compatible” in a precise way with $F$, we produce a (semi)functor $L_i(F; V): \pi_0(\mathcal{C}) \to \mathrm{Mod}_\bullet$ for each $i \ge 0$. We first give precise definitions of the categories involved in §2.1.1. The three steps of the construction are then described in §2.1.2–§2.1.4, and are put together in §2.1.5.

In §2.2 and §2.3 we then focus on step (1). This is the more geometric, or topological, part of our construction of representations, whereas step (2) is the more formal and algebraic part. We note that the semifunctors $\mathcal{F}: \mathcal{UD}_d \to \mathrm{Cov}_\bullet$ obtained as outputs of step (1) restrict, after passing to $\pi_0$, to all motion groups and all mapping class groups in dimension $d$. For this reason, we refer to them as *global* semifunctors.

In contrast, we then restrict each of these semifunctors $\mathcal{F}$ to a subcategory $\langle \mathcal{G}, \mathcal{M} \rangle \subset \mathcal{UD}_d$, where $\mathcal{G}$ and $\mathcal{M}$ are a pair of groupoids as in Hypothesis 1.69, satisfying further mild properties; see §2.2.3. This is in order to construct in §2.2.3 a continuous semifunctor $V = V_{\mathrm{col}}(\mathcal{F}) : \langle \mathcal{G}, \mathcal{M} \rangle \to \bullet \mathrm{Mod}_\bullet$ that is compatible with $\mathcal{F}$ in the above sense. It is constructed so that, under certain conditions, the functor $L_i(\mathcal{F}; V_{\mathrm{col}}(\mathcal{F})) : \pi_0(\langle \mathcal{G}, \mathcal{M} \rangle) \to \mathrm{Mod}_\bullet$ produced in step (2) has image contained in the subcategory $\mathrm{Mod}_R \subset \mathrm{Mod}_\bullet$ for a fixed ring $R$, in other words it is an *untwisted* functorial representation.

### 2.1. Elements of the construction

This section presents the elements of the construction of functorial homological representations, in the sense of step (2) above, and how to assemble them.
A minor subtlety is that the framework in §2.1 is set up in order to construct homological representation functors starting from a topologically-enriched semicategory $\mathcal{C}$, in order to be able to consider $\mathcal{C} := \mathcal{U}\mathcal{D}_d$ defined in §1. This is the reason why the construction produces a functor. However, in our applications of the construction in §2.2 and §2.3, the discrete semicategory $\pi_0(\mathcal{C})$ will always be a genuine category (i.e. with identities) and the induced functor from $\pi_0(\mathcal{C})$ will always be a genuine functor (i.e. preserving identities); see Remark 2.21.

### 2.1.1. Categories

In this subsection, we define the different categories involved in the construction.

**Remark 2.1** The topological spaces involved in the definitions of §2.1 will always be locally path-connected and semi-locally simply-connected.

**Definition 2.2** (The category of bicoverings.) The topologically-enriched category $\bullet\mathrm{Cov}_\bullet$ of spaces equipped with bicoverings is defined as follows. An object of $\bullet\mathrm{Cov}_\bullet$ consists of a pair of groups $Q_1, Q_2$ and a path-connected, based space $X$ admitting a universal covering (i.e. locally path-connected and semi-locally simply-connected), equipped with a surjective homomorphism $\phi: \pi_1(X) \to Q_1 \times Q_2$. For $i \in \{1, 2\}$, we denote by $\phi_i$ the further projection of $\phi$ onto $Q_i$. Via the correspondence between connected, regular coverings of $X$ and normal subgroups of $\pi_1(X)$, this is the same as specifying a pullback square of connected, regular coverings:

<figure><img src="image_9.png" /></figure>

A morphism in $\bullet\mathrm{Cov}_\bullet$ from $(Q_1, Q_2, X, \phi)$ to $(Q'_1, Q'_2, X', \phi')$ is a based, continuous map $f: X \to X'$ such that the induced homomorphism $\pi_1(f)$ sends $\ker(\phi_1)$ into $\ker(\phi'_1)$ and $\ker(\phi_2)$ into $\ker(\phi'_2)$. Equivalently, this condition may be phrased as saying that there exist (uniquely defined) homomorphisms $\alpha_1: Q_1 \to Q'_1$ and $\alpha_2: Q_2 \to Q'_2$ such that

$$ \phi'_1 \circ \pi_1(f) = \alpha_1 \circ \phi_1 \quad \text{and} \quad \phi'_2 \circ \pi_1(f) = \alpha_2 \circ \phi_2. $$

The morphism space of $\bullet\mathrm{Cov}_\bullet$ from $(Q_1, Q_2, X, \phi)$ to $(Q'_1, Q'_2, X', \phi')$ is topologised as a subspace of the space of continuous maps $X \to X'$ in the compact-open topology.

If $G$ is a group, the category $\mathcal{G}\mathrm{Cov}_\bullet$ is the subcategory of $\bullet\mathrm{Cov}_\bullet$ on those objects $(X, \phi)$ such that $Q_1 = G$ and those morphisms $f$ such that the induced homomorphism $\alpha_1: Q_1 \to Q'_1$ is equal to $\mathrm{id}_G$. Similarly, we have a subcategory $\bullet\mathrm{Cov}_G$. If $G$ is the trivial group, we drop it from the notation, and write $\mathrm{Cov}_\bullet$ and $\bullet\mathrm{Cov}_\bullet$ respectively for these subcategories of $\bullet\mathrm{Cov}_\bullet$. We note that $\bullet\mathrm{Cov}_\bullet$ and $\mathrm{Cov}_\bullet$ are abstractly isomorphic, but not equal as subcategories of $\bullet\mathrm{Cov}_\bullet$. As another variant, we write $\mathrm{Cov}_G^{\mathrm{tw}} \subset \mathrm{Cov}_\bullet$ for the full subcategory on those objects $(X, \phi)$ such that $Q_1 = G$. (The superscript $^{\mathrm{tw}}$ indicates that the morphisms are permitted to act on the transformation group $G$, i.e. we are considering twisted covering spaces.) We define the subcategories $\bullet\mathrm{Cov}_\bullet^{\mathrm{pr}}$, $\mathrm{Cov}_\bullet^{\mathrm{pr}}$, $\bullet\mathrm{Cov}^{\mathrm{pr}}$ and $\mathrm{Cov}_G^{\mathrm{pr,tw}}$ of $\bullet\mathrm{Cov}_\bullet$, $\mathrm{Cov}_\bullet$, $\bullet\mathrm{Cov}$ and $\mathrm{Cov}_G^{\mathrm{tw}}$, respectively, to have the same objects but only those morphisms that are proper as maps between spaces.

We recall that all of the rings that we consider are assumed to be associative and unital and that their morphisms preserve units.

**Definition 2.3** (The category of bimodules.) The category $\bullet\mathrm{Mod}_\bullet$ is the category of bimodules over rings. An object of $\bullet\mathrm{Mod}_\bullet$ is a pair of rings $(R, S)$ together with an $(R, S)$-bimodule $V$. A morphism from $(R, S, V)$ to $(R', S', V')$ is a pair of ring homomorphisms $\alpha: R \to R'$ and $\beta: S \to S'$, together with a morphism of $(R, S)$-bimodules $\theta: V \to (\alpha, \beta)^*(V')$.

The category $\bullet\mathrm{Mod}_\bullet$ is topologically-enriched by equipping each morphism space with the discrete topology.

For any ring $R$, we have a subcategory $\mathcal{R}\mathrm{Mod}_\bullet \subset \bullet\mathrm{Mod}_\bullet$ on those objects $(R', S', V')$ where $R' = R$ and those morphisms $(\alpha, \beta, \theta)$ where $\alpha = \mathrm{id}_R$ (note that this is generally not a full subcategory). If moreover $R = \mathbb{Z}$, we drop it from the notation, and write $\mathrm{Mod}_\bullet$ for $\mathcal{R}\mathrm{Mod}_\bullet$. Similarly, we have $\bullet\mathrm{Mod}_S$ and $\mathcal{R}\mathrm{Mod}_S$ for rings $R$ and $S$. We note that $\mathcal{R}\mathrm{Mod}_S$ is just the category
of $(R, S)$-bimodules, as usually defined. Furthermore, for a fixed ring $S$, we write $\mathrm{Mod}_S^{\mathrm{tw}} \subset \mathrm{Mod}_\bullet$ for the full subcategory on those objects $(S', V')$ where $S' = S$. (The superscript `tw` indicates that the morphisms are permitted to act on the underlying ring $S$, i.e. we are considering twisted modules.)

**Definition 2.4** (Bundles of $G$-sets and of $(R, S)$-bimodules.) For any space $X$ and group $G$, a bundle of left $G$-sets over $X$ is a functor $\pi_{\le 1}(X) \to G$Set, where $\pi_{\le 1}(X)$ is the fundamental groupoid of $X$ and $G$Set is the category of left $G$-sets. Similarly, for a pair of rings $(R, S)$, a bundle of $(R, S)$-bimodules over $X$ is a functor $\xi: \pi_{\le 1}(X) \to R\mathrm{Mod}_S$.

A morphism $[\tau] = [f, \alpha, \beta, \tau]: \xi \to \xi'$ of bundles of bimodules from $\xi: \pi_{\le 1}(X) \to R\mathrm{Mod}_S$ to $\xi': \pi_{\le 1}(X') \to R'\mathrm{Mod}_{S'}$ is a continuous map $f: X \to X'$, two ring homomorphisms $\alpha: R \to R'$ and $\beta: S \to S'$, and a natural transformation $\tau: \xi \Rightarrow (\alpha, \beta)^* \circ \xi' \circ \pi_{\le 1}(f)$, where $(\alpha, \beta)^*: R'\mathrm{Mod}_{S'} \to R\mathrm{Mod}_S$ denotes the restriction functor induced by $\alpha$ and $\beta$.

**Lemma 2.5** In notation of Definition 2.4, there is a well-defined, associative composition of bundles of bimodules, as follows. Let $[\tau_1] = [f_1, \alpha_1, \beta_1, \tau_1]: \xi \to \xi'$ and $[\tau_2] = [f_2, \alpha_2, \beta_2, \tau_2]: \xi' \to \xi''$ be bundles of bimodules. Their composition $[\tau_2] \circ [\tau_1]: \xi \to \xi''$ is the natural transformation

$$ [(\alpha_1, \beta_1)^* \circ \tau_2 \circ \pi_{\le 1}(f_1)] \circ \tau_1: \xi \Rightarrow (\alpha_2 \circ \alpha_1, \beta_2 \circ \beta_1)^* \circ \xi'' \circ \pi_{\le 1}(f_2 \circ f_1) \quad (2.1) $$

given by pasting together the natural transformations $\tau_1$ and $\tau_2$ in the diagram:

$$ \begin{tikzcd} \pi_{\le 1}(X) \arrow[r, "\xi"] \arrow[d, "\pi_{\le 1}(f_1)"'] & R\mathrm{Mod}_S \arrow[d, "\pi_{\le 1}(X')"'] \\ \pi_{\le 1}(f_2) \arrow[r, "\xi'"] & R'\mathrm{Mod}_{S'} \arrow[d, "\pi_{\le 1}(X'')"'] \arrow[u, "\xi''"'] \arrow[r, "\xi"'] & R''\mathrm{Mod}_{S''}. \end{tikzcd} \quad (2.2) $$

*Proof.* Since $(\alpha_1, \beta_1)^* \circ \tau_2 \circ \pi_{\le 1}(f_1)$ is a natural transformation (pre-composing or post-composing a natural transformation by a functor produces a natural transformation), the assignment (2.1) is well-defined as it is a composition of natural transformations. Associativity follows from associativity of composition of natural transformations, as well as functoriality of $\pi_{\le 1}(-)$: $\mathrm{Top} \to \mathrm{Cat}$ and of $\mathrm{_Mod}_-$: $\mathrm{Rings} \times \mathrm{Rings} \to \mathrm{Cat}$. $\square$

**Remark 2.6** For a path-connected space $X$, a bundle of left $G$-sets over $X$ is more often defined as a fibre bundle over $X$ with fibre a left $G$-set $T$ and structure group the automorphism group $\mathrm{Aut}_G(T)$ of $T$ as a left $G$-set; see [Ste51, §2] for example. With this definition, by unique path-lifting, a bundle of left $G$-sets over a path-connected space $X$ determines a functor $\pi_{\le 1}(X) \to G$Set. Moreover, whenever $X$ is locally path-connected and semi-locally simply-connected, this gives an identification between isomorphism classes of bundles of left $G$-sets over $X$ and isomorphism classes of functors $\pi_{\le 1}(X) \to G$Set. In all of our examples, the base space $X$ has these properties, and it is more convenient to use Definition 2.4 rather than this definition in terms of fibre bundles.

**Definition 2.7** (The category of bundles of bimodules.) The topologically-enriched category of bundles of bimodules $\bullet\mathrm{Top}_\bullet$ is defined as follows. An object of $\bullet\mathrm{Top}_\bullet$ is a space $X$ together with a pair of rings $(R, S)$ and a bundle $\xi$ of $(R, S)$-bimodules over $X$ in the sense of Definition 2.4. A morphism from $(X, R, S, \xi)$ to $(X', R', S', \xi')$ is a morphism of bundles of bimodules as described in Definition 2.4. The composition of morphisms and its associativity follow from Lemma 2.5. The category $\bullet\mathrm{Top}_\bullet$ is then topologically-enriched by topologising each morphism space from $(X, R, S, \xi)$ to $(X', R', S', \xi')$ as follows: a morphism is given by a four-tuple $[f, \alpha, \beta, \tau]$ (see Definition 2.4); we use the compact-open topology for the continuous map $f$ and the discrete topology for the other three components of this tuple.

For any ring $R$, we have a subcategory $R\mathrm{Top}_\bullet \subset \bullet\mathrm{Top}_\bullet$ on those objects $(X', R', S', \xi')$ where $R' = R$ and those morphisms $(f, \alpha, \beta, \tau)$ where $\alpha = \mathrm{id}_R$. If moreover $R = \mathbb{Z}$, we drop it from the notation, and write $\mathrm{Top}_\bullet$ for $R\mathrm{Top}_\bullet$. Similarly, we have $\bullet\mathrm{Top}_S$ and $R\mathrm{Top}_S$ for rings $R$ and $S$.
In addition, for a fixed ring $S$, we write $\text{Top}_S^{\text{tw}} \subset \text{Top}_\bullet$ for the full subcategory on those objects $(X', S', \xi')$ where $S' = S$. (The superscript `tw` indicates that the morphisms are permitted to act on the ground ring $S$, i.e. we are considering twisted bundles of bimodules.)

We define the subcategories $\bullet \text{Top}_\bullet^{\text{pr}}$, $\text{Top}_\bullet^{\text{pr}}$, $\bullet \text{Top}^{\text{pr}}$ and $\text{Top}_S^{\text{pr,tw}}$ of $\bullet \text{Top}_\bullet$, $\text{Top}_\bullet$, $\bullet \text{Top}$ and $\text{Top}_S^{\text{tw}}$, respectively, to have the same objects but only those morphisms whose underlying map of spaces is a proper map.

Note that $\bullet \text{Mod}_\bullet$ is equivalent to the full subcategory of $\bullet \text{Top}_\bullet$ on those objects whose underlying space is a point.

**Notation 2.8** Writing Rings for the category of (associative, unital) rings and Grp for the category of groups, there are obvious forgetful functors

$$ \bullet \text{Cov}_\bullet \xrightarrow{\mathcal{L}} \text{Grp} $$

$$ \bullet \text{Mod}_\bullet \subset \bullet \text{Top}_\bullet \xrightarrow{\mathcal{L}} \text{Rings} $$

that remember just the left (respectively right) underlying group or ring. For example, an object $(X, R, S, \xi)$ of $\bullet \text{Top}_\bullet$ is sent under $\mathcal{L}$ to $R$ and under $\mathcal{R}$ to $S$.

**Definition 2.9** Write $\bullet \text{Top}_\bullet \text{Mod}_\bullet$ for the pullback of the forgetful functors $\mathcal{L}: \bullet \text{Mod}_\bullet \to \text{Rings}$ and $\mathcal{R}: \bullet \text{Top}_\bullet \to \text{Rings}$:

$$ \begin{tikzcd} \bullet \text{Top}_\bullet \text{Mod}_\bullet \arrow[r] \arrow[d] & \bullet \text{Mod}_\bullet \arrow[d] \\ \bullet \text{Top}_\bullet \arrow[r, "\mathcal{L}"] \arrow[d] & \text{Rings} \end{tikzcd} \qquad (2.3) $$

Explicitly, an object of $\bullet \text{Top}_\bullet \text{Mod}_\bullet$ consists of a space $X$, a triple of rings $(R, S, T)$, a bundle of $(R, S)$-bimodules $\xi$ over $X$ and an $(S, T)$-bimodule $V$. A morphism from $(X, R, S, T, \xi, V)$ to $(X', R', S', T', \xi', V')$ consists of a morphism of bundles of bimodules as described in Definition 2.4 (a continuous map $f: X \to X'$, a pair of ring homomorphisms $(\alpha: R \to R, \beta: S \to S')$ and a natural transformation $\tau: \xi \Rightarrow (\alpha, \beta)^* \circ \xi' \circ \pi_{\le 1}(f)$), together with another ring homomorphism $\gamma: T \to T'$ and a morphism of $(S, T)$-bimodules $\theta: V \to (\beta, \gamma)^*(V')$. The category $\bullet \text{Top}_\bullet \text{Mod}_\bullet$ is then topologically-enriched by topologising each morphism space from $(X, R, S, \xi)$ to $(X', R', S', \xi')$ as follows: a morphism is given by a 6-tuple $[f, \alpha, \beta, \gamma, \tau, \theta]$; we use the compact-open topology for the continuous map $f$ and the discrete topology for the other five components of the tuple. We define the subcategory $\bullet \text{Top}_\bullet^{\text{pr}} \text{Mod}_\bullet$ via the analogue of (2.3), using the category $\bullet \text{Top}_\bullet^{\text{pr}}$ instead of $\bullet \text{Top}_\bullet$.

**Remark 2.10** The categories of bimodules $\bullet \text{Mod}_\bullet$ (see Definition 2.3), of bundles of bimodules $\bullet \text{Top}_\bullet$ (see Definition 2.7) and $\bullet \text{Top}_\bullet \text{Mod}_\bullet$ of Definition 2.9 may easily be generalised using a fixed ring $\mathbb{A}$ rather than $\mathbb{Z}$ as ground ring, $\mathbb{A}$-algebras $(R, S)$ rather than rings and the category $\mathbb{A}$-Alg rather than Rings. The work of §2.1.2–§2.1.5 repeats verbatim for this more general framework.

### 2.1.2. From bicoverings to bundles of bimodules

In this subsection, we introduce the linearisation continuous functor $\bullet \text{Cov}_\bullet \to \bullet \text{Top}_\bullet$.

**Proposition 2.11** There is a natural continuous functor $\bullet \text{Cov}_\bullet \to \bullet \text{Top}_\bullet$, taking bicoverings to bundles of bimodules, such that the squares

$$ \begin{tikzcd} \bullet \text{Cov}_\bullet \arrow[r] \arrow[d] & \bullet \text{Top}_\bullet \arrow[d] \\ \mathcal{L} \arrow[r, "\text{Grp}"] \arrow[d] & \mathcal{Z}[(-)^{\text{op}}] \arrow[d] \\ \text{Grp} \arrow[r, "\text{Z}[-]"'] \arrow[d] & \text{Rings} \end{tikzcd} \qquad (2.4) $$

commute, where $\mathcal{Z}[-]$ : $\text{Grp} \to \text{Rings}$ is the group ring functor and $(-)^{\text{op}}$: $\text{Grp} \to \text{Grp}$ takes a group to its opposite. By abuse of notation, we denote this functor $\bullet \text{Cov}_\bullet \to \bullet \text{Top}_\bullet$ also by $\mathcal{Z}[-]$. Furthermore, the restriction of the linearisation functor $\mathcal{Z}[-]$ to the subcategory $\text{Cov}_\bullet^{\text{pr}} \subset \text{Cov}_\bullet$ takes values in the subcategory $\text{Top}_\bullet^{\text{pr}} \subset \text{Top}_\bullet$.
Proof. On objects, this is defined as follows. Let $(Q_1, Q_2, X, x_0, \phi: \pi_1(X, x_0) \to Q_1 \times Q_2)$ be an object of $\bullet \mathrm{Cov}_\bullet$. The normal subgroup

$$ K := \ker(\phi_1) \cap \ker(\phi_2) = \ker(\phi) \triangleleft \pi_1(X, x_0) \qquad (2.5) $$

corresponds to a regular covering of $X$ with deck transformation group $Q := Q_1 \times Q_2$. In order to specify a particular regular covering of $X$, rather than just an isomorphism class of such, we will be slightly more careful. Start with the universal covering $\tilde{X}$ of $X$; more specifically, the standard model for $\tilde{X}$ whose underlying set consists of endpoint-preserving homotopy classes of paths in $X$ starting at $x_0$. This is equipped with an action of $\pi_1(X, x_0)$; take the quotient of $\tilde{X}$ by the action of the subgroup $K$. We denote this regular covering by $\xi_\phi: X^\phi = \tilde{X}/K \to X$. Whether deck transformations act on the left or the right is an arbitrary convention. We will consider them to act on the right, since this agrees with the typical convention that the structure group of a principal bundle (of which regular coverings are examples) acts on the total space on the right. This is a bundle of right $Q$-sets over $X$ (whose fibres all happen to be isomorphic to $Q$ itself). Since $Q$ is the product $Q_1 \times Q_2$, we may equally view $\xi_\phi$ as a bundle of $(Q_1^{\mathrm{op}}, Q_2)$-bisets over $X$, where a $(G, H)$-biset is a set equipped with a left $G$-action and a compatible right $H$-action.

Now replace each fibre of $\xi_\phi$ with the free $\mathbb{Z}$-module generated by that fibre; this forms a bundle of $(\mathbb{Z}[Q_1^{\mathrm{op}}], \mathbb{Z}[Q_2])$-bimodules over $X$. This operation of taking free $\mathbb{Z}$-modules fibrewise is simple to describe when viewing bundles of $(G, H)$-bisets over $X$ as functors $\pi_{\le 1}(X) \to G \mathrm{Set}_H$ to the category of $(G, H)$-bisets and bundles of $(R, S)$-bimodules over $X$ as functors $\pi_{\le 1}(X) \to R \mathrm{Mod}_S$ to the category of $(R, S)$-bimodules. Namely, the operation is given by post-composition with the functor $\mathbb{Z}[-]$ : $Q_1^{\mathrm{op}} \mathrm{Set}_{Q_2} \to \mathbb{Z}[Q_1^{\mathrm{op}}] \mathrm{Mod}_{\mathbb{Z}[Q_2]}$ defined by assigning the free $(\mathbb{Z}[Q_1^{\mathrm{op}}], \mathbb{Z}[Q_2])$-bimodule $\mathbb{Z}[S]$ to each $(Q_1^{\mathrm{op}}, Q_2)$-biset $S$. We denote the resulting bundle of bimodules by $\mathbb{Z}_{\mathrm{fib}}[\xi_\phi] : \mathbb{Z}_{\mathrm{fib}}[X^\phi] \to X$. This defines the linearisation functor $\mathbb{Z}[-] : \bullet \mathrm{Cov}_\bullet \to \bullet \mathrm{Top}_\bullet$ on objects:

$$ \mathbb{Z}[-](Q_1, Q_2, X, x_0, \phi) = (X, \mathbb{Z}[Q_1^{\mathrm{op}}], \mathbb{Z}[Q_2], \mathbb{Z}_{\mathrm{fib}}[\xi_\phi]). $$

In order to define the linearisation functor $\mathbb{Z}[-] : \bullet \mathrm{Cov}_\bullet \to \bullet \mathrm{Top}_\bullet$ on morphisms, we first note that, although we did not need it to define the functor on objects, the regular covering $\xi_\phi: X^\phi \to X$ associated to $(Q_1, Q_2, X, x_0, \phi)$ comes equipped with a particular choice of basepoint of $X^\phi$, covering the basepoint $x_0$ of $X$. This is because the standard construction of the universal cover $\tilde{X}$, which we used above, has a canonical basepoint (namely the constant path at $x_0$), and therefore so does its quotient $X^\phi$. Let us denote this basepoint by $\tilde{x}_0 \in X^\phi$.

Now suppose we have a morphism $(Q_1, Q_2, X, x_0, \phi) \to (Q'_1, Q'_2, Y, y_0, \phi')$ in $\bullet \mathrm{Cov}_\bullet$, that is, a continuous, based map $f: X \to Y$ such that $f_*(\ker(\phi_1)) \subseteq \ker(\phi'_1)$ and $f_*(\ker(\phi_2)) \subseteq \ker(\phi'_2)$. We recall from Definition 2.2 that this determines certain homomorphisms $\alpha_1: Q_1 \to Q'_1$ and $\alpha_2: Q_2 \to Q'_2$, which determine ring homomorphisms $\mathbb{Z}[\alpha_1^{\mathrm{op}}]: \mathbb{Z}[Q_1^{\mathrm{op}}] \to \mathbb{Z}[(Q'_1)^{\mathrm{op}}]$ and $\mathbb{Z}[\alpha_2]: \mathbb{Z}[Q_2] \to \mathbb{Z}[Q'_2]$. Let us write $\xi_\phi: X^\phi \to X$ and $\xi_{\phi'}: Y^\phi \to Y$ for the regular covering spaces corresponding to $\phi$ and $\phi'$ respectively. By covering space theory, for each point $\tilde{y} \in \xi_{\phi'}^{-1}(y_0)$, there is a unique continuous map $X^\phi \to Y^\phi$ that lifts the composition $f \circ \xi_\phi: X^\phi \to Y$ and that takes $\tilde{x}_0$ to $\tilde{y}$. We therefore obtain a uniquely-determined lift $\tilde{f}: X^\phi \to Y^\phi$ by requiring that $\tilde{f}(\tilde{x}_0) = \tilde{y}_0$. Extending this map $\mathbb{Z}$-linearly in each fibre results in a map

$$ \mathbb{Z}_{\mathrm{fib}}[\tilde{f}] : \mathbb{Z}_{\mathrm{fib}}[X^\phi] \to \mathbb{Z}_{\mathrm{fib}}[Y^\phi'] $$

of bundles of $\mathbb{Z}$-modules. Finally, one may check that $\mathbb{Z}_{\mathrm{fib}}[\tilde{f}]$ is a map of bundles of $(\mathbb{Z}[Q_1^{\mathrm{op}}], \mathbb{Z}[Q_2])$-bimodules, covering $f$, from $\mathbb{Z}_{\mathrm{fib}}[\xi_\phi]$ to $\mathbb{Z}_{\mathrm{fib}}[\xi_{\phi'}]$, where the latter is given the structure of a bundle of $(\mathbb{Z}[Q_1^{\mathrm{op}}], \mathbb{Z}[Q_2])$-bimodules via $\mathbb{Z}[\alpha_1^{\mathrm{op}}]$ and $\mathbb{Z}[\alpha_2]$. This uses the interpretation of morphisms of $\bullet \mathrm{Top}_\bullet$ from Remark 2.6. Hence we may define

$$ \mathbb{Z}[-](f) = (f, \mathbb{Z}[\alpha_1^{\mathrm{op}}], \mathbb{Z}[\alpha_2], \mathbb{Z}_{\mathrm{fib}}[\tilde{f}]). $$

The assignment for $\mathbb{Z}[-]$ on morphisms obviously satisfies the identity axiom, and one may straightforwardly check using the unique path lifting property that the composition axiom holds for the fourth component $\mathbb{Z}_{\mathrm{fib}}[\tilde{f}]$ of $\mathbb{Z}[-](f)$, this axiom being evident for the other components. Finally, the continuity of the linearisation functor $\mathbb{Z}[-] : \bullet \mathrm{Cov}_\bullet \to \bullet \mathrm{Top}_\bullet$ is immediate from the fact that it acts by the identity on the part of the input data given by the continuous map $f: X \to X'$. $\square$
**Notation 2.13** There are forgetful functors $\bullet Top. Mod. \to \bullet Top.$ and $\bullet Top. Mod. \to \bullet Mod.$ coming from the pullback square (2.3). For a continuous functor $F: C \to \bullet Top. Mod.$, we denote its compositions with these two forgetful functors by

$$F_1: C \to \bullet Top. \quad \text{and} \quad F_2: C \to \bullet Mod.$$

respectively. In this notation, we denote the composite functor $\otimes \circ F = (2.6) \circ F$ by $F_1 \otimes F_2$.

### 2.1.4. Twisted homology functor

Over a fixed ring $R$, we view local coefficient systems on a space $X$ as bundles of right $R$-modules over $X$. From this viewpoint, homology with local coefficients (defined over $R$ and in degree $i \ge 0$) is a continuous functor

$$H_i: \mathrm{Top}_R \to \mathrm{Mod}_R. \qquad (2.9)$$

It is defined on objects by sending $(X, \xi)$ to the $R$-module $H_i(X; \mathcal{L}_\xi)$, where $\mathcal{L}_\xi$ is the local system defined by the bundle of $R$-modules $\xi: \pi \le 1(X) \to \mathrm{Mod}_R$. See, for example, [DK01, §5.4] or [Pal18, §5.1]. The following fact may be proven by directly generalising the usual construction of singular twisted homology, keeping careful track of the variable bimodule structure.

**Proposition 2.14** In any degree $i \ge 0$, homology with local coefficients extends to a continuous functor

$$H_i: \bullet Top. \to \bullet Mod. \qquad (2.10)$$

such that the square

$$\begin{tikzcd}
& & H_i \arrow[r] \arrow[d] & \bullet Mod. \arrow[d] \\
\bullet Top. \arrow[r] \arrow[ur] & H_i \arrow[r] \arrow[ur] & Mod_R \arrow[r] \arrow[ur] & \bullet Top. \arrow[r] \arrow[ur] \\
\bullet Top_R & Mod_R & &
\end{tikzcd} \qquad (2.11)$$

commutes for any ring $R$. These statements repeat mutatis mutandis for Borel-Moore homology with local coefficients, providing a continuous functor $H_i^{\mathrm{BM}}: \bullet Top_i^{\mathrm{pr}} \to \bullet Mod_i^{\mathrm{pr}}$ in any degree $i \ge 0$.

*Proof.* The continuous functor $H_i$ is defined on objects by sending $(X, R, S, \xi)$ to $(R, S, H_i(X; \mathcal{L}_\xi))$, where $\mathcal{L}_\xi$ is the local system corresponding to the bundle of $(R, S)$-bimodules $\xi$ over $X$. For a morphism $(f, \alpha, \beta, \tau)$ with $\alpha = \mathrm{id}_R$ and $\beta = \mathrm{id}_S$, there is a well-defined map $H_i(X; \mathcal{L}_\xi) \to H_i(X'; \mathcal{L}_{\xi'})$ of $(R, S)$-bimodules, induced by $f$ for the spaces and by $\tau$ for the local coefficients, constructed in [DK01, Th. 5.11 and Ex. 83]. For general $\alpha$ and $\beta$, the same construction produces a well-defined map $H_i(X; \mathcal{L}_\xi) \to (\alpha, \beta)^*H_i(X'; \mathcal{L}_{\xi'})$ of $(R, S)$-bimodules that is compatible with the canonical map $(\alpha, \beta)^*H_i(X'; \mathcal{L}_{\xi'}) \to H_i(X'; \mathcal{L}_{(\alpha, \beta)*\xi'})$ of the universal coefficient theorem. It is straightforward to check from this construction that it respects composition of morphisms. For Borel-Moore homology with local coefficients, the construction is the straightforward analogue of this, generalising [Bre97, Chap. V, §4]. $\square$

**Remark 2.15** (Twisted homology as homology of covering spaces.) A regular covering $\rho: \hat{X} \to X$ with deck transformation group $G$ induces a bundle of $\mathbb{Z}[G]$-modules $\xi(\rho): \pi \le 1(X) \to \mathrm{Mod}_{\mathbb{Z}[G]}$ by unique path lifting, which in turn determines a local system $\mathcal{L}_{\xi(\rho)}$ on $X$. By Shapiro's lemma for covering spaces, there is a canonical isomorphism $H_*(X; \mathcal{L}_{\xi(\rho)}) \cong H_*(\hat{X}; \mathbb{Z})$. If $\rho$ is a finite covering (not in general if it is infinite), we also have $H_*^{\mathrm{BM}}(X; \mathcal{L}_{\xi(\rho)}) \cong H_*^{\mathrm{BM}}(\hat{X}; \mathbb{Z})$. In the applications (see §3) of our general construction of homological representations, the relevant local systems will typically arise from infinite regular coverings, so when we use ordinary homology (but not when we use Borel-Moore homology) we may interpret them as actions on the untwisted homology of covering spaces.