### 2.1.5. The general construction

We now put together the linearisation functor (§2.1.2), the fibrewise tensor product functor (§2.1.3) and the twisted homology functor (§2.1.4) to give a general construction (taking two continuous semifunctors as input) of homological representation functors, described in Theorem 2.19 and Definition 2.20 below.

Throughout §2.1.5, we assume that the projection semifunctor $C \to \pi_0(C)$ is continuous, in other words that the path-components of the morphism spaces of $C$ are open. In particular, this assumption is always satisfied when considering $C = \langle G, M \rangle$ as in Hypothesis 1.69; see Remark 1.70. We first record the following observation, saying that continuous semifunctors with discrete codomain factor through $\pi_0$ of their domain.

**Lemma 2.16** Let $C$ be a topologically-enriched semicategory and $D$ a discrete semicategory. Any continuous semifunctor $M: C \to D$ factors uniquely into the projection semifunctor $C \to \pi_0(C)$ followed by a semifunctor $\pi_0(M): \pi_0(C) \to D$.

*Proof.* The categories $C$ and $\pi_0(C)$ have the same objects, so we may define $\pi_0(M)(c) := M(c)$ for each $c \in \text{ob}(C)$. For each pair of objects $(x, y)$ of $C$ and morphisms $\phi$ and $\phi'$ in $C(x, y)$ such that $\pi_0(\phi) = \pi_0(\phi')$, we have $M(\phi) = M(\phi')$ since $M$ is continuous and $D$ is discrete, so in particular the path-components of its morphism spaces are points. We may therefore define $\pi_0(M)$ on morphisms by assigning $\pi_0(M)([\phi]) = M(\phi)$ for each morphism $[\phi] \in \pi_0(C)(x, y)$ with $\phi \in C(x, y)$ such that $\pi_0(\phi) = [\phi]$. Composition of morphisms and the associativity axiom for $\pi_0(M)$ follow from those for $M$. This defines a semifunctor $\pi_0(M)$ with the required property, which is, moreover, uniquely determined by $M$. $\square$

We recall (see Notation 2.8) that $\mathcal{L}$ and $\mathcal{R}$ denote the semifunctors

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

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

that remember just the first (respectively second) underlying group or ring. We use the same notation for their restrictions to the subcategories $\bullet \text{Cov.}^{\text{pr}}$ and $\bullet \text{Top.}^{\text{pr}}$.

We are now ready to describe the general construction of homological representation functors. The input for the construction consists of:

* a topologically-enriched (small) semicategory $C$,

* a continuous semifunctor $F: C \to \text{Cov.}$,

* a continuous semifunctor $V: C \to \bullet \text{Mod.}$,

* a positive integer $i \ge 0$,

where $F$ and $V$ satisfy the following condition.

**Condition 2.17** The continuous semifunctors $F$ and $V$ are required to be compatible in the sense that we have

$$ \mathbb{Z}[\mathcal{R} \circ F] = \mathcal{L} \circ V, $$

where $\mathbb{Z}[-]$ denotes the group ring functor $\text{Grp} \to \text{Rings}$.

**Example 2.18** For fixed $F: C \to \text{Cov.}$, a choice of $V: C \to \bullet \text{Mod.}$ for which Condition 2.17 is automatically satisfied, is the trivial coefficient system $V = V_{\text{tr}}(F)$, defined as follows. We denote by tr: $\text{Rings} \to \bullet \text{Mod.}$ the functor that sends $R$ to the bimodule $(R, R, R)$. The trivial coefficient system $V_{\text{tr}}(F)$ is the continuous semifunctor defined as the composition $\text{tr} \circ \mathbb{Z}[-] \circ \mathcal{R} \circ F$. In particular, using the notational convention of Notation 2.13, we have $(\mathbb{Z}[-] \circ F) \otimes V_{\text{tr}}(F) \cong \mathbb{Z}[-] \circ F$; this is the sense in which the choice $V = V_{\text{tr}}(F)$ is trivial for $F$. It follows from this definition that Condition 2.17 is satisfied for $F$ and $V = V_{\text{tr}}(F)$.

Another important (and non-trivial) choice of $V: C \to \bullet \text{Mod.}$ constructed from $F: C \to \text{Cov.}$ (together with a subsemicategory of $C$) is the colimit coefficient system $V_{\text{col}}(F)$ introduced in §2.2.3. In contrast with $V_{\text{tr}}(F)$, it is not defined for any category $C$: it requires one to consider $C = \langle G, M \rangle$, where $G$ and $M$ are a pair of groupoids as in Hypothesis 1.69, satisfying further mild properties; see §2.2.3.
**Theorem 2.19** Continuous semifunctors F and V satisfying Condition 2.17 induce a continuous
semifunctor $C \to \text{Top}_\bullet\text{Mod}_\bullet$. The composition of this with the fibrewise tensor product functor $\otimes$
(§2.1.3) and the twisted homology functor $H_i$ (§2.1.4) then induces a semifunctor $\pi_0(C) \to \text{Mod}_\bullet$.

If F takes values in the subcategory Cov<sub>•</sub><sup>pr</sup>, then the above procedure repeats verbatim using the twisted Borel-Moore homology functor H<sub>i</sub><sup>BM</sup> in place of H<sub>i</sub>.

Proof. Condition 2.17 implies the identity $\mathcal{L} \circ V = \mathcal{R} \circ \mathbb{Z}[-] \circ F$, where $\mathbb{Z}[-]$ denotes the linearisation functor of Proposition 2.11. Hence the continuous semifunctors $\mathbb{Z}[-] \circ F$ and $V$ jointly determine a continuous semifunctor $C \to \text{Top}_\bullet \text{Mod}_\bullet$ by the universal property of the pullback, proving the first statement. This may then be composed with the fibrewise tensor product $\otimes$ and twisted homology $H_i$ to obtain a continuous semifunctor of the form $C \to \text{Mod}_\bullet$. Since $\text{Mod}_\bullet$ is a discrete category, Lemma 2.16 implies that this factors uniquely through a semifunctor $\pi_0(C) \to \text{Mod}_\bullet$, which is the output of the construction. This may be summarised diagrammatically as follows, where the input is red and the output is blue. (We note that the right-hand square of diagram (2.4) appears as a subdiagram here, where the top $\mathbb{Z}[-]$ is the linearisation functor introduced in Proposition 2.11 while the bottom $\mathbb{Z}[-]$ denotes the group ring functor.)

<figure><img src="image_10.png" /></figure>

If $F$ takes values in the subcategory $\mathrm{Cov}_{\bullet}^{\mathrm{pr}}$, the alternative construction is similar, restricting both the linearisation functor $\mathbb{Z}[-]$ and the fibrewise tensor product functor $\otimes$ to the subcategories of their (co)domains with superscript $^{\mathrm{pr}}$ and using $H_i^{\mathrm{BM}}$ in place of $H_i$. $\square$

**Definition 2.20** (The general construction.) Given continuous semifunctors F and V satisfying Condition 2.17, define $L_i(F; V)$ to be the semifunctor given by Theorem 2.19, called the homological representation semifunctor associated to F and V. Using the notational convention described in Notation 2.13, this may be written symbolically as

$$
L_i(F; V) = H_i \circ ((\mathbb{Z}[-] \circ F) \otimes V).
$$

Similarly, the homological representation semifunctor $H_i^{\text{BM}} \circ ((\mathbb{Z}[-] \circ F) \otimes V)$ given by Theorem 2.19 using Borel-Moore homology is denoted by $L_i^{\text{BM}}(F; V)$.

Given a continuous semifunctor $F: C \to \mathrm{Cov}_\bullet$, we may apply Theorem 2.19 to $F$, and a fortiori Definition 2.20, along with the coefficient systems of Example 2.18: hence the key ingredient that we must construct is the semifunctor $F: C \to \mathrm{Cov}_\bullet$, which is the aim of §2.2–§2.3.

**Remark 2.21** Although the output $L_i(F; V)$ of Definition 2.20 is a *semifunctor*, we check in §2.2.2.2 that it is a *genuine functor* under some standard conditions; see Corollary 2.46. In particular, this will be the case in all of the situations that we address in our applications of §3.

**2.1.6.** **Functorial quotients of groups**

The geometric input for the general construction of §2.1.5 is the semifunctor $F: C \to \mathrm{Cov}_\bullet$; this
will be the subject of §2.2 and §2.3. The definition of $F$ will depend on a number of parameters, one
of which is a *functorial quotient of groups*. Here, we introduce this notion and study its behaviour
with respect to split short exact sequences. Recall that $\mathrm{Grp}$ denotes the category of groups.

**Definition 2.22** A functorial quotient of groups is a functor $Q: \mathrm{Grp} \to \mathrm{Grp}$ which is equipped with a natural transformation $q: \mathrm{id}_{\mathrm{Grp}} \Rightarrow Q$ such that $q(G) : G \to Q(G)$ is a quotient homomorphism for each group $G$.
**Example 2.23** (Canonical examples.) The trivial functor $0$ (constant at the trivial group) and the identity functor $\mathrm{id}_{\mathrm{Grp}}$ provide obvious examples of functorial quotients of groups. Some more sophisticated examples of functorial quotients of groups naturally arise in families. A functorial descending normal series is a sequence indexed by integers $\ell \ge 1$ of endofunctors $s_\ell$: $\mathrm{Grp} \to \mathrm{Grp}$ with $s_1 = \mathrm{id}_{\mathrm{Grp}}$, such that $s_{\ell+1}(G) \subseteq s_\ell(G)$ and these inclusions assemble to a natural transformation $s_{\ell+1} \Rightarrow s_\ell$ for each $\ell \ge 1$, and each inclusion $s_\ell(G) \subseteq G$ is normal. For each $\ell$, we then define the associated functorial quotient of groups $Q_{s_\ell}$: $\mathrm{Grp} \to \mathrm{Grp}$ by the natural transformation $G \mapsto G/s_\ell(G)$. For instance, the lower central series $\{\Gamma_\ell(-)\}_{\ell \ge 1}$ and the derived series are functorial descending normal series. In particular, we will use functorial quotients of groups defined from the lower central series in the applications of §3.

We now prove two crucial properties of functorial quotients of groups, concerning their interactions with split short exact sequences. They will have a decisive role in the underlying structure of the homological representation functors of §2.2 and §2.3.

**Lemma 2.24** For any functorial quotient of groups $Q$: $\mathrm{Grp} \to \mathrm{Grp}$ and split short exact sequence

$$1 \longrightarrow A \longrightarrow B \xrightarrow{k} g \longrightarrow C \longrightarrow 1, \qquad (2.13)$$

we have $\mathrm{Im}(q(B) \circ f: A \to B \to Q(B)) = \ker(Q(g): Q(B) \to Q(C))$. Denoting this group by $Q^*(A)$, we have an induced commutative diagram

$$\begin{array}{ccc} 1 & \longrightarrow & A \longrightarrow B \xrightarrow{k} g \longrightarrow C \longrightarrow 1 \\ \Big\downarrow & & \Big\downarrow & & \Big\downarrow \\ 1 & \longrightarrow & Q^*(A) \longrightarrow Q(B) \longrightarrow Q(C) \longrightarrow 1 \end{array} \qquad (2.14)$$

in which both rows are split short exact sequences and the middle and right-hand vertical maps are given by the natural transformation $q$. Furthermore, there is an action of $C$ on the quotient $Q^*(A)$ induced by the conjugation action of the semi-direct product defined by (2.13).

*Proof.* The second statement is clear, given the first one: we define the right-hand square by applying the functor $Q$ to the map $g$ and its given section and by applying the natural transformation $q$ to the groups $B$ and $C$. We then fill in the left-hand square by factoring $q(B) \circ f$ uniquely as a surjection followed by an injection. The final thing to check is exactness in the middle of the bottom row, which is precisely the first statement of the lemma.

To prove the first statement, first note that the inclusion $\mathrm{Im}(q(B) \circ f) \subseteq \ker(Q(g))$ follows immediately from exactness of the top row. To prove the opposite inclusion, let $x \in Q(B)$ with $Q(g)(x) = 1$; we need to find a lift $y \in B$ of $x$ such that $y \in f(A)$. To do this, first pick any lift $y' \in B$ of $x$ and set $z = g(y') \in C$. Denoting the given section of $g$ by $s$, note that $s(z)$ projects to $1 \in Q(B)$, since $z$ projects to $1 \in Q(C)$. Thus $y = y' \cdot s(z)^{-1} \in B$ is another lift of $x$, and moreover $g(y) = g(y') \cdot z^{-1} = z \cdot z^{-1} = 1$, so $y \in f(A)$ by exactness of the top row.

Finally, the action of $C$ on $Q^*(A)$ is formally defined either by lifting elements of $Q^*(A)$ along the vertical map $A \to Q^*(A)$ and using the conjugation action of the semi-direct product on the top row of (2.14), or equivalently by projecting along the vertical map $C \to Q(C)$ to the bottom-right group of (2.14) and then using the conjugation action of the semi-direct product on the bottom row. $\square$

**Notation 2.25** We denote by $Q^*(A)_C$ the coinvariants of the group $Q^*(A)$ under the action of $C$ shown in Lemma 2.24, i.e. the largest quotient of $Q^*(A)$ that collapses the orbits of the $C$-action.

Let us use the notation $[A, B, C]$ for a split short exact sequence of the form (2.13). Recall that a morphism of split short exact sequences $[A, B, C] \to [A', B', C']$ is a triple $\Psi = [\Psi_A, \Psi_B, \Psi_C]$ of group homomorphisms $\Psi_A: A \to A'$, $\Psi_B: B \to B'$ and $\Psi_C: C \to C'$ making the obvious diagram commute. In particular, we denote by $[q^*(A), q(B), q(C)]$ the morphism of split short exact sequences defined by diagram (2.14). In this notation, we have the following naturality property of the construction of Lemma 2.24 illustrated in diagram (2.14):
**Lemma 2.26** For a morphism of split short exact sequences $\Psi: [A, B, C] \to [A', B', C']$, there exists a unique group homomorphism $\phi: Q^*(A) \to Q^*(A')$ such that there is a morphism of split short exact sequences $Q(\Psi) := [\phi, Q(\Psi_B), Q(\Psi_C)]$ making the following diagram commute:

$$ \begin{tikzcd} [A, B, C] \arrow[r, "\Psi"] \arrow[d, "[q^*(A), q(B), q(C)]"'] & [A', B', C'] \arrow[d, "[q^*(A'), q(B'), q(C')]"'] \\ [Q^*(A), Q(B), Q(C)] \arrow[r, "Q(\Psi)"'] & [Q^*(A'), Q(B'), Q(C')] \arrow[d, "[q^*(A'), q(B'), q(C')]"'] \end{tikzcd} \quad (2.15) $$

Furthermore, there exists a unique group homomorphism $\phi_{\text{coinv}}: Q^*(A)_C \to Q^*(A')_{C'}$ such that the following diagram (where the vertical maps are the canonical projections onto the coinvariants) is commutative:

$$ \begin{tikzcd} Q^*(A) \arrow[r, "\phi"] \arrow[d, "\downarrow"'] & Q^*(A') \arrow[d, "\downarrow"'] \\ Q^*(A)_C \arrow[r, "\phi_{\text{coinv}}"'] & Q^*(A')_{C'} \arrow[d, "\downarrow"'] \end{tikzcd} \quad (2.16) $$

*Proof.* Since $q: \text{id} \Rightarrow Q$ is a natural transformation, each of the morphisms $g: B \to C, g': B' \to C', \Psi_B: B \to B'$ and $\Psi_C: C \to C'$ induces a commutative square of the form of the right-hand side of (2.14). We then obtain a cubical commutative diagram with these four commutative squares along with the one defined by $(g, g', \Psi_B, \Psi_C)$ (the top of the cube) and its image under the functor $Q$ (the bottom of the cube), which is of the form

$$ \begin{tikzcd} Q(B) \arrow[r, "\sim", "Q(g)"'] \arrow[d, "\sim", "Q(\Psi_B)"'] & Q(C) \arrow[d, "\sim", "Q(\Psi_C)"'] \\ Q(B') \arrow[r, "\sim", "Q(g')"'] & Q(C'), \arrow[d, "\sim", "Q(\Psi_C)"'] \end{tikzcd} \quad (2.17) $$

where the dotted arrows are the images under $Q$ of the given sections of $g$ and $g'$. Commutativity of (2.17) along with the universal property of $Q^*(A')$ as the kernel of $Q(g')$ implies the existence of a unique group homomorphism $\phi: Q^*(A) \to Q^*(A')$ such that $Q(\Psi) := [\phi, Q(\Psi_B), Q(\Psi_C)]$ is a well-defined morphism of split short exact sequences $[Q^*(A), Q(B), Q(C)] \to [Q^*(A'), Q(B'), Q(C')]$. Moreover, since the cube diagram of which (2.17) is the bottom face is commutative, the universal property of $Q^*(A')$ as the kernel of $Q(g')$ ensures the existence of a unique group homomorphism $\phi': A \to Q^*(A')$ such that

$$ [\phi', q(B') \circ \Psi_B, q(C') \circ \Psi_C] = [\phi', Q(\Psi_B) \circ q(B), Q(\Psi_C) \circ q(C)] \quad (2.18) $$

is a well-defined morphism of short exact sequences. The assignment $\phi' = q^*(A') \circ \Psi_A$ makes the left-hand side of (2.18) a well-defined morphism of short exact sequences, whereas the assignment $\phi' = \phi \circ q^*(A)$ makes the right-hand side a well-defined morphism of short exact sequences. By uniqueness of $\phi'$, we conclude that $q^*(A') \circ \Psi_A = \phi' = \phi \circ q^*(A)$, so diagram (2.15) is commutative.

For a semi-direct product $H \times G$, we denote the conjugation action $G \times H \to H$ by $\text{conj}_G(H)$. By the commutativity of diagram (2.15), we have $\phi \circ \text{conj}_G(Q^*(A)) = \text{conj}_{C'}(Q^*(A')) \circ (\Psi_C, \phi)$. The universal property of the quotient $Q^*(A)_C$ then provides the unique group homomorphism $\phi_{\text{coinv}}: Q^*(A)_C \to Q^*(A')_{C'}$ making the diagram (2.16) commute. $\square$

## 2.2. First constructions of homological representation functors

In this section, we describe the first version of our recipe for constructing homological representation functors on the category $\mathcal{U}\pi_0(\mathcal{D}_d) \cong \pi_0(\mathcal{U}\mathcal{D}_d)$ (see Lemma 1.71) via the general construction of §2.1.5 above. This consists in specifying the input $\mathcal{F}: \mathcal{U}\mathcal{D}_d \to \text{Cov}_\bullet$ for the general construction, which is done in Theorem 2.28 below, and depends on a closed submanifold $Z \subset \mathbb{R}^d$ and open subgroup $G \le \text{Diff}(Z)$, as well as a functorial quotient of groups $Q$ (see §2.1.6).
Throughout §2.2, we fix an integer $d \ge 2$, $d \ne 4$ (cf. Remark 1.54) and consider the Quillen bracket semicategories of manifolds $\mathcal{U}\mathcal{D}_d$ and $\mathcal{U}\mathcal{D}_d^+$. In addition, we consider groupoids $\mathcal{G}$ and $\mathcal{M}$ satisfying the conditions of Hypothesis 1.69. By Lemma 1.64, there is a natural inclusion of semicategories

$$ \langle \mathcal{G}, \mathcal{M} \rangle \to \mathcal{U}\mathcal{D}_d, \qquad (2.19) $$

or $\langle \mathcal{G}, \mathcal{M} \rangle \to \mathcal{U}\mathcal{D}_d^+$ when considering *orientedly* decorated $d$-manifolds. We recall that this is given by the inclusion of the objects of $\mathcal{M}$ into the objects of $\mathcal{D}_d$ (or $\mathcal{D}_d^+$), while on morphisms it is an inclusion of embedding spaces, under the identification of Corollary 1.80. In this section, we construct *global* semifunctors on the categories $\mathcal{U}\mathcal{D}_d$ or $\mathcal{U}\mathcal{D}_d^+$. We may then restrict them to subcategories of the form $\langle \mathcal{G}, \mathcal{M} \rangle$ along (2.19) in order to consider the homological representations for a specific family of groups; see §3.2.1.

We fix a closed submanifold $Z \subset \mathbb{R}^d$ (that is orientable, if we are working with $\mathcal{U}\mathcal{D}_d^+$). We also consider an open subgroup $\mathcal{G}$ of $\mathrm{Diff}(Z)$, which must lie in the subgroup $\mathrm{Diff}^+(Z) \subset \mathrm{Diff}(Z)$ if we are working with $\mathcal{U}\mathcal{D}_d^+$. We then construct, in Theorem 2.28, continuous semifunctors associated to each functorial quotient of groups $\mathcal{Q}$:

$$ \mathcal{F}_{(Z,G,Q)} \text{ and } \mathcal{F}_{(Z,G,Q)}: \mathcal{U}\mathcal{D}_d \to \mathrm{Cov}_{\bullet}, \qquad (2.20) $$

or of the form $\mathcal{U}\mathcal{D}_d^+ \to \mathrm{Cov}_{\bullet}$ when considering *orientedly* decorated $d$-manifolds. We also construct variants $\mathcal{F}_{(Z,G,Q)}^u$ and $\mathcal{F}_{(Z,G,Q)}^u$ by taking *coinvariants* at a certain point during the construction (see §2.2.1.1). These continuous semifunctors are the essential geometric input in our general construction of §2.1.5 (see Definition 2.20). Their construction occupies §2.2.1 and their elementary properties are established in §2.2.2.

Restricting along (2.19) to any subcategory $\langle \mathcal{G}, \mathcal{M} \rangle$, we construct in §2.2.3 a *colimit* coefficient system $V$: $\langle \mathcal{G}, \mathcal{M} \rangle \to \bullet\mathrm{Mod}_{\bullet}$ adapted to each semifunctor (2.20). These are devised so that, under certain conditions, the construction of §2.1.5 leads to *untwisted* functorial homological representations; see Proposition 2.59.

**The homological representation functors.** Having constructed the continuous semifunctors (2.20), the construction of homological representation functors is then immediate. Namely, we apply the machinery of §2.1 (summarised in Definition 2.20) to the input $F = (2.20)$ or one of its variants, possibly restricted along (2.19) to a subcategory $\langle \mathcal{G}, \mathcal{M} \rangle \subset \mathcal{U}\mathcal{D}_d$, together with a *coefficient system* $V$ adapted to $F$, for instance the *trivial* coefficient system $V_{\mathrm{tr}}(F)$ of Example 2.18 or the *colimit* coefficient system $V_{\mathrm{col}}(F)$ introduced in §2.2.3 below. Before stating the core construction of Theorem 2.28, we record this consequence.

**Corollary 2.27** For groupoids $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69, the semifunctors of Theorem 2.28 provide semifunctors of the form

$$ \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \mathrm{Mod}_{\bullet}, \qquad (2.21) $$

where $V$ is any continuous functor satisfying Condition 2.17, such as $V_{\mathrm{tr}}(F)$ or $V_{\mathrm{col}}(F)$. We denote them by $L_i(\mathcal{F}_{(Z,G,Q)}; V)$, $L_i(\mathcal{F}_{(Z,G,Q)}^u; V)$, $L_i(\mathcal{F}_{(Z,G,Q)}; V)$ and $L_i(\mathcal{F}_{(Z,G,Q)}^u; V)$. If the *semifunctor* $\pi_0(V)$ induced from $V$ by Lemma 2.16 is a genuine functor, then the semifunctors (2.21) upgrade to genuine functors.

*Proof.* The construction of the semifunctors (2.21) is a direct application of Definition 2.20, where we note that $\pi_0(\langle \mathcal{G}, \mathcal{M} \rangle)$ is identified canonically with $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$ by Lemma 1.71. The upgrade of the resulting homological representation semifunctors into genuine functors is a consequence of Corollary 2.46 below. $\square$

### 2.2.1. The construction of the functors $\mathcal{F}$ and $\mathcal{F}$

The goal of this section is to prove the following result, defining the input for the construction of homological representation functors:

**Theorem 2.28** For any integer $d \ge 2$, closed submanifold $Z \subset \mathbb{R}^d$ and open subgroup $\mathcal{G}$ of $\mathrm{Diff}(Z)$, each functorial quotient of groups $\mathcal{Q}$ determines continuous semifunctors

$$ \mathcal{F}_{(Z,G,Q)}, \mathcal{F}_{(Z,G,Q)}^u, \mathcal{F}_{(Z,G,Q)} \text{ and } \mathcal{F}_{(Z,G,Q)}^u: \mathcal{U}\mathcal{D}_d \to \mathrm{Cov}_{\bullet}. \qquad (2.22) $$
Similar continuous semifunctors are defined on the semicategory $\mathcal{UD}_d^+$ if $Z$ is orientable and $G$ is contained in $\mathrm{Diff}^+(Z)$.

The proof of Theorem 2.28 occupies the remainder of §2.2.1. Namely, in §2.2.1.1 and §2.2.1.2, we describe the construction of the semifunctors of Theorem 2.28 (on objects and on morphisms respectively) in its $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ (“open”) variants. The modifications involved in defining the “closed” variants $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ of (2.22) are then summarised in §2.2.1.3. Throughout §2.2.1.1–§2.2.1.3, we focus on the functors defined on the category $\mathcal{UD}_d$, with the oriented setting (functors defined on $\mathcal{UD}_d^+$) being exactly analogous, as we record here:

**Proposition 2.29** All the work of §2.2.1.1–§2.2.1.3 below repeats mutatis mutandis considering $\mathcal{D}_d^+$ instead of $\mathcal{D}_d$ and assuming that $Z$ and $G$ are orientable and orientation-preserving respectively.

### 2.2.1.1. The construction of the functor $\mathcal{F}$ on objects

We define the continuous semifunctors $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ on the objects of $\mathcal{UD}_d$, which are the same as the objects of $\mathcal{D}_d$, namely decorated $d$-dimensional manifolds. Let $(M, A) \in \mathrm{ob}(\mathcal{D}_d)$. We shall associate to this:

(i) a based, path-connected space $X_{(Z,G)}(M, A)$ that admits a universal cover,

(ii) two surjective homomorphisms $\phi_{(Z,G,Q)}(M, A): \pi_1(X_{(Z,G)}(M, A)) \to \mathcal{Q}_{(Z,G,Q)}(M, A)$ and $\phi_{(Z,G,Q)}^u(M, A): \pi_1(X_{(Z,G)}(M, A)) \to \mathcal{Q}_{(Z,G,Q)}^u(M, A)$.

Together, these data determine two objects of $\mathrm{Cov.}$ (corresponding to $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ respectively). To simplify the notation, since the choice of $(Z, G, Q)$ is fixed throughout this construction, we will drop the subscripts, denoting the space by $X(M, A)$ and the two surjective homomorphisms by $\phi(M, A): \pi_1(X(M, A)) \to \mathcal{Q}(M, A)$ and $\phi^u(M, A): \pi_1(X(M, A)) \to \mathcal{Q}^u(M, A)$.

**The space.** Recall from Notation 1.26 that we write $\tilde{M}$ for the interior of $\tilde{M} = M\mathfrak{h}\mathbb{B}_1^d$, where $\mathfrak{h}$ denotes the boundary connected sum along boundary-cylinder-germs (i.e. the semi-monoidal structure of $\mathcal{D}_d$), and that we have fixed an identification of the interior of $\mathbb{B}_1^d$ with $\mathbb{R}^d$, so that there is a preferred embedding $\mathbb{R}^d \hookrightarrow \tilde{M}$. Its image is disjoint from $\tilde{M}$, hence in particular disjoint from $A \subset \tilde{M}$. Thus, restricting this embedding to $Z \subset \mathbb{R}^d$, we obtain a preferred embedding $Z \hookrightarrow \tilde{M} \setminus A$, which determines a basepoint of the relative embedding space

$$ \mathcal{E}_G(Z, \tilde{M} \setminus A) = \mathrm{Emb}(Z, \tilde{M} \setminus A)/\mathcal{G}. $$

**Definition 2.30** We define $X(M, A)$ to be the path-component of $\mathcal{E}_G(Z, \tilde{M} \setminus A)$ containing the basepoint.

To complete step (i) of the construction, we have to show that $X(M, A)$ admits a universal cover: this is the result of Corollary 2.33 below. To prove this, we need the following point-set topological result:

**Lemma 2.31** Let $f: X \to Y$ be a surjective fibre bundle and suppose that $X$ is semi-locally simply-connected. Then $Y$ is also semi-locally simply-connected.

*Proof.* Let $y \in Y$ and let $U$ be an open neighbourhood of $y$ in $Y$. We need to find a smaller open neighbourhood $V \subseteq U$ of $y$ such that any loop in $V$ based at $y$ is nullhomotopic in $U$. First, choose a smaller open neighbourhood $U' \subseteq U$ such that $f$ is trivialisable over $U'$, and choose a trivialisation $\varphi: f^{-1}(U') \cong U' \times F$ (where $F = f^{-1}(y)$). Also, since $f$ is surjective, we can choose a point $z \in F$. Since $X$ is semi-locally simply-connected, we may find an open neighbourhood $W \subseteq f^{-1}(U')$ of $\tilde{y} = \varphi^{-1}(y, z)$ such that any loop in $W$ based at $\tilde{y}$ is nullhomotopic in $f^{-1}(U')$. By the definition of the product topology, we may then find open subsets $V \subseteq U'$ and $F' \subseteq F$ such that $y \in V$, $z \in F'$ and $\varphi^{-1}(V \times F') \subseteq W$. Now let $\gamma$ be any loop in $V$ based at $y$. Then $\tilde{\gamma} = \varphi^{-1} \circ (\gamma \times \{z\})$ is a loop in $W$ based at $\tilde{y}$. By above, we may find a nullhomotopy of $\tilde{\gamma}$ in $f^{-1}(U')$. Composing this nullhomotopy with $f$, it becomes a nullhomotopy of $\gamma$ in $U' \subseteq U$. $\square$

**Proposition 2.32** The space $X(M, A)$ is locally path-connected and semi-locally simply-connected.
group $\mathcal{Q}(M, A)$ is discrete, this action factors through an action of $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A))$ on $\mathcal{Q}(M, A)$. Pre-composing with the canonical map (1.20), this gives an action of $\pi_1(\mathcal{E}(A, \dot{M}))$ on $\mathcal{Q}(M, A)$.

We also have an action of $\pi_1(\mathcal{E}(A, \dot{M}))$ (the top-right group of (2.24)) on $\mathcal{Q}(M, A)$ (the bottom-left group of (2.24)) described in Lemma 2.24. Concretely, this action is induced by the splittings in diagram (2.24). Pre-composing with the isomorphism $\pi_1(\mathcal{E}(A, \dot{M})) \cong \pi_1(\mathcal{E}(A, \dot{M}))$ induced by the inclusion $\dot{M} \subset \check{M}$, we thus have another action of $\pi_1(\mathcal{E}(A, \dot{M}))$ on $\mathcal{Q}(M, A)$.

**Proposition 2.35** *The two actions of $\pi_1(\mathcal{E}(A, \dot{M}))$ on $\mathcal{Q}(M, A)$ described above are equal.*

*Proof.* Since both of these actions are induced via the quotient $\phi(M, A)$ from actions defined on $\pi_1(\mathcal{E}_G(Z, \check{M} \setminus A))$, it suffices to prove that the two actions agree on this group, before passing to the quotient. This is precisely part (ii) of Proposition 1.48. $\square$

**Notation 2.36** We denote by $\mathcal{Q}^u(M, A)$ the coinvariants $\mathcal{Q}(M, A)_{\pi_1(\mathcal{E}(A, \dot{M}))}$ associated to this action, and by $\phi^u(M, A)$ the composite $\pi_1(X(M, A)) \to \mathcal{Q}(M, A) \to \mathcal{Q}^u(M, A)$, where the first arrow is $\phi(M, A)$ while the second one is the canonical projection onto the coinvariants.

We then define $\check{\mathcal{F}}_{(Z,G,Q)}^u$ on objects by:

$$ (M, A) \mapsto (X(M, A), \phi^u(M, A)). \qquad (2.26) $$

**Remark 2.37** By construction, any quotient of $\mathcal{Q}(M, A)$ on which the group $\pi_1(\mathcal{E}(A, \dot{M}))$ acts trivially factors through the quotient onto $\mathcal{Q}^u(M, A)$. In particular, the covering spaces encoded by $\check{\mathcal{F}}_{(Z,G,Q)}^u$ have a trivial action of these motion groups. In this sense they are *untwisted* (in a universal way), whence the superscript u in the notation for this variant.

### 2.2.1.2. The construction of the semifactor $\check{\mathcal{F}}$ on morphisms

We now define the semifactors $\check{\mathcal{F}}_{(Z,G,Q)}$ and $\check{\mathcal{F}}_{(Z,G,Q)}^u$ on morphisms of $\mathcal{UD}_d$ and check the composition axiom. Recall from Corollary 1.80 that the morphisms of $\mathcal{UD}_d$ may be described as embeddings of manifolds satisfying the three properties of Definition 1.49, with composition induced by the evident composition of embeddings (see Definition 1.51); this description will be used throughout §2.2.1.2. By Proposition 1.81, each morphism $\varphi: (M, A) \to (N, B)$ in $\mathcal{UD}_d$ induces a map of split homotopy fibration sequences $\mathcal{E}_\varphi: (1.5)_{(M,A)} \to (1.5)_{(N,B)}$, in particular a map

$$ f_{\varphi}: \mathcal{E}_G(Z, \check{M} \setminus A) \to \mathcal{E}_G(Z, \check{N} \setminus B). $$

Notation of the form $(1.5)_{(N,B)}$ means the sequence (1.5) with each instance of $(M, A)$ replaced by $(N, B)$. This map preserves basepoints and therefore restricts to a based map

$$ f_{\varphi}^{X}: X(M, A) \to X(N, B). \qquad (2.27) $$

This defines the actions of $\check{\mathcal{F}}_{(Z,G,Q)}$ and $\check{\mathcal{F}}_{(Z,G,Q)}^u$ on the morphism $\varphi$ at the level of spaces. To verify that (2.27) determines well-defined morphisms in $\mathrm{Cov}_\bullet$ (see Definition 2.2), we must verify that the homomorphism $\pi_1(f_{\varphi}^X)$ descends along the quotients $\phi(M, A)$ and $\phi(N, B)$ and also along the quotients $\phi^u(M, A)$ and $\phi^u(N, B)$.

The map of split homotopy fibration sequences $\mathcal{E}_\varphi: (1.5)_{(M,A)} \to (1.5)_{(N,B)}$ induces a map of split short exact sequences $\Psi_\varphi: (1.7)_{(M,A)} \to (1.7)_{(N,B)}$. It then follows from Lemma 2.26 that $\Psi_\varphi$ induces a map of diagrams (2.24) $(M,A) \to (2.24)_{(N,B)}$, in particular homomorphisms

$$ \theta_\varphi: \mathcal{Q}(M, A) \to \mathcal{Q}(N, B) \quad \text{and} \quad \theta_\varphi^u: \mathcal{Q}^u(M, A) \to \mathcal{Q}^u(N, B). \qquad (2.28) $$

In particular, this means that

$$ \theta_{\varphi} \circ \phi(M, A) = \phi(N, B) \circ \pi_1(f_{\varphi}^{X}) \quad \text{and} \quad \theta_{\varphi}^{u} \circ \phi^{u}(M, A) = \phi^{u}(N, B) \circ \pi_1(f_{\varphi}^{X}), $$

i.e. that $\pi_1(f_{\varphi}^X)$ descends along the quotients $\phi(M, A)$ and $\phi(N, B)$ and along the quotients $\phi^u(M, A)$ and $\phi^u(N, B)$. Equivalently, it sends $\ker(\phi(M, A))$ into $\ker(\phi(N, B))$ and $\ker(\phi^u(M, A))$ into $\ker(\phi^u(N, B))$. So $f_{\varphi}^X$ is a morphism in $\mathrm{Cov}_\bullet$ from $(X(M, A), \phi(M, A))$ to $(X(N, B), \phi(N, B))$ and also a morphism in $\mathrm{Cov}_\bullet$ from $(X(M, A), \phi^u(M, A))$ to $(X(N, B), \phi^u(N, B))$.

In summary, we define both $\check{\mathcal{F}}_{(Z,G,Q)}$ and $\check{\mathcal{F}}_{(Z,G,Q)}^u$ on morphisms by $\varphi \mapsto f_{\varphi}^X$, interpreted as a morphism in $\mathrm{Cov}_\bullet$ in these two different ways.
**Continuity and composition axioms.** We first prove the continuity of the construction at the level of hom-spaces.

**Lemma 2.38** The function $\mathcal{U}\mathcal{D}_d((M, A), (N, B)) \to \text{Cov}_{\bullet}(\hat{\mathcal{F}}_{(Z,G,Q)}(M, A), \hat{\mathcal{F}}_{(Z,G,Q)}(N, B))$ and its analogue for $\hat{\mathcal{F}}^u$ defined above are continuous functions of hom-spaces.

*Proof.* The function $\mathcal{U}\mathcal{D}_d((M, A), (N, B)) \to \text{Map}_*(\mathcal{E}_G(Z, \check{M} \setminus A), \mathcal{E}_G(Z, \tilde{N} \setminus B))$ given by $\varphi \mapsto f_\varphi$ is continuous, since it is the adjoint of $\text{Emb}_{\text{dec}}(M, N) \times \mathcal{E}_G(Z, \check{M} \setminus A) \to \mathcal{E}_G(Z, \tilde{N} \setminus B)$, which is a quotient of a restriction of the composition map $\text{Emb}((\bar{M}, A), (\bar{N}, B)) \times \text{Emb}(Z, \check{M} \setminus A) \to \text{Emb}(Z, \tilde{N} \setminus B)$, which is continuous in the Whitney topology since $Z$ is compact (see [Mat69, §2, Prop. 1]). Restriction to a subspace is always continuous in the compact-open topology, so it follows that the function $\mathcal{U}\mathcal{D}_d((M, A), (N, B)) \to \text{Map}_*(X(M, A), X(N, B))$ given by $\varphi \mapsto f_\varphi^X$ is also continuous.

Now, the hom-space $\text{Cov}_{\bullet}(\hat{\mathcal{F}}_{(Z,G,Q)}(M, A), \hat{\mathcal{F}}_{(Z,G,Q)}(N, B))$ is by definition the subspace of $\text{Map}_*(X(M, A), X(N, B))$ of those maps whose induced homomorphism on $\pi_1$ sends $\text{ker}(\phi(M, A))$ into $\text{ker}(\phi(N, B))$ and we know by construction that the function $\varphi \mapsto f_\varphi^X$ takes values in this subspace. Hence, by restricting the codomain, we conclude that the function $\mathcal{U}\mathcal{D}_d((M, A), (N, B)) \to \text{Cov}_{\bullet}(\hat{\mathcal{F}}_{(Z,G,Q)}(M, A), \hat{\mathcal{F}}_{(Z,G,Q)}(N, B))$ is continuous, as claimed. The proof for $\hat{\mathcal{F}}^u$ is identical; we just restrict in the last step of the proof to the subspace of $\text{Map}_*(X(M, A), X(N, B))$ of those maps whose induced homomorphism on $\pi_1$ sends $\text{ker}(\phi^u(M, A))$ into $\text{ker}(\phi^u(N, B))$. $\square$

The composition axiom thus remains the only point to check in order to prove that $\hat{\mathcal{F}}_{(Z,G,Q)}$ and $\hat{\mathcal{F}}_{(Z,G,Q)}^u$ are well-defined continuous semifunctors. Let us consider morphisms $\varphi: (M, A) \to (N, B)$ and $\varphi': (N, B) \to (P, C)$ in $\mathcal{U}\mathcal{D}_d$, viewed as embeddings via Corollary 1.80.

**Lemma 2.39** There is an equality $f_{\varphi' \circ \varphi}^X = f_{\varphi' \circ \varphi}^X$.

*Proof.* The composite embedding $\varphi' \circ \varphi$ clearly induces the same map of split homotopy fibration sequences $(1.5)_{(M,A)} \to (1.5)_{(P,C)}$ as that obtained by concatenating the maps induced by the two embeddings $\varphi$ and $\varphi'$ separately. In particular, there is an equality $f_{\varphi' \circ \varphi} = f_{\varphi' \circ \varphi}$ of maps $\mathcal{E}_G(Z, \check{M} \setminus A) \to \mathcal{E}_G(Z, \check{P} \setminus C)$. This identity is preserved when restricting to path-components of embedding spaces, so we obtain the claimed equality. $\square$

By the definition above, Lemma 2.39 tells us that $\hat{\mathcal{F}}_{(Z,G,Q)}(\varphi' \circ \varphi) = \hat{\mathcal{F}}_{(Z,G,Q)}(\varphi') \circ \hat{\mathcal{F}}_{(Z,G,Q)}(\varphi)$ and $\hat{\mathcal{F}}_{(Z,G,Q)}^u(\varphi' \circ \varphi) = \hat{\mathcal{F}}_{(Z,G,Q)}^u(\varphi') \circ \hat{\mathcal{F}}_{(Z,G,Q)}^u(\varphi)$; we thus have well-defined semifunctors.

This concludes the proof of Theorem 2.28 for the “open” variants $\hat{\mathcal{F}}_{(Z,G,Q)}$ and $\hat{\mathcal{F}}_{(Z,G,Q)}^u$.

### 2.2.1.3. The closed variant $\mathcal{F}$

The construction of the “closed” variants $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ of Theorem 2.28 is a slight modification of the construction above of the “open” variants $\hat{\mathcal{F}}_{(Z,G,Q)}$ and $\hat{\mathcal{F}}_{(Z,G,Q)}^u$. To do this, we simply replace $\mathcal{E}_G(Z, \check{M} \setminus A)$ with the slightly larger embedding space $\mathcal{E}_G(Z, \bar{M} \setminus A)$. This has the effect of replacing $X(M, A)$ with another space $X'(M, A)$ that is homotopy equivalent to $X(M, A)$ by Proposition 1.31. In particular, we have a canonical isomorphism $\pi_1(X(M, A)) \cong \pi_1(X'(M, A))$.

**Remark 2.40** We note that the spaces $X'(M, A)$ and $X(M, A)$, although homotopy equivalent, are not proper homotopy equivalent, which is why the open and closed variants of (2.20) give rise to different homological representations when using Borel-Moore homology (see Remark 2.42).

The only small technical difficulty arises when checking that $X'(M, A)$ has good local properties, so it admits a universal cover (see Corollary 2.33): the argument is exactly analogous, except that one has to use a variant of [Pal21, Prop. 4.15] where the target manifold is allowed to have non-empty boundary. This may be proved by a small adaptation of the proof of Proposition 4.15 of [Pal21], using ideas of [Cer61] (where all manifolds are allowed to have corners of any codimension).

The quotient $\phi'(M, A): \pi_1(X'(M, A)) \to \mathcal{Q}(M, A)$ is defined to be equal to the quotient $\phi(M, A): \pi_1(X(M, A)) \to \mathcal{Q}(M, A)$ under the isomorphism $\pi_1(X(M, A)) \cong \pi_1(X'(M, A))$ induced by the homotopy equivalence of Proposition 1.31. The semifunctor $\mathcal{F}_{(Z,G,Q)}$ is defined on objects by $(M, A) \mapsto (X'(M, A), \phi'(M, A))$.
To define the *untwisted* version, we quotient further onto the coinvariants of $\mathcal{Q}(M, A)$ by the action of $\pi_1(\mathcal{E}(A, M))$. This is identical to the action of $\pi_1(\mathcal{E}(A, \mathring{M}))$ under the isomorphism $\pi_1(\mathcal{E}(A, \mathring{M})) \cong \pi_1(\mathcal{E}(A, M))$ induced by the inclusion $\mathcal{E}(A, \mathring{M}) \subset \mathcal{E}(A, M)$, so this further quotient is precisely $\mathcal{Q}^u(M, A)$. Let us write $\phi^{u,}(M, A)$ for the quotient map $\phi^u(M, A)$ pre-composed with the inverse of the isomorphism $\pi_1(X(M, A)) \cong \pi_1(X'(M, A))$. The semifunctor $\mathcal{F}_{(Z,G,Q)}^u$ is then defined on objects by $(M, A) \mapsto (X'(M, A), \phi^{u,}(M, A))$.

On morphisms, the construction is exactly the same as the construction in the “open” setting (§2.2.1.2), using the fact that the actions of embeddings of decorated manifolds by post-composition on $\mathcal{E}_G(Z, \mathring{M} \setminus A)$ and on $\mathcal{E}_G(Z, \bar{M} \setminus A)$ commute with the inclusion $\mathcal{E}_G(Z, \mathring{M} \setminus A) \subset \mathcal{E}_G(Z, \bar{M} \setminus A)$. This concludes the proof of Theorem 2.28.

To finish §2.2.1, we note the following relationship between the “open” and “closed” semifunctors that we have constructed. We note that a *homotopy* in $\operatorname{Cov}_\bullet$ between two morphisms $f_1, f_2: (X, \phi: \pi_1(X) \to Q) \to (X', \phi': \pi_1(X') \to Q')$ is a homotopy of based maps $X \to X'$ from $f_1$ to $f_2$; in particular it can only exist if the homomorphisms $Q \to Q'$ induced by $f_1$ and $f_2$ are equal. Then, a *homotopy equivalence* in $\operatorname{Cov}_\bullet$ from $(X, \phi: \pi_1(X) \to Q)$ to $(X', \phi': \pi_1(X') \to Q')$ is a based homotopy equivalence $X \to X'$ such that the induced homomorphism $Q \to Q'$ is an isomorphism. Finally, a *natural homotopy equivalence* is a natural transformation between functors with $\operatorname{Cov}_\bullet$ as target category that is componentwise a homotopy equivalence.

**Lemma 2.41** There are natural homotopy equivalences between the semifunctors of Theorem 2.28 $\check{\mathcal{F}}_{(Z,G,Q)} \Rightarrow \mathcal{F}_{(Z,G,Q)}$ and $\check{\mathcal{F}}_{(Z,G,Q)}^u \Rightarrow \mathcal{F}_{(Z,G,Q)}^u$.

**Remark 2.42** These natural homotopy equivalences imply that, when using *ordinary twisted homology* (or any other homotopy invariant flavour of twisted homology) in our general construction, it does not matter whether we use the *open* or the *closed* variants of the semifunctors. However, this remark does not apply if we use *Borel-Moore homology*; see Remark 2.40.

*Proof of Lemma 2.41.* Our goal is to construct based homotopy equivalences $X(M, A) \to X'(M, A)$, naturally in $(M, A) \in \mathcal{U}\mathcal{D}_d$, so that the two induced endomorphisms $\mathcal{Q}(M, A) \to \mathcal{Q}(M, A)$ and $\mathcal{Q}^u(M, A) \to \mathcal{Q}^u(M, A)$ are isomorphisms (in fact they will both be the identity).

By Proposition 1.31, the inclusion $\mathcal{E}_G(Z, \mathring{M} \setminus A) \subset \mathcal{E}_G(Z, \bar{M} \setminus A)$ is a based homotopy equivalence. It is clearly natural with respect to embeddings of decorated manifolds, i.e. natural in $(M, A) \in \mathcal{U}\mathcal{D}_d$. Restricting to path-components of the basepoints, we obtain the desired natural based homotopy equivalence $X(M, A) \to X'(M, A)$. The fact that it induces the identity endomorphism of $\mathcal{Q}(M, A)$ is immediate, since the quotient $\phi'(M, A)$ was defined to coincide with the quotient $\phi(M, A)$ under the identification $\pi_1(X(M, A)) \cong \pi_1(X'(M, A))$ induced by this homotopy equivalence. For the same reason $(\phi^{u,}(M, A)$ was defined to coincide with $\phi^u(M, A)$ under this identification), the induced endomorphism of $\mathcal{Q}^u(M, A)$ is also the identity. $\square$

### 2.2.2. Elementary properties

We now record some fundamental properties of the continuous semifunctors of Theorem 2.28.

#### 2.2.2.1. Uniqueness up to isotopy

The continuous semifunctors constructed in Theorem 2.28 depend only on the isotopy class of the submanifold $Z \subset \mathbb{R}^d$, in the following precise sense.

**Definition 2.43** Let $Z, Z' \subset \mathbb{R}^d$ be two closed submanifolds and let $G \leq Diff(Z)$ and $G' \leq Diff(Z')$ be open subgroups. We say that $(Z, G)$ and $(Z', G')$ are *isotopic* if there is a diffeomorphism $Z \cong Z'$ such that the induced isomorphism $Diff(Z) \cong Diff(Z')$ sends $G$ onto $G'$ and the induced bijection of path-components $\pi_0(Emb(Z, \mathbb{R}^d)/G) \cong \pi_0(Emb(Z', \mathbb{R}^d)/G')$ sends $\mathrm{incl}_Z$ to $\mathrm{incl}_{Z'}$.

**Proposition 2.44** If $(Z, G)$ is isotopic to $(Z', G')$, there is a natural isomorphism of continuous semifunctors $\check{\mathcal{F}}_{(Z,G,Q)} \cong \check{\mathcal{F}}_{(Z',G',Q)}$, and similarly for each of the other variants of the construction.

*Proof.* The assumption that $(Z, G)$ is isotopic to $(Z', G')$ implies that we have a homeomorphism $\eta: \mathcal{E}_G(Z, \mathbb{R}^d) \cong \mathcal{E}_{G'}(Z', \mathbb{R}^d)$ that sends the basepoint $\mathrm{incl}_Z$ of the domain into the path-component containing the basepoint $\mathrm{incl}_{Z'}$ of the codomain. Let us choose a path in $\mathcal{E}_{G'}(Z', \mathbb{R}^d)$ from $\mathrm{incl}_{Z'}$ to $\eta(\mathrm{incl}_Z)$. Since the restriction map $\mathrm{Diff}_c(\mathbb{R}^d) \to \mathcal{E}_{G'}(Z', \mathbb{R}^d)$ is a fibre bundle (by the same
argument as in the proof of Proposition 1.33), we may lift this to a path in $Diff_c(\mathbb{R}^d)$ from the identity to a diffeomorphism $\varphi \in Diff_c(\mathbb{R}^d)$ whose pre-composition with $\eta([\text{incl}_Z])$ is $[\text{incl}_{Z'}]$. For each $(M, A) \in \text{ob}(\mathcal{U}\mathcal{D}_d)$, pre-composition with (the inverse of) the diffeomorphism $Z \cong Z'$ also defines a homeomorphism $\eta_{(M,A)}: \mathcal{E}_G(Z, \tilde{M} \setminus A) \cong \mathcal{E}_{G'}(Z', \tilde{M} \setminus A)$, which agrees with $\eta$ on $[\text{incl}_Z]$ under the preferred embedding $\mathbb{R}^d \subset \tilde{M} \setminus A$ (see Notation 1.26). Denoting by $\varphi_{(M,A)}$ the extension of $\varphi \in Diff_c(\mathbb{R}^d)$ to a diffeomorphism of $\tilde{M} \setminus A$ by setting it to be the identity outside of $\mathbb{R}^d \subset \tilde{M} \setminus A$, we therefore have a composite homeomorphism

$$ \mathcal{E}_G(Z, \tilde{M} \setminus A) \xrightarrow{\eta_{(M,A)}} \mathcal{E}_{G'}(Z', \tilde{M} \setminus A) \xrightarrow{\varphi_{(M,A)} \circ -} \mathcal{E}_{G'}(Z', \tilde{M} \setminus A) $$

that sends $[\text{incl}_Z]$ to $[\text{incl}_{Z'}]$, i.e. is basepoint-preserving. Restricting to the path-components of the basepoints, we therefore have a based homeomorphism $\zeta_{(M,A)}: X_{(Z,G)}(M, A) \cong X_{(Z',G')}(M, A)$. This induces an isomorphism on $\pi_1$ that extends to an isomorphism of split short exact sequences of the form (1.7). By Lemma 2.26, this induces an isomorphism of commutative diagrams of the form (2.24), which implies that $\pi_1(\zeta_{(M,A)})$ descends to an isomorphism $\mathcal{Q}_{(Z,G,Q)}(M, A) \cong \mathcal{Q}_{(Z',G',Q)}(M, A)$. Thus $\zeta_{(M,A)}$ is an isomorphism $\mathcal{F}_{(Z,G,Q)}(M, A) \cong \mathcal{F}_{(Z',G',Q)}(M, A)$ in $\text{Cov}_\bullet$. This isomorphism is moreover natural with respect to morphisms of $\mathcal{U}\mathcal{D}_d$, since they act by post-composition with embeddings of decorated manifolds, whereas $\zeta_{(M,A)}$ acts by pre-composition with (the inverse of) a diffeomorphism $Z \cong Z'$ and post-composition with a diffeomorphism $\varphi$ supported in $\mathbb{R}^d \subset \mathbb{B}_1^d$ (where all embeddings of decorated manifolds act by the identity). $\square$

### 2.2.2.2. Functors of path-components

Note that Theorem 2.28 defines only semifunctors, since $\mathcal{U}\mathcal{D}_d$ and $\mathcal{U}\mathcal{D}_d^+$ are only semicategories; see §1.4.2. However, this technical issue does not matter once we pass to $\pi_0$:

**Proposition 2.45** The continuous semifunctors of Theorem 2.28 induce genuine functors on $\pi_0$.

*Proof.* We consider any one of the semifunctors (2.22) of Theorem 2.28, which we denote by $\mathcal{F}$; the proof repeats verbatim for the semifunctors defined on the category $\mathcal{U}\mathcal{D}_d^+$. Let $(M, A)$ be an object of $\pi_0(\mathcal{U}\mathcal{D}_d)$ (i.e. an object of $\mathcal{D}_d$, a decorated manifold). We have to show that $\pi_0(\mathcal{F})$ sends $\text{id}_{(M,A)}$ to an identity morphism of $\pi_0(\text{Cov}_\bullet)$. To do this, we first have to identify the identity morphism of $(M, A)$ in $\pi_0(\mathcal{U}\mathcal{D}_d)$.

Let $e_1: \mathbb{D}^{d-1} \times [0, 1] = \mathbb{B}_1^d \hookrightarrow M$ denote one of the boundary cylinders that $(M, A)$ is equipped with (more precisely, a representative of one of the germs of boundary cylinders that $(M, A)$ is equipped with). The boundary connected sum $(\mathbb{B}_1^d, \emptyset)\#(M, A)$ may be viewed as the union of $\mathbb{D}^{d-1} \times [-1, 1]$ with $M$ along $\mathbb{D}^{d-1} \times [0, 1]$ via the embedding $e_1$. Choose a diffeomorphism $[-1, 1] \to [0, 1]$ that is given by $t \mapsto t+1$ on $[-1, -1+\epsilon]$ and by $t \mapsto t$ on $[1-\epsilon, 1]$ for some $\epsilon > 0$. Taking the product with $\mathbb{D}^{d-1}$ (acting by the identity on this factor of the product) and extending by the identity over $M \setminus \text{Im}(e_1)$, this determines an isomorphism of decorated manifolds

$$ v_{(M,A)}: (\mathbb{B}_1^d, \emptyset)\#(M, A) \to (M, A). \qquad (2.29) $$

Consider the endomorphism of $(M, A)$ in $\mathcal{U}\mathcal{D}_d$ given by $\Upsilon_{(M,A)} = [(\mathbb{B}_1^d, \emptyset), v_{(M,A)}]$. One may check that, for any endomorphism $\varphi$ of $(M, A)$ in $\mathcal{U}\mathcal{D}_d$, the compositions $\Upsilon_{(M,A)} \circ \varphi$ and $\varphi \circ \Upsilon_{(M,A)}$ are both isotopic to $\varphi$. Hence $\Upsilon_{(M,A)}$ is the identity of $(M, A)$ in the category $\pi_0(\mathcal{U}\mathcal{D}_d)$. Under the identification of Corollary 1.80, this corresponds to the self-embedding of $(M, A)$ given by restricting the diffeomorphism (2.29) to the submanifold $(M, A) \subset (\mathbb{B}_1^d, \emptyset)\#(M, A)$. Since this self-embedding is isotopic to the identity, the induced self-map of embedding spaces $X(M, A) \to X(M, A)$ is homotopic to the identity, and hence is an identity morphism in $\pi_0(\text{Cov}_\bullet)$. $\square$

Recall that, apart from the continuous semifunctors provided by Theorem 2.28, the other input for the general construction of §2.1.5 is a continuous semifunctor $V: \mathcal{U}\mathcal{D}_d \to \bullet\text{Mod}_\bullet$ satisfying Condition 2.17. By Lemma 2.16, it induces an (abstract) semifunctor $\pi_0(V): \pi_0(\mathcal{U}\mathcal{D}_d) \to \bullet\text{Mod}_\bullet$.

**Corollary 2.46** If the semifunctor $\pi_0(V)$ is a genuine functor, then the associated homological representation semifunctor (2.21) of Corollary 2.27 upgrades to a genuine functor.
*Proof.* We consider any one of the semifunctors of Theorem 2.28, which we denote by $\mathcal{F}$. First, it follows from Lemma 2.16 that the linearisation (genuine) functor $\mathbb{Z}[-]$ of §2.1.2, the fibrewise tensor product (genuine) functor $\otimes$ of §2.1.3 and the twisted homology (genuine) functor $H_i$ of §2.1.4 induce (genuine) functors $\pi_0(\mathbb{Z}[-])$, $\pi_0(\otimes)$ and $\pi_0(H_i)$ on path-components, which commute with the canonical projections $\mathcal{C} \to \pi_0(\mathcal{C})$ for the categories $\mathcal{C}$ involved. Also, by Proposition 2.45, the composite $\pi_0(\mathbb{Z}[-]) \circ \pi_0(\mathcal{F})$ is a genuine functor. Now, consider the commutative diagram (2.12) defining the homological representation semifunctor $L_i(\mathcal{F}; V)$. The functor $\pi_0(\mathbb{Z}[-]) \circ \pi_0(\mathcal{F})$ and the semifunctor $\pi_0(V)$ determine a semifunctor $\pi_0(\mathcal{F}, V)$: $\pi_0(\mathcal{U}\mathcal{D}_d) \to \pi_0(\mathrm{Top}.\mathrm{Mod}.)$ by the universal property of the pullback. But then, since we assume that $\pi_0(V)$ is a genuine functor, so is $\pi_0(\mathcal{F}, V)$. Finally, $L_i(\mathcal{F}; V)$ is defined by the composite of $\pi_0(\mathcal{F}, V)$ with the genuine functors $\pi_0(\otimes)$ and $\pi_0(H_i)$. $\square$

### 2.2.2.3. Borel-Moore homology variant

We finally show that the closed variants of the semifunctors of Theorem 2.28 take values in the subcategory $\mathrm{Cov}^{\mathrm{pr}} \subset \mathrm{Cov}.;$ see Lemma 2.47. By Theorem 2.19, these semifunctors therefore work equally well as an input for our general construction when we use twisted Borel-Moore homology, which is functorial only with respect to proper maps of spaces.

**Lemma 2.47** The continuous semifunctors $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ of Theorem 2.28 and their analogues defined on $\mathcal{U}\mathcal{D}_d^+$ take values in the subcategory $\mathrm{Cov}^{\mathrm{pr}} \subset \mathrm{Cov}.$

*Proof.* Whichever closed semifunctor of Theorem 2.28 we consider, the proof goes as follows. We have to show that, for any morphism $\varphi: (M, A) \to (N, B)$ of decorated manifolds, the induced map of spaces $X'(M, A) \to X'(N, B)$ (in the closed variant of the construction in §2.2.1.3) is a proper map (preimages of compact subspaces are compact). This map is (a restriction to particular path-components of) the inclusion of embedding spaces $\mathcal{E}_G(Z, \bar{M} \setminus A) \hookrightarrow \mathcal{E}_G(Z, \bar{N} \setminus B)$ induced by an embedding of decorated manifolds $(M, A) \hookrightarrow (N, B)$ satisfying the three properties of Definition 1.49 (by Proposition 1.76). In particular, the third property implies that the inclusion $\bar{M} \setminus A \hookrightarrow \bar{N} \setminus B$ has closed image, which implies that the inclusion of embedding spaces above also has closed image (this holds for the compact-open topology, hence also for the Whitney topology, which is finer); any closed inclusion is a proper map. $\square$

**Remark 2.48** The continuous semifunctor $\mathcal{F}_{(Z,G,Q)}$ of Theorem 2.28 does not take values in the subcategory $\mathrm{Cov}^{\mathrm{pr}} \subset \mathrm{Cov}.$, since proof of Lemma 2.47 breaks down in this setting as the inclusion of $\bar{M} \setminus A$ into $\bar{N} \setminus B$ does not have closed image. However, the restriction of the semifunctor $\mathcal{F}_{(Z,G,Q)}$ to the underlying groupoid $\mathcal{D}_d$ of $\mathcal{U}\mathcal{D}_d$ does take values in $\mathrm{Cov}^{\mathrm{pr}}$, since the underlying groupoid of $\mathrm{Cov}.$ is contained in $\mathrm{Cov}^{\mathrm{pr}}$ (because homeomorphisms are proper maps). Therefore, using Borel-Moore homology together with the semifunctor $\mathcal{F}_{(Z,G,Q)}$ in our general construction is not fully functorial: namely, it is functorial only for the isomorphisms $\mathcal{D}_d$ of $\mathcal{U}\mathcal{D}_d$, so we just obtain representations of the individual groups in this case.

Furthermore, the natural homotopy equivalences of Lemma 2.41 are not proper. Thus, when using Borel-Moore twisted homology – which is invariant only under proper homotopy equivalences – in the general construction of §2.1.5, the two choices $\mathcal{F}_{(Z,G,Q)}$ and $\mathcal{F}_{(Z,G,Q)}^u$ will lead to a priori different homological representations. As noted above, the homological representations obtained using Borel-Moore homology are defined on all of $\mathcal{U}\mathcal{D}_d$ when using $\mathcal{F}_{(Z,G,Q)}$, but only on $\mathcal{D}_d$ when using $\mathcal{F}_{(Z,G,Q)}$, so it is only on this subgroupoid that we may compare them.

### 2.2.3. Colimit coefficient systems and untwisted representations

Throughout this section, we fix a pair of groupoids $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69. Let us consider any semifunctor $F: (\mathcal{G}, \mathcal{M}) \to \mathrm{Cov}.$ that induces a genuine functor on $\pi_0$, for example (see Proposition 2.45) one of those constructed in Theorem 2.28 or its restriction along (2.19) to a subcategory $(\mathcal{G}, \mathcal{M}) \subseteq \mathcal{U}\mathcal{D}_d$. In this section we construct an associated colimit coefficient system $V_{\mathrm{col}}(F): (\mathcal{G}, \mathcal{M}) \to \bullet\mathrm{Mod}.$. Under certain conditions on the subcategory $(\mathcal{G}, \mathcal{M})$, if $F$ is the restriction to $(\mathcal{G}, \mathcal{M}) \subseteq \mathcal{U}\mathcal{D}_d$ of one of the two semifunctors $\mathcal{F}_{(Z,G,Q)}^u$ or $\mathcal{F}_{(Z,G,Q)}^u$ of Theorem 2.28, the homological representation functor $L_i(F; V_{\mathrm{col}}(F))$: $(\pi_0(\mathcal{G}), \pi_0(\mathcal{M})) \to \mathrm{Mod}.$ resulting from the general construction of §2.1.5 takes values in a subcategory of $\mathrm{Mod}.$ of the form $\mathrm{Mod}_{\mathbb{Z}[Q]}$ for a fixed group $Q$; see Proposition 2.59 below.
The idea to construct $V_{\text{col}}(F)$ involves “stabilising” along the poset of canonical morphisms in the category $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$, so we first fix notation for this poset.

**Definition 2.49** A morphism $\varphi: M \to N$ of $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$ is called *canonical* if $N = L \ltimes M$ and $\varphi = [L, \text{id}_{L \ltimes M}]$ for some object $L$ of $\pi_0(\mathcal{G})$. Write $\mathcal{P}(\mathcal{G}, \mathcal{M})$ for the subcategory of $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$ with the same objects and only the canonical morphisms. There is at most one canonical morphism between any pair of objects, so this subcategory is a poset. For an object $M$, we also write $\mathcal{P}(\mathcal{G}, \mathcal{M})_{\ge M}$ for the subposet on all objects $N \ge M$, in other words all objects of the form $L \ltimes M$. Finally, we write $(-\ltimes M): \mathcal{P}(\mathcal{G}, \mathcal{M}) \cong \mathcal{P}(\mathcal{G}, \mathcal{M})_{\ge M}$ for the canonical isomorphism of posets.

**Remark 2.50** The fact that the subcategory $\mathcal{P}(\mathcal{G}, \mathcal{M})$ of Definition 2.49 is a poset is equivalent to the statement that if $L \ltimes M = L' \ltimes M$ then $L = L'$, which is tautologically true by construction. To clarify, we note that the corresponding statement with $\cong$ in place of $=$ is false in general, but that is not relevant here. We also recall from Remark 1.62 that $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$ is a small category, so $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is really a set and not a proper class.

The basic construction is the following operation on functors $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Grp}$. For a functor $T: \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Grp}$, since $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$ is a small category and the category of groups $\text{Grp}$ is cocomplete, we may define

$$ T_{\text{col}}(M) := \varinjlim_{\mathcal{P}(\mathcal{G}, \mathcal{M})} (T \circ (-\ltimes M)). \qquad (2.30) $$

Each morphism $\varphi: M \to N$ induces a natural transformation $T \circ (-\ltimes M) \Rightarrow T \circ (-\ltimes N)$ given by $T(\text{id}_L \ltimes \varphi)$ on each object $L$ of $\mathcal{P}(\mathcal{G}, \mathcal{M})$. Natural transformations of diagrams induce morphisms of colimits, so we may define $T_{\text{col}}(\varphi)$ to be the morphism $T_{\text{col}}(M) \to T_{\text{col}}(N)$ induced by this natural transformation. The identity and composition axioms are straightforward to check from the definition, and so we prove:

**Lemma 2.51** For a functor $T: \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Grp}$, the above assignments define a functor $T_{\text{col}}: \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Grp}$.

Moreover, there is a natural transformation $T \Rightarrow T_{\text{col}}$ given on each object $M$ by the morphism $T(M) \to T_{\text{col}}(M)$ induced by the morphisms $T([L, \text{id}_{L \ltimes M}]): T(M) \to T(L \ltimes M)$ for all $L$. An important observation is the following.

**Lemma 2.52** If the poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set, then for any $T: \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Grp}$, the functor $T_{\text{col}}$ is isomorphic to a functor that is constant on $\mathcal{P}(\mathcal{G}, \mathcal{M}) \subset \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$, in other words it sends every object to a fixed group $\mathcal{T}$ and it sends every canonical morphism to $\text{id}_{\mathcal{T}}$.

*Proof.* This follows immediately from the fact that, when $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set, all of the sub- posets $\mathcal{P}(\mathcal{G}, \mathcal{M})_{\ge M}$ are cofinal in each other, so all of the colimits (2.30) are canonically isomorphic to each other; see [Sta23, Lemma 04E7]. Thus they are all canonically isomorphic to a fixed choice of group $\mathcal{T}$ representing this isomorphism class. Moreover, since the colimit in (2.30) is taken over all canonical morphisms, the canonical morphisms are sent to $\text{id}_{\mathcal{T}}$ under these identifications. Hence these canonical isomorphisms assemble into an isomorphism of functors from $T_{\text{col}}$ to one having the properties described. $\square$

**Notation 2.53** When the poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set, the group (2.30) (up to isomorphism) is independent of $M$, by the proof of Lemma 2.52, and we denote it by $\mathcal{Q}_{\text{col}}(T)$.

The technical result that we shall need is the following strengthening of Lemma 2.52 under an additional condition.

**Definition 2.54** A subgroupoid $\mathcal{H} \subseteq \mathcal{D}_d$ (respectively $\mathcal{H}' \subseteq \mathcal{D}_d^+$) is called a *motion groupoid* if, for each object $(M, A)$ of $\mathcal{H}$, the subgroup $\text{Aut}_{\mathcal{H}}(M, A) \subseteq \text{Diff}_{\text{dec}}(M, A)$ (resp. $\text{Aut}_{\mathcal{H}'}(M, A) \subseteq \text{Diff}_{\text{dec}}^+(M, A)$) lies in $\text{Diff}_{\text{dec}}^{\text{br}}(M, A)$ (resp. in $\text{Diff}_{\text{dec}}^{\text{br,+}}(M, A)$). (The motivation for this terminology is Proposition 1.42.)

**Proposition 2.55** Suppose that we have groupoids $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69 and let $F = \mathcal{F}^u$ be one of the untwisted semifunctors of Theorem 2.28, restricted along the inclusion (2.19). Assume that the poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set and that $\mathcal{M}$ is a motion groupoid. Then the functor $(\mathcal{R} \circ \mathcal{F}^u)_{\text{col}}: \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Grp}$ is equivalent to a constant functor.
Proof. By Corollary 1.84, we may find a skeleton of $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$ that is itself a Quillen bracket category. That is, we may find skeleta $\mathcal{G}_0 \subset \pi_0(\mathcal{G})$ and $\mathcal{M}_0 \subset \pi_0(\mathcal{M})$, together with a monoidal structure on $\mathcal{G}_0$ and a left action of $\mathcal{G}_0$ on $\mathcal{M}_0$, such that $\langle\mathcal{G}_0, \mathcal{M}_0\rangle$ is a skeleton of $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$, i.e. the inclusion $i: \langle\mathcal{G}_0, \mathcal{M}_0\rangle \hookrightarrow \langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$ is an equivalence. By construction, every morphism in $\langle\mathcal{G}_0, \mathcal{M}_0\rangle$ is the composition of a canonical morphism and an isomorphism of $\mathcal{M}_0$ (via the canonical embedding of $\mathcal{M}_0$ as the underlying groupoid of $\langle\mathcal{G}_0, \mathcal{M}_0\rangle$); since $\mathcal{M}_0$ is skeletal, this isomorphism must be an automorphism.

We will prove below that each automorphism of $\pi_0(\mathcal{M})$ (thus in particular of $\mathcal{M}_0$) is sent by $T = \mathbb{R}\circ\mathcal{F}^u$ to an identity morphism. This implies that $T_{\text{col}}$ also sends each automorphism of $\pi_0(\mathcal{M})$ to an identity morphism, since $T_{\text{col}}(\varphi)$ is induced by a natural transformation whose component morphisms are of the form $T(\text{id}_L\varphi)$, and $\text{id}_L\varphi$ is an automorphism if $\varphi$ is an automorphism.

By Lemma 2.52, $T_{\text{col}}$ is isomorphic to a functor $T'_{\text{col}}$ that sends every object to a fixed group $\mathcal{T}$ and every canonical morphism to $\text{id}_{\mathcal{T}}$. By the previous paragraph, $T_{\text{col}}$ sends every automorphism of $\pi_0(\mathcal{M})$ to an identity morphism; it follows that $T'_{\text{col}}$ sends every automorphism of $\pi_0(\mathcal{M})$ to $\text{id}_{\mathcal{T}}$. Thus $T'_{\text{col}}$ is constant on objects, canonical morphisms and automorphisms of $\pi_0(\mathcal{M})$ (in particular automorphisms of $\mathcal{M}_0$). As noted above, every morphism in $\langle\mathcal{G}_0, \mathcal{M}_0\rangle$ is the composition of a canonical morphism and an automorphism of $\mathcal{M}_0$, so $T'_{\text{col}}$ is constant on $\langle\mathcal{G}_0, \mathcal{M}_0\rangle$.

Let $r: \langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle \to \langle\mathcal{G}_0, \mathcal{M}_0\rangle$ be an inverse equivalence for the inclusion $i$. Then $T_{\text{col}}$ is equivalent to $T_{\text{col}}\circ i\circ r$. By above, $T_{\text{col}}$ is isomorphic to $T'_{\text{col}}$, which has the property that $T'_{\text{col}}\circ i$ is constant. It thus follows that $T_{\text{col}}$ is equivalent to $T'_{\text{col}}\circ i\circ r$, which is constant.

It remains to prove the claim above that $T$ sends every automorphism $\varphi: (M, A) \to (M, A)$ of $\pi_0(\mathcal{M})$ to an identity morphism. We first recall that, by the definition of $\mathcal{F}^u$ in §2.2.1, the functor $T = \mathbb{R}\circ\mathcal{F}^u$ sends the object $(M, A)$ to $\mathcal{Q}^u(M, A)$, which is the coinvariants of $\mathcal{Q}(M, A)$ under the action of $\pi_1(\mathcal{E}(A, \mathring{M}))$ induced by the splittings in diagram (2.24). It also sends the automorphism $\varphi: (M, A) \to (M, A)$ to the automorphism $\theta_{\varphi}^u: \mathcal{Q}^u(M, A) \to \mathcal{Q}^u(M, A)$ of (2.28), which is the map on coinvariants induced by the map $\theta_{\varphi}: \mathcal{Q}(M, A) \to \mathcal{Q}(M, A)$, which is part of the map of diagrams of the form (2.24) induced by $\varphi$, as explained in §2.2.1.2. Our goal is thus to prove that $\theta_{\varphi}^u$ is the identity.

By the assumption that $\mathcal{M}$ is a motion groupoid, $\varphi$ lies in the subgroup $\pi_0(\text{Diff}_{\text{dec}}^{\text{br}}(M, A)) \subseteq \pi_0(\text{Diff}_{\text{dec}}(M, A))$. We may therefore choose a lift $\tilde{\varphi} \in \pi_1(\mathcal{E}(A, \mathring{M}))$ of $\varphi$ under the surjection (1.12) of Proposition 1.42. Via the isomorphism $\pi_1(\mathcal{E}(A, \mathring{M})) \cong \pi_1(\mathcal{E}(A, \mathring{M}))$ induced by the inclusion $\mathring{M} \subset \tilde{M}$ we may view $\tilde{\varphi}$ as an element of $\pi_1(\mathcal{E}(A, \mathring{M}))$. Via the splittings in diagram (2.24), it induces an automorphism $\tilde{\varphi}_{\sharp}$ of $\mathcal{Q}(M, A)$. Proposition 2.35 tells us that $\theta_{\varphi} = \tilde{\varphi}_{\sharp}$. By definition of $\mathcal{Q}^u(M, A)$ as coinvariants, the automorphism of $\mathcal{Q}^u(M, A)$ induced by $\tilde{\varphi}_{\sharp}$ is the identity. But $\theta_{\varphi}^u$ is precisely the automorphism of $\mathcal{Q}^u(M, A)$ induced by $\theta_{\varphi} = \tilde{\varphi}_{\sharp}$, so it is the identity. $\square$

**The colimit coefficient system $V_{\text{col}}(F)$.** Given any semifunctor $F: (\mathcal{G}, \mathcal{M}) \to \text{Cov}_{\bullet}$ that induces a genuine functor on $\pi_0$, we now define the colimit coefficient system semifunctor; see Definition 2.56. Composing with the forgetful functor $\mathbb{R}: \text{Cov}_{\bullet} \to \text{Grp}$ of Notation 2.8, we have a semifunctor $\mathbb{R}\circ F: (\mathcal{G}, \mathcal{M}) \to \text{Grp}$, which induces by Lemma 2.16 a semifunctor $\pi_0(\langle\mathcal{G}, \mathcal{M}\rangle) \to \text{Grp}$. Since $F$ induces a genuine functor on $\pi_0$ (by assumption), this is a genuine functor. Composing with the isomorphism of Lemma 1.71, we have a functor $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle \to \text{Grp}$, which we also denote by $\mathbb{R}\circ F$. Applying Lemma 2.51, we may therefore consider the functor $(\mathbb{R}\circ F)_{\text{col}}$ and the natural transformation $\mathbb{R}\circ F \Rightarrow (\mathbb{R}\circ F)_{\text{col}}$. Composing with the group ring functor $\mathbb{Z}[-]$; $\text{Grp} \to \text{Rings}$, this is a natural transformation of functors of the form $\langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle \to \text{Rings}$. Hence, we obtain:

**Definition 2.56** For $F: (\mathcal{G}, \mathcal{M}) \to \text{Cov}_{\bullet}$ a semifunctor that induces a genuine functor on $\pi_0$, we define the colimit coefficient system semifunctor:

$$V_{\text{col}}(F): (\mathcal{G}, \mathcal{M}) \to \bullet \text{Mod}_{\bullet} \qquad (2.31)$$

to be the composite $(\mathbb{Z}[\mathbb{R}\circ F] \times \mathbb{Z}[(\mathbb{R}\circ F)_{\text{col}}] \times \mathbb{Z}[(\mathbb{R}\circ F)_{\text{col}}]) \circ \Delta_3 \circ \pi_0$, where $\Delta_3$ is the 3-ary diagonal functor and $\pi_0: (\mathcal{G}, \mathcal{M}) \to \langle\pi_0(\mathcal{G}), \pi_0(\mathcal{M})\rangle$ is the projection onto path-components composed with the isomorphism of Lemma 1.71. Concretely, on objects, $V_{\text{col}}(F)$ sends $M$ to the group ring $\mathbb{Z}[(\mathbb{R}\circ F)_{\text{col}}(M)]$ considered as a right module over itself and as a left module over $\mathbb{Z}[\mathbb{R}\circ F(M)]$ via the canonical group homomorphism $\mathbb{R}\circ F(M) \to (\mathbb{R}\circ F)_{\text{col}}(M)$.
**Lemma 2.57** The assignments for (2.31) determine a continuous semifunctor, such that Condition 2.17 is satisfied and the semifunctor $\pi_0(V_{\text{col}}(F))$ is a genuine functor.

*Proof.* For each object $M$ of $\langle \mathcal{G}, \mathcal{M} \rangle$, the group ring $\mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(M)]$ has a canonical bimodule structure over the pair of rings $(\mathbb{Z}[\mathcal{R} \circ F(M)], \mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(M)])$: the action of $\mathbb{Z}[\mathcal{R} \circ F(M)]$ is induced by the canonical group homomorphism $\mathcal{R} \circ F(M) \to (\mathcal{R} \circ F)_{\text{col}}(M)$ and left multiplication, while the action of $\mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(M)]$ is right multiplication. Thus $V_{\text{col}}(F)(M)$ is an object of $\bullet\text{Mod}_{\bullet}$. For each morphism $\varphi: M \to N$ of $\langle \mathcal{G}, \mathcal{M} \rangle$, the ring homomorphisms $\mathbb{Z}[\mathcal{R} \circ F(\varphi)]$ and $\mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(\varphi)]$ are well-defined and the latter is moreover also a bimodule homomorphism $\mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(M)] \to (\mathbb{Z}[\mathcal{R} \circ F(\varphi)], \mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(\varphi)])^*(\mathbb{Z}[\langle \mathcal{R} \circ F \rangle_{\text{col}}(N)])$. Thus $V_{\text{col}}(F)(\varphi)$ is a morphism of $\bullet\text{Mod}_{\bullet}$.

It is routine to verify that $V_{\text{col}}(F)$ respects composition, so it is a well-defined semifunctor. It is continuous since the projection of $\langle \mathcal{G}, \mathcal{M} \rangle$ onto its path-components is continuous (see Remark 1.70) and this is then post-composed with functors between discrete categories to obtain $V_{\text{col}}(F)$.

Applying the forgetful functor $\mathcal{L}$ of Notation 2.8, it follows from the definition of $V_{\text{col}}(F)$ that $\mathcal{L} \circ V_{\text{col}}(F)$ is equal to $\mathbb{Z}[\mathcal{R} \circ F]$, thus Condition 2.17 automatically holds.

Let $\Upsilon_M$ be an endomorphism of $M$ in $\langle \mathcal{G}, \mathcal{M} \rangle$ whose image in $\pi_0(\langle \mathcal{G}, \mathcal{M} \rangle)$ is the identity of $M$. (For example, we could take the explicit endomorphism defined in the proof of Proposition 2.45.) By Proposition 2.45, we have $F(\Upsilon_M) = \text{id}_{F(M)}$ and it follows from our definition of $V_{\text{col}}(F)$ on morphisms that $V_{\text{col}}(F)(\Upsilon_M)$ is an identity morphism. Thus $\pi_0(V_{\text{col}}(F))$ preserves identities, i.e. it is a genuine functor. $\square$

Hence each semifunctor $F: \langle \mathcal{G}, \mathcal{M} \rangle \to \text{Cov}_{\bullet}$ (inducing a genuine functor on $\pi_0$) automatically induces a homological representation functor $L_i(F; V_{\text{col}}(F))$: $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \text{Mod}_{\bullet}$ for each $i \ge 0$ by the construction of §2.1.5 applied to $F$ and its associated colimit coefficient system (2.31). The fundamental “untwisting” results of this section are the following two propositions:

**Proposition 2.58** Suppose that we have groupoids $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69 and assume that the poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set. Then, for each $i \ge 0$, the homological representation functor $L_i(F; V_{\text{col}}(F))$ is isomorphic to a functor taking values in the subcategory $\text{Mod}_{\mathbb{Z}[\mathcal{Q}]}^{\text{tw}} \subset \text{Mod}_{\bullet}$ for a fixed group $\mathcal{Q}$.

*Proof.* By Lemma 2.52, the functor $(\mathcal{R} \circ F)_{\text{col}}$ is isomorphic to a functor that sends all objects to a fixed group $\mathcal{Q}$. By construction it follows that, up to isomorphism, $V_{\text{col}}(F)$ takes values in the subcategory $\bullet\text{Mod}_{\mathbb{Z}[\mathcal{Q}]}^{\text{tw}} \subset \bullet\text{Mod}_{\bullet}$. It then follows directly from the construction of §2.1.5 that the resulting homological representation functor $L_i(F; V_{\text{col}}(F))$, up to isomorphism, takes values in the subcategory $\text{Mod}_{\mathbb{Z}[\mathcal{Q}]}^{\text{tw}} \subset \text{Mod}_{\bullet}$. $\square$

**Proposition 2.59** Suppose that we have groupoids $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69 and let $F = \mathcal{F}^u$ be one of the untwisted semifunctors of Theorem 2.28, restricted along the inclusion (2.19). Assume that the poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set and that $\mathcal{M}$ is a motion groupoid. Then, for each $i \ge 0$, the homological representation functor $L_i(\mathcal{F}^u; V_{\text{col}}(\mathcal{F}^u))$ is isomorphic to a functor taking values in the subcategory $\text{Mod}_{\mathbb{Z}[\mathcal{Q}]} \subset \text{Mod}_{\bullet}$ for a fixed group $\mathcal{Q}$.

*Proof.* By Proposition 2.55, the functor $(\mathcal{R} \circ \mathcal{F}^u)_{\text{col}}$ is isomorphic to the constant functor at a fixed group $\mathcal{Q}$. By construction it follows that, up to isomorphism, $V_{\text{col}}(\mathcal{F}^u)$ takes values in the subcategory $\bullet\text{Mod}_{\mathbb{Z}[\mathcal{Q}]} \subset \bullet\text{Mod}_{\bullet}$. It then follows directly from the construction of §2.1.5 that the resulting homological representation functor $L_i(\mathcal{F}^u; V_{\text{col}}(\mathcal{F}^u))$, up to isomorphism, takes values in the subcategory $\text{Mod}_{\mathbb{Z}[\mathcal{Q}]} \subset \text{Mod}_{\bullet}$. $\square$

**Example 2.60** (Low-degree lower central series.) We note that, in some examples of interest, we have $\mathcal{F} = \mathcal{F}^u$ for the semifunctors of Theorem 2.28, which means that Proposition 2.59 applies to the homological representation functors $L_i(\mathcal{F}; V_{\text{col}}(\mathcal{F}))$ for any of the semifunctors $\mathcal{F}$ of Theorem 2.28.

This is true exactly when, for each object $(M, A)$, the $\pi_1(\mathcal{E}(A, \mathring{M}))$-action on $\mathcal{Q}(M, A)$ is trivial, so that $\mathcal{Q}^u(M, A) = \mathcal{Q}(M, A)$; see Notation 2.36. An obvious example of this is when $\mathcal{Q}$ is the quotient onto the trivial group, since $\mathcal{Q}(M, A)$ is then always trivial. A less obvious but important example is when $\mathcal{Q}$ is the abelianisation functor: in this case, all groups on the bottom row of diagram (2.24) are abelian, so the semi-direct product is a direct product, and so the $\pi_1(\mathcal{E}(A, \mathring{M}))$-action on $\mathcal{Q}(M, A)$ is trivial. These are precisely the $\ell \in \{1, 2\}$ cases of Example 2.23 applied to the
lower central series, since $Q_{\Gamma_1}$ is the quotient onto the trivial group and $Q_{\Gamma_2}$ is the abelianisation functor.

## 2.3. Further constructions of homological representation functors

The construction of the functors $\mathcal{F}: \mathcal{U}\mathcal{D}_d \to \text{Mod}_\bullet$ in §2.2 (see Theorem 2.28) is well-adapted for motion groups in the sense that, if one restricts to a subcategory of $\mathcal{U}\mathcal{D}_d$ of the form $\langle \mathcal{G}, \mathcal{M} \rangle$, where $\mathcal{M}$ is a motion groupoid (see Definition 2.54 and Proposition 1.42), then Proposition 2.59 implies that the induced homological representation functor $L_i(\mathcal{F}; V_{\text{col}}(\mathcal{F}))$ takes values in the subcategory $\text{Mod}_{\mathbb{Z}[\mathcal{Q}]} \subset \text{Mod}_\bullet$ for a fixed group $\mathcal{Q}$, as long as the underlying poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is directed and if $\mathcal{F}$ is one of the two untwisted semifunctors constructed in Theorem 2.28. Thus the construction of §2.2 is effective for producing families of untwisted representations of motion groups (or braided mapping class groups; see Remark 1.43). However, for the full mapping class groups, the construction of §2.2 typically produces twisted representations.

In this subsection, we describe a variant of the construction of §2.2, based on the split short exact sequence (1.9) instead of (1.7), which has good properties for the full mapping class groups. More precisely, this variant satisfies an analogue of Proposition 2.59 where one does not have to assume that $\mathcal{M}$ is a motion groupoid; see Proposition 2.65 below.

Since the constructions and proofs are very similar to those of §2.2, we review the adaptation of the results of §2.2, pointing out and focusing on the differences.

### 2.3.1. Construction of the homological representation functors

As in §2.2, we fix an integer $d \ge 2$, $d \ne 4$ (cf. Remark 1.54), a closed submanifold $Z \subset \mathbb{R}^d$, an open subgroup $\mathcal{G}$ of $\text{Diff}(Z)$ and a functorial quotient of groups $\mathcal{Q}$. We explain in the following paragraphs the construction of continuous semifunctors analogous to (2.22), which we denote by $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}$, $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}_{\text{u}}$, $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}$ and $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}_{\text{u}}$ to distinguish them from those of Theorem 2.28. The result of this construction is summarised in Theorem 2.62 below.

**The semifunctor $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}$.** The construction of the semifunctor $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}$ is similar to that of $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}$ in §2.2.1. On objects, we follow the construction of §2.2.1.1; in particular we use the same space $X(M, A)$ as in §2.2.1.1 (see Definition 2.30). However, the quotient $\phi(M, A): \pi_1(X(M, A)) \to \mathcal{Q}(M, A)$ is defined using the following 6-term commutative diagram instead of (2.24).

$$
\begin{tikzcd}
& & \pi_1(X(M, A)) \arrow[d, "\cong"'] \\
1 \arrow[r] & \pi_1(\mathcal{E}_G(Z, \mathcal{M} \setminus A)) \arrow[r] \arrow[d, "\cong"'] & \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A, Z|\mathcal{G})) \arrow[r, "\subsetneq"'] \arrow[d, "\cong"'] & \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A)) \arrow[r] \arrow[d, "\cong"'] & 1 \\
& \phi(M, A) \arrow[u] \arrow[r, "\twoheadleftarrow"'] & q(M, A) \arrow[u] \arrow[r, "\twoheadleftarrow"'] & \bar{q}(M, A) \arrow[u] \arrow[r, "\subsetneq"'] & Q(\pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A, Z|\mathcal{G}))) \arrow[r, "\cong"'] & 1
\end{tikzcd}
\qquad (2.32)
$$

The top row is the split short exact sequence (1.9) of Corollary 1.34. The rest of the diagram is induced from this just as in §2.2.1.1, using Lemma 2.24. The construction of $\mathcal{F}^{\mathcal{G}, \mathcal{Q}}$ on morphisms is then exactly as in §2.2.1.2, using the fact that the commutative diagram (2.32) is functorial in the morphisms of $\mathcal{U}\mathcal{D}_d$ thanks to Proposition 1.81 and Lemma 2.26. In particular, Lemmas 2.38 and 2.39 and their proofs repeat verbatim.

**Untwisted and closed variants.** By Proposition 1.81 and Lemma 2.26, the commutative diagram (2.32) is functorial in the morphisms of $\mathcal{U}\mathcal{D}_d$, so there is an action of $\text{Diff}_{\text{dec}}(M, A) = \text{Aut}_{\mathcal{U}\mathcal{D}_d}((M, A))$ on the bottom-left group $\mathcal{Q}(M, A)$. Since $\mathcal{Q}(M, A)$ is discrete, this factors through an action of $\pi_0(\text{Diff}_{\text{dec}}(M, A))$ on $\mathcal{Q}(M, A)$. On the other hand, the top-left group $\pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A))$ of (2.32) acts on $\mathcal{Q}(M, A)$ either by lifting elements of $\mathcal{Q}(M, A)$ along $\phi(M, A)$ and using the conjugation action of the semi-direct product on the top row, or equivalently by projecting along $\bar{q}(M, A)$ to the bottom-right group of (2.32) and then using the conjugation action of the semi-direct product on the bottom row. Via the canonical isomorphism $\pi_0(\text{Diff}_{\text{dec}}(M, A)) \cong \pi_0(\text{Diff}_{\text{dec}}(\bar{M}, A))$,
induced by extending diffeomorphisms by the identity on $\bar{M} \setminus M$, this gives another action of $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A))$ on $\mathcal{Q}(M, A)$. The following is the analogue of Proposition 2.35.

**Proposition 2.61** The two actions of $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A))$ on $\mathcal{Q}(M, A)$ described above are equal.

*Proof.* Since both of these actions are induced via the quotient $\phi(M, A)$ from actions defined on $\pi_1(\mathcal{E}_G(Z, \tilde{M} \setminus A))$, it suffices to prove that the two actions agree on this group, before passing to the quotient. This is precisely part (i) of Proposition 1.48. $\square$

Similarly to Notation 2.36, we denote by $\mathcal{Q}^u(M, A)$ the coinvariants $\mathcal{Q}(M, A)_{\pi_0(\mathrm{Diff}_{\mathrm{dec}}(M, A))}$ associated to this action, and by $\phi^u(M, A)$ the composite $X(M, A) \to \mathcal{Q}(M, A) \to \mathcal{Q}^u(M, A)$, where the first arrow is $\phi(M, A)$ and the second is the canonical projection onto the coinvariants. As in Remark 2.37, this quotient group $\mathcal{Q}^u(M, A)$ is the optimal one for the associated homological representations to be untwisted; see Proposition 2.65 below. We thus define the untwisted variant $\tilde{\mathcal{F}}^u_{(Z,G,Q)}$ by $(M, A) \mapsto (X(M, A), \phi^u(M, A))$ on objects; on morphisms it is defined just as above, using functoriality of the diagram (2.32) by Proposition 1.81 and Lemma 2.26.

Finally, the closed variants $\tilde{\mathcal{F}}_{(Z,G,Q)}$ and $\tilde{\mathcal{F}}^u_{(Z,G,Q)}$ are constructed by making the same modification as in §2.2.1.3, using the (homotopy equivalent but not proper homotopy equivalent) space $X'(M, A)$ in place of $X(M, A)$.

The discussion above proves the following analogue of Theorem 2.28.

**Theorem 2.62** For any integer $d \ge 2$, closed submanifold $Z \subset \mathbb{R}^d$ and open subgroup $G$ of $\mathrm{Diff}(Z)$, each functorial quotient of groups $Q$ determines, via (2.32), continuous semifunctors

$$ \tilde{\mathcal{F}}_{(Z,G,Q)}, \tilde{\mathcal{F}}^u_{(Z,G,Q)}, \tilde{\mathcal{F}}_{(Z,G,Q)} \text{ and } \tilde{\mathcal{F}}^u_{(Z,G,Q)}: \mathcal{U}\mathcal{D}_d \to \mathrm{Cov}_{\bullet}. \qquad (2.33) $$

Similar continuous semifunctors are defined on the semicategory $\mathcal{U}\mathcal{D}_d^+$ if $Z$ is orientable and $G$ is contained in $\mathrm{Diff}^+(Z)$.

**Remark 2.63** There is a natural morphism of diagrams (2.24) $\to$ (2.32) induced by Lemma 2.26 from the map of split short exact sequences from the top row to the middle row in diagram (1.10) of Proposition 1.36. This induces a natural transformation of semifunctors (2.22) $\Rightarrow$ (2.33) for each of the four versions and for each fixed $(Z, G, Q)$.

**Corollary 2.64** The semifunctors of Theorem 2.62 provide semifunctors of the form

$$ \langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle \to \mathrm{Mod}_{\bullet}, \qquad (2.34) $$

which we denote by $L_i(\tilde{\mathcal{F}}_{(Z,G,Q)}; V)$, $L_i(\tilde{\mathcal{F}}^u_{(Z,G,Q)}; V)$, $L_i(\tilde{\mathcal{F}}_{(Z,G,Q)}; V)$ and $L_i(\tilde{\mathcal{F}}^u_{(Z,G,Q)}; V)$, where $V$ is any continuous semifactor satisfying Condition 2.17. If the semifactor $\pi_0(V)$ defined from $V$ by Lemma 2.16 is a genuine functor, then the semifunctors (2.34) upgrade to genuine functors.

*Proof.* This is a direct application of the general construction of §2.1.5 (see Definition 2.20), together with the identification of $\pi_0(\langle \mathcal{G}, \mathcal{M} \rangle)$ with $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$ by Lemma 1.71. The upgrade of (2.34) to genuine functors follows from the analogues of Proposition 2.45 and Corollary 2.46, whose proofs are identical. $\square$

### 2.3.2. Elementary properties

The properties of §2.2.2 also hold for the semifunctors of Theorem 2.62. In particular, the proof of Proposition 2.44 repeats verbatim to prove that, for fixed $Q$, each of the four versions of (2.33) depends (up to natural isomorphism) on $(Z, G)$ only up to isotopy in the sense of Definition 2.43. In addition, it is routine to adapt the proofs of §2.2.2 to show that the continuous semifunctors (2.33) induce genuine functors on $\pi_0$ (viz. Proposition 2.45), that the closed variants $\tilde{\mathcal{F}}_{(Z,G,Q)}$ and $\tilde{\mathcal{F}}^u_{(Z,G,Q)}$ take values in $\mathrm{Cov}_{\bullet}^{\mathrm{pr}}$ (viz. Lemma 2.47) and that there are natural homotopy equivalences $\tilde{\mathcal{F}}_{(Z,G,Q)} \Rightarrow \tilde{\mathcal{F}}_{(Z,G,Q)}$ and $\tilde{\mathcal{F}}^u_{(Z,G,Q)} \Rightarrow \tilde{\mathcal{F}}^u_{(Z,G,Q)}$ (viz. Lemma 2.41). Indeed, the only difference between the constructions of the semifunctors of Theorem 2.28 and those of Theorem 2.62 is that, in the former, the groups $\mathcal{Q}(M, A)$ (and the homomorphisms $\mathcal{Q}(\varphi)$) are those obtained from diagram (2.24) (and its functoriality), whereas in the latter they are obtained from diagram (2.32) (and its functoriality).
### 2.3.3. Colimit coefficient systems and untwisted representations

Recall from §2.2.3 that each semifunctor $F: \langle \mathcal{G}, \mathcal{M} \rangle \to \mathrm{Cov}_{\bullet}$ that induces a genuine functor on $\pi_0$ has an associated colimit coefficient system (semifunctor) $V_{\mathrm{col}}(F): \langle \mathcal{G}, \mathcal{M} \rangle \to \bullet \mathrm{Mod}_{\bullet}$; see Definition 2.56 and Lemma 2.57. Since the semifunctors (2.33) induce genuine functors on $\pi_0$ (see §2.3.2), we may apply this construction to $F = (2.33)$. Proposition 2.58 applies directly to this setting, since it makes no further assumptions on $F$. The analogue of Proposition 2.59 is the following, in which we notably drop the assumption that $\mathcal{M}$ is a motion groupoid.

**Proposition 2.65** Suppose that we have groupoids $\mathcal{G}$ and $\mathcal{M}$ as in Hypothesis 1.69 and let $F = \mathfrak{F}^u$ be one of the untwisted semifunctors of Theorem 2.62, restricted along the inclusion (2.19). Assume that the poset $\mathcal{P}(\mathcal{G}, \mathcal{M})$ is a directed set. Then, for each $i \ge 0$, the homological representation functor $L_i(\mathfrak{F}^u; V_{\mathrm{col}}(\mathfrak{F}^u))$ is isomorphic to a functor taking values in the subcategory $\mathrm{Mod}_{\bullet}[\mathbb{Q}] \subset \mathrm{Mod}_{\bullet}$ for a fixed group $\mathbb{Q}$.

*Proof.* Just as in the proof of Proposition 2.59, this will follow immediately from the definition of the colimit coefficient system $V_{\mathrm{col}}(\mathfrak{F}^u)$ and the general construction of §2.1.5, once we have proven the analogue of Proposition 2.55, in which we consider $\mathfrak{F}^u$ in place of $\mathcal{F}^u$ (and we do not assume that $\mathcal{M}$ is a motion groupoid). The proof of this analogue is identical to that of Proposition 2.55 itself, the only difference being that we use Proposition 2.61 instead of Proposition 2.35 to understand the action of automorphisms of $\pi_0(\mathcal{M})$ on $\mathcal{Q}(\mathcal{M}, A)$. Since Proposition 2.61 applies to the whole mapping class group $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(\mathcal{M}, A))$ (whereas Proposition 2.35 applies only to the subgroup $\pi_0(\mathrm{Diff}^{\mathrm{br}}_{\mathrm{dec}}(\mathcal{M}, A))$, the braided mapping class group), we do not need to assume that $\mathcal{M}$ is a motion groupoid, as we had to for Proposition 2.59. $\square$

## 3. Applications for mapping class groups and motion groups

In this section, we study in detail the representations obtained as outputs of the general construction of §2.1.5 (see Definition 2.20), applied following the methods of §2.2–§2.3 to some important families of mapping class groups and motion groups, namely the families of classical braid groups, braid groups on surfaces and loop braid groups in §3.2.1, and the families of mapping class groups of surfaces in §3.2.2. Beforehand, we detail the categorical framework coming from §1 for these families of groups in §3.1.

### 3.1. Categories for mapping class groups and motion groups

The mapping class groups and motion groups that we shall study in §3.2.1 and §3.2.2 generally come in families, in the sense that they naturally form a set $\{G_n\}_{n \in \mathbb{N}}$ along with canonical maps $g_n: G_n \to G_{n+1}$. A good way to treat these objects systematically, extracting their essential structure, is to package them using the Quillen bracket construction framework of §1.4, as follows.

**Construction 3.1** Let $\{g_n: G_n \to G_{n+1}\}_{n \in \mathbb{N}}$ be a family of groups of the following form.

(i) There is a fixed dimension $d \ge 2$ such that, for each $n \in \mathbb{N}$, the group $G_n$ is isomorphic either to the mapping class group $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(M_n, A_n))$ (see Notation 1.17) or to the braided mapping class group $\pi_0(\mathrm{Diff}^{\mathrm{br}}_{\mathrm{dec}}(M_n, A_n))$ (see Definition 1.38), where $(M_n, A_n)$ is a decorated manifold (see Definition 1.11); in particular, $M_n$ is a smooth $d$-manifold with non-empty boundary and $A_n$ is a closed submanifold of its interior $\dot{M}_n$. Moreover, the homomorphism $g_n$ is induced by an embedding in $\mathrm{Emb}_{\mathrm{dec}}((M_n, A_n), (M_{n+1}, A_{n+1}))$ (see Definition 1.49), sending an isotopy class of diffeomorphisms of $(M_n, A_n)$ to the isotopy class of diffeomorphisms of $(M_{n+1}, A_{n+1})$ given by applying this embedding and extending it by the identity on the complement of its image.

(ii) There is a decorated manifold $(M, A) \in \mathcal{D}_d$ such that, for each $n \in \mathbb{N}$, the decorated manifold $(M_{n+1}, A_{n+1})$ is isomorphic to the boundary connected sum $(M, A)\#(M_n, A_n)$ and, under this isomorphism, the preferred embedding of $(M_n, A_n)$ into $(M_{n+1}, A_{n+1})$ corresponds to the inclusion into $(M, A)\#(M_n, A_n)$.

(iii) In the case when $G_n$ is a *braided* mapping class group in point (i), we also assume that the mapping class group $\pi_0(\mathrm{Diff}_{\mathrm{dec}}((M, \emptyset)^{\sharp n}))$ is trivial for all $n \in \mathbb{N}$. This implies that we have an equality $\pi_0(\mathrm{Diff}^{\mathrm{br}}_{\mathrm{dec}}((M, A)^{\sharp n})) = \pi_0(\mathrm{Diff}_{\mathrm{dec}}((M, A)^{\sharp n}))$.
In order to encode the family of groups {$g_n : G_n \to G_{n+1}$}$_{n \in \mathbb{N}}$, we carry out the following steps.

(1) We first define $\mathcal{M}$ to be a certain subgroupoid of $\mathcal{D}_d$ whose objects are all decorated manifolds that are isomorphic to $(M_n, A_n)$ for some $n \in \mathbb{N}$. In the case when $G_n$ is a full mapping class group, we define $\mathcal{M}$ to be the full subgroupoid on these objects. In the case when $G_n$ is a *braided* mapping class group, we must be slightly more careful. For each decorated manifold $(N, \emptyset)$ in the isomorphism class of $(M_n, \emptyset)$, let us fix an isomorphism $i_{(N,\emptyset)} : (N, \emptyset) \cong (M_n, \emptyset)$; in the case $(N, \emptyset) = (M_n, \emptyset)$ we take it to be the identity. We then define the morphisms of $\mathcal{M}$ to be those isomorphisms $(N, B) \cong (N', B')$ that become isotopic to $i_{(N',\emptyset)}^{-1} \circ i_{(N,\emptyset)}$ after forgetting the submanifolds $B$ and $B'$. (We call these *braided isomorphisms*, generalising Definition 1.38 to isomorphisms between distinct objects.) This ensures that $\mathcal{M}$ is a well-defined subgroupoid of $\mathcal{D}_d$ and that its automorphism groups are braided diffeomorphism groups.

(2) We then define $\mathcal{G}$ to be the full subgroupoid of $\mathcal{D}_d$ on all objects that are isomorphic to $(M, A)^{\otimes n}$ for some $n \in \mathbb{N}$. This is evidently closed under the semi-monoidal structure $\flat$ of $\mathcal{D}_d$ (see Definition 1.22). Moreover, the subgroupoid $\mathcal{M} \subseteq \mathcal{D}_d$ is closed under the left action of $\mathcal{G}$ induced by $\flat$; in the case when the groups $G_n$ are full mapping class groups (where we defined $\mathcal{M}$ to be a full subgroupoid in step (1)) this is clear, whereas in the case when they are *braided* mapping class groups it follows from the assumption in point (iii).

(3) Then, by Corollaries 1.80 and 1.84, there exists a skeleton $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle_0$ of the Quillen bracket construction $\langle \pi_0(\mathcal{G}), \pi_0(\mathcal{M}) \rangle$; this encodes the family of groups {$g_n : G_n \to G_{n+1}$}$_{n \in \mathbb{N}}$.

**Remark 3.2** Recall from Convention 1.2 that the topologically-enriched groupoids $\mathcal{D}_d$ and $\mathcal{D}_d^+$ are essentially small, and that we have implicitly replaced them with equivalent small groupoids. All of the subgroupoids of $\mathcal{D}_d$ and $\mathcal{D}_d^+$ that we shall consider are therefore also small, as are the corresponding discrete groupoids after applying the functor $\pi_0$ and the corresponding (discrete or topologically-enriched) Quillen bracket categories.

In the remainder of this subsection, we describe and discuss some properties of the subgroupoids $\mathcal{G}$ and $\mathcal{M}$ defined as in Construction 3.1, corresponding to mapping class groups of surfaces in §3.1.1, to surface braid groups in §3.1.2 and to loop braid groups and extended loop braid groups in §3.1.3. In §3.2 we then apply the general construction of §2 to these examples to obtain coherent representations of these families of groups.

### 3.1.1. Mapping class groups of surfaces

We denote by $\mathbb{D}^2$ the closed unit 2-disc. Let $\Sigma_{0,1}^1$ denote the cylinder $\mathbb{S}^1 \times [0, 1]$ (this notation is more commonly used to denote a once-punctured disc, which is consistent with the fact that the non-distinguished boundary component of $\Sigma_{0,1}^1$ may be moved freely by diffeomorphisms), let $\Sigma_{1,1}$ denote the torus with one boundary component $(\mathbb{S}^1 \times \mathbb{S}^1) \setminus \mathbb{D}^2$ and let $N_{1,1}$ denote a Möbius band. For $g \ge 0$, we denote by $\Sigma_{g,1}^s$ the boundary connected sum $(\flat_s \Sigma_{0,1}^1) \flat(\flat_g \Sigma_{1,1})$ and by $\Gamma_{g,1}^s$ the mapping class group $\pi_0(\mathrm{Diff}_\partial(\Sigma_{g,1}^s))$. For $h \ge 0$, we denote by $\mathcal{N}_{h,1}^s$ the boundary connected sum $(\flat_s \Sigma_{0,1}^1) \flat(\flat_h \mathcal{N}_{1,1})$ and by $\mathcal{N}_{h,1}^s$ the mapping class group $\pi_0(\mathrm{Diff}_\partial(\mathcal{N}_{h,1}^s))$. When $s=0$, we omit it from the notation. Let $\mathcal{M}_2^{+,t}$ and $\mathcal{M}_2^{-,t}$ be the full subgroupoids of $\mathcal{D}_2$ on decorated surfaces of the form $(S, \emptyset, e_1, e_2)$, where $S$ is homeomorphic to $\Sigma_{g,1}$ for some $g \ge 0$ (for $\mathcal{M}_2^{+,t}$) or to $\mathcal{N}_{h,1}$ for some $h \ge 0$ (for $\mathcal{M}_2^{-,t}$) and the boundary-cylinders $e_1$ and $e_2$ both lie on the same boundary component of $S$. The groupoids $\mathcal{M}_2^{+,t}$ and $\mathcal{M}_2^{-,t}$ clearly both inherit the semi-monoidal structure $\flat$ from $\mathcal{D}_2$ (see Definition 1.22). The choices $\mathcal{G} = \mathcal{M} = \mathcal{M}_2^{+,t}$ and $\mathcal{G} = \mathcal{M} = \mathcal{M}_2^{-,t}$ thus each fit into Hypothesis 1.69.

We denote by $\mathcal{M}_2^+$ and $\mathcal{M}_2^-$ the path-components $\pi_0(\mathcal{M}_2^{+,t})$ and $\pi_0(\mathcal{M}_2^{-,t})$ respectively. Since $\mathcal{M}_2^{+,t}$ and $\mathcal{M}_2^{-,t}$ contain the solid cylinder $(\mathbb{B}_1^2, \emptyset, \mathrm{id}, r)$, the groupoids $\mathcal{M}_2^+$ and $\mathcal{M}_2^-$ have, by Lemma 1.24, a monoidal structure induced by the boundary connected sum $\flat$ of Definition 1.19. Hence we may apply Quillen’s bracket construction and by Lemma 1.71 we have isomorphisms of categories $\pi_0(\mathcal{U}\mathcal{M}_2^{+,t}) \cong \mathcal{U}\mathcal{M}_2^+$ and $\pi_0(\mathcal{U}\mathcal{M}_2^{-,t}) \cong \mathcal{U}\mathcal{M}_2^-$. By Proposition 1.82, we may pass to monoidal skeleta of $(\mathcal{M}_2^+, \flat)$ and $(\mathcal{M}_2^-, \flat)$ and the categories $\mathcal{U}\mathcal{M}_2^+$ and $\mathcal{U}\mathcal{M}_2^-$ are thus also skeletal by Corollary 1.84. For diffeomorphisms of surfaces, the condition of fixing pointwise two (neighbourhoods of) intervals in a boundary component is equivalent to fixing pointwise the whole boundary component. So the morphisms in $\mathcal{M}_2^+$ and $\mathcal{M}_2^-$ may be identified with the isotopy classes
of diffeomorphisms of the surfaces $S$ that that fix pointwise the preferred boundary component of $S$, i.e. their automorphism groups are the mapping class groups $\pi_0(\mathrm{Diff}_\partial(S))$ of $S$.

### 3.1.2. Surface braid groups

Let $S$ be a compact, connected, smooth surface with a chosen boundary component $\partial_0 S$. For each non-negative integer $n$, we denote by $\underline{n}$ a closed submanifold of $S$ consisting of $n$ distinct points in the interior of $S$. Let $\mathcal{B}r^S$ be the subgroupoid of $\mathcal{D}_2$ with objects all decorated surfaces of the form $(S', \underline{n}', e_1, e_2)$, where the embedded 1-discs $e_1(b\mathbb{B}_1^2)$ and $e_2(b\mathbb{B}_1^2)$ lie on the same boundary component $\partial_0 S'$, $n$ is any non-negative integer and there is a diffeomorphism $S \cong S'$ taking $\partial_0 S$ onto $\partial_0 S'$ and $\underline{n}$ onto $\underline{n}'$. The morphisms of $\mathcal{B}r^S$ are the *braided isomorphisms* of $\mathcal{D}_2$, as described in step (1) of Construction 3.1; in particular its automorphism groups are given by the *braided* diffeomorphism groups of Definition 1.38. Since the condition of being a *braided* is invariant under deformations along paths in the space of all isomorphisms in $\mathcal{D}_2$, the groupoid $\mathcal{B}r^S$ is 0-full in $\mathcal{D}_2$. If $S = \mathcal{D}^2$, this collection of objects is closed under the semi-monoidal structure $\mathfrak{h}$ of Definition 1.22, so in this case $\mathcal{B}r^{\mathcal{D}^2}$ is a semi-monoidal subgroupoid of $\mathcal{D}_2$. Moreover, in this case, $\mathcal{B}r^{\mathcal{D}^2}$ is full (not just 0-full) in $\mathcal{D}_2$, since the mapping class group $\pi_0(\mathrm{Diff}_{I \sqcup I}(\mathcal{D}^2))$ is trivial by Lemma 3.3 below. In general, the groupoid $\mathcal{B}r^S$ is closed under the left action of $\mathcal{B}r^{\mathcal{D}^2}$ via $\mathfrak{h}$. Thus the choice of $\mathcal{G} = \mathcal{B}r^{\mathcal{D}^2}$ and $\mathcal{M} = \mathcal{B}r^S$ fits into Hypothesis 1.69.

**Lemma 3.3** Let $I \sqcup I$ be a pair of disjoint closed intervals in the boundary of the disc $\mathcal{D}^2$ and write $\mathrm{Diff}_{I \sqcup I}(\mathcal{D}^2)$ for the group of diffeomorphisms of $\mathcal{D}^2$ that fix $I \sqcup I$ pointwise. Then $\mathrm{Diff}_{I \sqcup I}(\mathcal{D}^2)$ is weakly contractible, hence path-connected, so the mapping class group $\pi_0(\mathrm{Diff}_{I \sqcup I}(\mathcal{D}^2))$ is trivial.

*Proof.* By [Cer61, §II.2.2.2, Cor. 2] (see also [Pal60b; Lim63]), the map

$$ \mathrm{Diff}_{I \sqcup I}(\mathcal{D}^2) \to \mathrm{Diff}_{\partial}(I \sqcup I) \cong (\mathrm{Diff}_{\partial}(I))^2 $$

that remembers just the action of a diffeomorphism restricted to the *complementary* pair of intervals $\partial\mathcal{D}^2 \setminus (I \sqcup I) \cong I \sqcup I$ is a fibre bundle. Its fibre over the identity is $\mathrm{Diff}_{\partial}(\mathcal{D}^2)$, the group of diffeomorphisms of $\mathcal{D}^2$ fixing all of $\partial\mathcal{D}^2$ pointwise. The diffeomorphism group $\mathrm{Diff}_{\partial}(I)$ is easily seen to be contractible, and the diffeomorphism group $\mathrm{Diff}_{\partial}(\mathcal{D}^2)$ was shown to be contractible by Smale [Sma59]. The long exact sequence of the above fibre bundle then implies that $\mathrm{Diff}_{I \sqcup I}(\mathcal{D}^2)$ is weakly contractible. $\square$

Let $\beta^S$ be the groupoid $\pi_0(\mathcal{B}r^S)$. For each non-negative integer $n$, the automorphism group of its object $(S, \underline{n})$ is the surface braid group $\pi_0(\mathrm{Diff}_{\mathrm{dec}}^{\mathrm{br}}(S, \underline{n}))$ of $S$, which we denote by $\mathbf{B}_n(S)$. When $S = \mathcal{D}^2$, we abbreviate $\beta^{\mathcal{D}^2}$ to $\beta$ for brevity. Since $\mathcal{B}r^{\mathcal{D}^2}$ contains the solid cylinder $(\mathbb{B}_1^2, \emptyset, \mathrm{id}, r)$, Lemma 1.24 implies that the groupoid $\beta$ inherits a monoidal structure from the boundary connected sum $\mathfrak{h}$ of $\mathcal{D}_2$, which also induces on $\beta^S$ a left module structure over the monoidal groupoid $\beta$. Then, the Quillen bracket construction defines categories $\mathfrak{U}\mathcal{B}r^{\mathcal{D}^2}$, $\langle\mathcal{B}r^{\mathcal{D}^2}, \mathcal{B}r^S\rangle$, $\mathfrak{U}\beta$ and $\langle\beta, \beta^S\rangle$. By Lemma 1.71, we have isomorphisms of categories $\pi_0(\mathfrak{U}\mathcal{B}r^{\mathcal{D}^2}) \cong \mathfrak{U}\beta$ and $\pi_0(\langle\mathcal{B}r^{\mathcal{D}^2}, \mathcal{B}r^S\rangle) \cong \langle\beta, \beta^S\rangle$. By Proposition 1.82, we may pass to monoidal skeleta of $(\beta, \mathfrak{h})$ and $(\beta^S, \mathfrak{h})$; after doing this, the groupoid $\beta$ becomes equivalent to the well-known *braid groupoid*; see [Mac98, Chap. XI, §4] for instance. The categories $\mathfrak{U}\beta$ and $\langle\beta, \beta^S\rangle$ are then also skeletal by Corollary 1.84.

An alternative viewpoint on surface braid groups is as fundamental groups of configuration spaces. Fix a non-negative integer $n$ and a compact, connected surface $S$. Then the embedding space $\mathrm{Emb}(\underline{n}, \hat{S})$ is the *ordered configuration space* of $n$ points in $\hat{S}$, denoted by $F_n(\hat{S})$. For a partition $\lambda = \{\lambda_1; \dots; \lambda_r\} \vdash n$, the quotient space $F_n(\hat{S})/\mathfrak{S}_\lambda$, induced by the natural action of $\mathfrak{S}_\lambda := \mathfrak{S}_{\lambda_1} \times \dots \times \mathfrak{S}_{\lambda_r} \subseteq \mathfrak{S}_n$ on the coordinates, is the $\lambda$-partitioned configuration space of $n$ points in $\hat{S}$, denoted by $C_\lambda(\hat{S})$. When $\lambda$ is the trivial partition $\{n\}$ (which we simply denote by $n$), this is the *unordered configuration space* and is denoted by $C_n(\hat{S})$. The fundamental group of $C_n(\hat{S})$ is isomorphic to $\mathbf{B}_n(S)$ (this is a classical fact, or Proposition 1.42 with $M = S$ and $Z = \underline{n}$). More generally, following Definition 1.32, the fundamental group $\pi_1(C_\lambda(\hat{S}))$, which is isomorphic to $\pi_0(\mathrm{Diff}_{\mathrm{dec}}^{\mathrm{br}}(S, \lambda_1, \dots, \lambda_r))$ by Proposition 1.42, is called the $\lambda$-partitioned braid group $\mathbf{B}_\lambda(S)$.

### 3.1.3. Loop braid groups

We now focus on the families of extended and non-extended loop braid groups. Their definitions are recalled here and we refer to [Dam17] for a complete introduction to these groups. Let us
denote by $\mathbb{D}^3$ the closed 3-disc. For each non-negative integer $n$, we denote by $n\mathbb{S}^1$ a closed sub-manifold of $\mathbb{D}^3$ consisting of a collection of $n$ disjoint circles forming a trivial link of $n$ components in the interior of $\mathbb{D}^3$. The notation $n\mathbb{S}_+^1$ denotes the same unlink as $n\mathbb{S}^1$, but it indicates that we shall require any diffeomorphism on this unlink to be orientation-preserving; otherwise orientation-reversing diffeomorphisms are permitted. Choosing also two germs of boundary-cylinders (which we elide from the notation), the pair $(\mathbb{D}^3, n\mathbb{S}^1)$ forms a decorated manifold, which we sometimes denote by $\mathbb{D}_n^3$ for simplicity. Let $\text{Diff}_\partial(\mathbb{D}^3, n\mathbb{S}^1)$ be the group of self-diffeomorphisms of $\mathbb{D}^3$ that fix $\partial\mathbb{D}^3$ pointwise and $n\mathbb{S}^1$ setwise. The extended loop braid group $\mathbf{LB}'_n$ is the group of isotopy classes $\pi_0(\text{Diff}_\partial(\mathbb{D}^3, n\mathbb{S}^1))$. Let $\text{Diff}_\partial(\mathbb{D}^3, n\mathbb{S}_+^1)$ denote the subgroup of diffeomorphisms that also preserve the orientation of $n\mathbb{S}^1$. The (non-extended) loop braid group $\mathbf{LB}_n$ is the group of isotopy classes $\pi_0(\text{Diff}_\partial(\mathbb{D}^3, n\mathbb{S}_+^1))$.

We now set up a categorical framework for handling these families of groups. Let $\mathcal{L}\mathbf{B}'$ (respectively $\mathcal{L}\mathbf{B}$) be the full subgroupoid of $\mathcal{D}_3$ (respectively $\mathcal{D}_3^+$) on all decorated manifolds $(M, A, e_1, e_2)$ such that the pair $(M, A)$ is diffeomorphic to the 3-disc relative to an embedded $n$-component unlink. Both $\mathcal{L}\mathbf{B}'$ and $\mathcal{L}\mathbf{B}$ are closed under the semi-monoidal structure $\natural$ of Definition 1.22, so they inherit a semi-monoidal structure from $\mathcal{D}_3$ and $\mathcal{D}_3^+$. In particular, either choice $G = \mathcal{M} = \mathcal{L}\mathbf{B}$ or $G = \mathcal{M} = \mathcal{L}\mathbf{B}'$ fits into Hypothesis 1.69. We denote $\pi_0(\mathcal{L}\mathbf{B}')$ and $\pi_0(\mathcal{L}\mathbf{B})$ by $\mathcal{L}\mathbf{B}'$ and $\mathcal{L}\mathbf{B}$ respectively. Since $\mathcal{L}\mathbf{B}'$ and $\mathcal{L}\mathbf{B}$ both contain the solid cylinder $(\mathbb{B}_1^3, \emptyset, \text{id}, r)$, the groupoids $\mathcal{L}\mathbf{B}'$ and $\mathcal{L}\mathbf{B}$ both inherit a monoidal structure from the semi-monoidal structures of $\mathcal{D}_3$ and $\mathcal{D}_3^+$ by Lemma 1.24. We may pass to monoidal skeleta of $(\mathcal{L}\mathbf{B}', \natural)$ and $(\mathcal{L}\mathbf{B}, \natural)$ by Proposition 1.82. Hence Quillen's bracket construction defines categories $\mathfrak{U}\mathcal{L}\mathbf{B}'$ and $\mathfrak{U}\mathcal{L}\mathbf{B}$, which are skeletal by Corollary 1.84. By Lemma 1.71, we have isomorphisms of categories $\pi_0(\mathfrak{U}\mathcal{L}\mathbf{B}') \cong \mathfrak{U}\mathcal{L}\mathbf{B}'$ and $\pi_0(\mathfrak{U}\mathcal{L}\mathbf{B}) \cong \mathfrak{U}\mathcal{L}\mathbf{B}$.

Let us show that there are isomorphisms $\text{Aut}_{\mathcal{L}\mathbf{B}'}(\mathbb{D}_n^3) \cong \mathbf{LB}'_n$ and $\text{Aut}_{\mathcal{L}\mathbf{B}}(\mathbb{D}_n^3) \cong \mathbf{LB}_n$. We show this for the first case, the other one following by an identical argument. The automorphism group of $\mathbb{D}_n^3$ in $\mathcal{L}\mathbf{B}'$ is $\pi_0(\text{Diff}_{\text{dec}}(\mathbb{D}_n^3))$, where $\text{Diff}_{\text{dec}}(\mathbb{D}_n^3)$ is the topological group of diffeomorphisms of $\mathbb{D}^3$ that send the embedded $n$-component unlink onto itself and that restrict to the identity on a neighbourhood of two disjoint 2-discs in $\partial\mathbb{D}^3$. The condition of fixing a neighbourhood of two discs in the boundary is equivalent (up to homotopy equivalence, so in particular on $\pi_0$) to fixing just the two discs themselves. On the other hand, $\mathbf{LB}'_n$ has the same description except that diffeomorphisms must fix the whole boundary $\partial\mathbb{D}^3$. To show that these two groups are isomorphic, it therefore suffices to show the following.

**Lemma 3.4** Let $M$ be a 3-manifold with a boundary component $\partial_0 M \cong \mathbb{S}^2$. For isotopy classes of diffeomorphisms of $M$, fixing two disjoint 2-discs in $\partial_0 M$ is equivalent to fixing all of $\partial_0 M$.

*Proof.* Let $\text{Diff}(M, \partial_0 M)$ be the group of diffeomorphisms of $M$ that send $\partial_0 M$ onto itself. The restriction map $\text{Diff}(M, \partial_0 M) \to \text{Diff}(\partial_0 M) \cong \text{Diff}(\mathbb{S}^2)$ is a fibre bundle, by [Cer61, p. 294, §II.2.2.2, Cor. 2]. Hence its restriction $\text{Diff}_{\mathbb{D}^2 \times \mathbb{D}^2}(M) \to \text{Diff}_{\mathbb{D}^2 \times \mathbb{D}^2}(\mathbb{S}^2) \cong \text{Diff}_{\partial C}(C)$ is also a fibre bundle, where the subscript $\mathbb{D}^2 \times \mathbb{D}^2$ means that diffeomorphisms must restrict to the identity on a given pair of disjoint discs in $\partial_0 M \cong \mathbb{S}^2$ and $C$ denotes the 2-dimensional cylinder $\mathbb{S}^1 \times [0, 1]$. The fibre is $\text{Diff}_{\partial_0 M}(M)$ and we obtain an exact sequence

$$ \dots \to \pi_1(\text{Diff}_{\partial C}(C)) \xrightarrow{(*)} \pi_0(\text{Diff}_{\partial_0 M}(M)) \to \pi_0(\text{Diff}_{\mathbb{D}^2 \times \mathbb{D}^2}(M)) \xrightarrow{(*)} \pi_0(\text{Diff}_{\partial C}(C)). $$

By [Gra73, Théorème 1], $\text{Diff}_{\partial C}(C)$ is contractible, and hence (*) is a bijection. $\square$

<figure><img src="image_11.png" /></figure>

Since their automorphism groups are the (extended) loop braid groups, we call $\mathcal{L}\mathbf{B}'$ and $\mathcal{L}\mathbf{B}$ respectively the extended loop braid groupoid and the (non-extended) loop braid groupoid. Also, recall (see Definition 1.38) the braided diffeomorphism group $\text{Diff}_{\text{dec}}^{\text{br}}(M, A)$, consisting of all of those (decorated) diffeomorphisms that become isotopic to the identity after forgetting $A$. Since $\pi_0(\text{Diff}_{\text{dec}}(\mathbb{D}^3, \emptyset))$ is trivial by [Cer68] (and Lemma 3.4), this condition is vacuous when $M = \mathbb{D}^3$, so we have $\mathbf{LB}'_n \cong \text{Aut}_{\mathcal{L}\mathbf{B}'}(\mathbb{D}_n^3) = \pi_0(\text{Diff}_{\text{dec}}(\mathbb{D}_n^3)) = \pi_0(\text{Diff}_{\text{dec}}^{\text{br}}(\mathbb{D}_n^3))$ and $\mathbf{LB}_n \cong \text{Aut}_{\mathcal{L}\mathbf{B}}(\mathbb{D}_n^3) = \pi_0(\text{Diff}_{\text{dec}}^{+}(\mathbb{D}_n^3)) = \pi_0(\text{Diff}_{\text{dec}}^{\text{br},+}(\mathbb{D}_n^3))$. Hence $\mathbf{LB}'_n$ and $\mathbf{LB}_n$ may be thought of as *braided mapping class groups* (although the adjective *braided* is redundant in this case).

Finally, we introduce here further generalisations of loop braid groups. We fix non-negative integers $n$ and $k$ and a partition $\lambda = \{\lambda_1; \dots; \lambda_r\} \vdash k$. Now, consider the embedding spaces $F_k(\mathbb{D}_n^3) = \text{Emb}(k, \mathbb{D}^3 \setminus n\mathbb{S}^1)$ and $\bar{U}_k(\mathbb{D}_n^3) = \text{Emb}^{\text{unl}}(k\mathbb{S}^1, \mathbb{D}^3 \setminus n\mathbb{S}^1)$, where the superscript unl
means the path-component of the embedding space corresponding to $(n + k)$-component unlinks.
We set

$$C_{\lambda}(\mathbb{D}_n^3) = F_k(\mathbb{D}_n^3)/\mathfrak{S}_{\lambda}, \quad U_{\lambda}^{+}(\mathbb{D}_n^3) = \bar{U}_k(\mathbb{D}_n^3)/\mathrm{Diff}^{+}(\lambda\mathbb{S}^1), \quad U_{\lambda}(\mathbb{D}_n^3) = \bar{U}_k(\mathbb{D}_n^3)/\mathrm{Diff}(\lambda\mathbb{S}^1), \quad (3.1)$$

where $\lambda\mathbb{S}^1 = (\lambda_1\mathbb{S}^1, \dots, \lambda_r\mathbb{S}^1)$. The first is the $\lambda$-partitioned configuration space of $k$ points in the unlink-complement $\tilde{\mathbb{D}}^3 \setminus n\mathbb{S}^1$. The middle space is the space of $\lambda$-partitioned oriented $k$-component unlinks in $\tilde{\mathbb{D}}^3 \setminus n\mathbb{S}^1$ such that the resulting $(n+k)$-component link is again trivial. The right-hand space is similar, except that the $k$-component unlinks are unoriented. By Proposition 1.42, we have isomorphisms $\pi_1(U_n^+(\mathbb{D}^3)) \cong \mathbf{LB}_n$ and $\pi_1(U_n(\mathbb{D}^3)) \cong \mathbf{LB}'_n$ and, more generally, the groups $\pi_1(U_{\lambda}^+(\mathbb{D}^3))$ and $\pi_1(U_{\lambda}(\mathbb{D}^3))$ are isomorphic to the partitioned versions $\mathbf{LB}_{\lambda}$ and $\mathbf{LB}'_{\lambda}$ respectively.

## 3.2. Examples of homological representation functors

Recall from §2.2–§2.3 that the input to construct a homological representation functor is a continuous semifunctor of the form $\tilde{F}_{(Z,G,Q)}$, for $Z$ a closed submanifold of $\mathbb{R}^d$, $G$ an open subgroup of $\mathrm{Diff}(Z)$ and $Q$ a functorial quotient of groups. (Here, $F$ refers either to the semifunctors denoted by $\mathcal{F}$ in Theorem 2.28 or the semifunctors denoted by $\mathfrak{F}$ in Theorem 2.62.) Throughout §3.2, we consider an integer $k \ge 1$, which will determine the closed submanifold $Z \subset \mathbb{R}^d$, as well as a partition $\lambda = \{\lambda_1; \dots; \lambda_r\} \vdash k$, which will determine the group $G$. We denote by $r'$ the number of indices $1 \le i \le r$ such that $\lambda_i \ge 2$.

Furthermore, we confine our study throughout §3.2 to the homological representation functors with the following restrictions on the inputs of the construction of Definition 2.20.

* For conclusion, we mainly present the “open” versions $\tilde{F}_{(Z,G,Q)}$ rather than the “closed” variants $F_{(Z,G,Q)}$ (see §2.2.1.3 and its analogue in §2.3.1), although everything in §3.2 repeats verbatim using the closed variants $F_{(Z,G,Q)}$.

* We always take the lower central series functorial quotients of groups $Q_{\Gamma_l}$: $\mathrm{Grp} \to \mathrm{Grp}$ (see Example 2.23) for the parameter $Q$, and we simplify the indexing notation of $\tilde{F}$ using $l$ instead of $Q_{\Gamma_l}$. We choose to focus on constructions with these functorial quotients of groups since the lower central series of (partitioned) surface braid groups and loop braid groups are well-understood (see for example [DPS22]), which helps us to understand the specific examples of homological representations fitting into this framework. We recall from Example 2.60 that the *untwisted* variant semifunctor $\tilde{F}_{(Z,G,l)}^u$ is equal to $\tilde{F}_{(Z,G,l)}$ for the parameter $l \in \{1, 2\}$.

* For simplicity, we always take the continuous semifunctor $V: C \to \bullet\mathrm{Mod}_\bullet$ in the inputs for Definition 2.20 to be equal to the colimit coefficient system $V = V_{\mathrm{col}}(\tilde{F}_{(Z,G,l)}^u)$ associated to $\tilde{F}_{(Z,G,l)}^u$ (see §2.2.3 and §2.3.3). This allows us to apply Propositions 2.58, 2.59 and 2.65 to deduce that the homological representation functors that we construct take values in $\mathrm{Mod}_{\mathbb{Z}[Q]}^{\mathrm{tw}}$ or $\mathrm{Mod}_{\mathbb{Z}[Q]}$ for a fixed transformation group $Q$. We recall that the colimit coefficient system $V_{\mathrm{col}}(\tilde{F}_{(Z,G,l)}^u)$ and the transformation group $Q$ are determined by the semifunctor $\tilde{F}_{(Z,G,l)}^u$ and the choice of subcategory $\langle G, M \rangle \subseteq \mathcal{U}\mathcal{D}_d$ to which we have restricted it.

* The sequence of construction in each setting is thus as follows: we select the parameters $k \ge 1$, $\lambda \vdash k$ and $l \ge 1$, determining the data $Z$, $G$ and $Q$ respectively; we restrict the continuous semifunctor $\tilde{F} = \tilde{F}_{(Z,G,l)}^u$ of Theorem 2.28 or Theorem 2.62 to a subcategory $\langle G, M \rangle \subseteq \mathcal{U}\mathcal{D}_d$; this determines the colimit coefficient system $V_{\mathrm{col}}(\tilde{F})$ by §2.2.3 or §2.3.3; the data of $\tilde{F}$ and $V_{\mathrm{col}}(\tilde{F})$ then determine a homological representation functor $L_i(\tilde{F}; V_{\mathrm{col}}(\tilde{F}))$ for each $i \ge 0$.

### 3.2.1. Motion groups

We apply the general construction of §2.2 to some families of motion groups (see Definition 1.39). Namely, in §3.2.1.1–§3.2.1.3, we study examples of the homological representation functors introduced in §2.2, defined on the subcategories of $\mathcal{U}\mathcal{D}_2$ and $\mathcal{U}\mathcal{D}_3$ that are relevant for classical braid groups (see §3.2.1.1), surface braid groups (see §3.2.1.2) and loop braid groups (see §3.2.1.3). These subcategories are introduced in §3.1.2 and §3.1.3.

#### 3.2.1.1. Classical braid groups

We denote by $\tilde{F}_{(k, \mathfrak{S}_{\lambda}, l)}(\mathbb{D})$ and $\tilde{F}_{(k, \mathfrak{S}_{\lambda}, l)}^u(\mathbb{D})$ the restrictions to the subcategory $\mathcal{U}\mathcal{B}r_{\mathbb{D}^2} \subset \mathcal{U}\mathcal{D}_2$ of the continuous semifunctors of Theorem 2.28, where we take $Z := k$ a finite set of $k \ge 1$ points,
G := S_λ and Q := Q_{Γ_ℓ} for some ℓ ≥ 1. Recall from §3.1.2 that the automorphism groups of BΓ<sup>2</sup> are the braided diffeomorphism groups Diff<sub>dec</sub><sup>br</sup>(D<sup>2</sup>, n) (see Definition 1.38), so it is a motion groupoid (see Definition 2.54). Also, by definition of the braid groupoid β (see §3.1.2), the poset P(β, β) (see Definition 2.49) is isomorphic to the poset of natural numbers N with its usual order, so in particular it is a directed set. Therefore, using the colimit coefficient systems (2.31) associated to the semifunctors under consideration, we obtain from Corollary 2.27, Proposition 2.58 and Proposition 2.59 the following homological representation functors for all i ≥ 0:

$$
L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(\mathbb{D})}): \mathfrak{U}\beta \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\lambda, \ell)(\mathbb{D})}]}^{\mathrm{tw}} \quad \text{and} \quad L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(\mathbb{D})}^{\mathrm{u}}): \mathfrak{U}\beta \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\lambda, \ell)(\mathbb{D})}^{\mathrm{u}}]}.
\quad (3.2)
$$

Here, $\mathcal{Q}_{(\lambda,\ell)}(\mathbb{D})$ and $\mathcal{Q}_{(\lambda,\ell)}^{\mathrm{u}}(\mathbb{D})$ denote the colimit group $\mathcal{Q}_{\mathrm{col}}(\mathcal{R} \circ \mathcal{F})$ (Notation 2.53) associated to $\mathcal{F} = \mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(\mathbb{D})}$ and $\mathcal{F} = \mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(\mathbb{D})}^{\mathrm{u}}$ respectively. For $\ell = 2$ and the trivial partition $\lambda = \{k\}$, the representations arising from the functors (3.2) are related to the well-known families of Lawrence-Bigelow representations, originally introduced by Lawrence [Law90] as representations of Hecke algebras and recovered by Bigelow [Big01, §2] following a more geometric method. Namely, they define a $\mathbf{B}_n$-representation $\mathfrak{L}\mathfrak{B}_k(n)$ for each $k \ge 1$ and $n \ge 0$, called the $k$-th Lawrence-Bigelow representation. The most famous among these are the Burau representations, first introduced in [Bur35], and the Lawrence-Krammer-Bigelow representations, which Bigelow [Big01, Th. 1.1] and Krammer [Kra02] independently proved to be faithful. The following result may be straightforwardly verified by unwinding the construction of §2.2 and comparing it to that of [Big01, §2].

**Theorem 3.5** (The Lawrence-Bigelow representations.) Let $k \ge 1$ and $n \ge 0$ be integers. The $\mathbf{B}_n$-representation encoded by the functor $L_k(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,2})(\mathbb{D})})$, i.e. (3.2) with $i = k = \lambda$ and $\ell = 2$, is isomorphic to the Lawrence-Bigelow representation $\mathfrak{L}\mathfrak{B}_k(n)$.

In addition, we compute that $\mathcal{Q}_{(\lambda,2)}(\mathbb{D}) \cong \mathbb{Z}^{r'} \times \mathbb{Z}^{r(r-1)/2} \times \mathbb{Z}^r$; see Lemma 4.3. As far as the authors know, there are no representations of the braid groups in the literature whose ground rings are of the form $\mathbb{Z}[\mathbb{Z}^{r'} \times \mathbb{Z}^{r(r-1)/2} \times \mathbb{Z}^r]$ for $r \ge 2$. Hence the functors (3.2) for $\ell = 2$ appear to define new representations of the braid groups.

Furthermore, if $\lambda$ is a partition of the form $\{2; \lambda'\}$ or $\{1; 1; 1; \lambda''\}$ where $\lambda'$ and $\lambda''$ are any partitions of $k-2$ and $k-3$ respectively, we prove in the sequel [PS22b] that $L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(\mathbb{D})}) \neq L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell+1})(\mathbb{D})})$ for each $\ell \ge 2$; see [PS22b, Table 2]. The transformation group $\mathcal{Q}_{\{2; \lambda'\}, \ell}(\mathbb{D})$ is computed in Proposition 4.5 for $\ell \ge 2$ when the partition $\lambda' \vdash (k-2)$ is such that $\lambda'_l \ge 3$ for all $1 \le l \le r-1$. Also, one may easily deduce from [Big01, Th. 1.1] that each functor $L_k(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{2; \lambda' \ell})(\mathbb{D})})$ for $\ell \ge 2$ encodes a faithful representation of $\mathbf{B}_n$ for each $n$; see [PS22b, Rem. 4.8].

3.2.1.2. Surface braid groups

We consider the continuous semifunctors $\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}$ and $\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\mathrm{u}}$ of Theorem 2.28 where we take $Z := \underline{k}$ a finite set of $k \ge 1$ points, $G := \mathfrak{S}_{\lambda}$ and $Q := Q_{\Gamma_{\ell}}$ for some $\ell \ge 1$. Let $S$ be a compact, connected surface with boundary, different from the 2-disc: it is therefore homeomorphic to either $\Sigma_{g,1}$ or $\mathcal{N}_{h,1}$ for $g$ or $h \ge 1$. We denote by $\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(S)$ and $\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\mathrm{u}}(S)$ the restrictions of these semifunctors to the subcategory $\langle \mathfrak{B}\Gamma^{\mathrm{D}^2}, \mathfrak{B}r^S \rangle \subset \mathfrak{U}\mathcal{D}_2$. Recall from §3.1.2 that the automorphism groups of $\beta^S$ are the braided diffeomorphism groups $\mathrm{Diff}_{\mathrm{dec}}^{\mathrm{br}}(S, \underline{n})$ (see Definition 1.38), so it is a motion groupoid (see Definition 2.54). Also, by definition of the groupoid $\beta^S$ (see §3.1.2), the poset $\mathcal{P}(\beta, \beta^S)$ (see Definition 2.49) is isomorphic to the poset of natural numbers $\mathbb{N}$ with its usual order, so in particular it is a directed set. By Corollary 2.27, Proposition 2.58 and Proposition 2.59, using the colimit coefficient systems (2.31) associated to these semifunctors, we have the following homological representation functors for all $i \ge 0$:

$$
L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(S)}): \langle \beta, \beta^S \rangle \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\lambda, \ell)(S)}]}^{\mathrm{tw}} \quad \text{and} \quad L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(S)}^{\mathrm{u}}): \langle \beta, \beta^S \rangle \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\lambda, \ell)(S)}^{\mathrm{u}}]}.
\quad (3.3)
$$

Here, $\mathcal{Q}_{(\lambda,\ell)}(S)$ and $\mathcal{Q}_{(\lambda,\ell)}^{\mathrm{u}}(S)$ denote the colimit group $\mathcal{Q}_{\mathrm{col}}(\mathcal{R} \circ \mathcal{F})$ (Notation 2.53) associated to
$\mathcal{F} = \mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(S)}$ and $\mathcal{F} = \mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)(S)}^{\mathrm{u}}$ respectively. For $\ell \in \{2, 3\}$, we compute that $\mathcal{Q}_{(\lambda,2)}(S) \cong$
($\mathbb{Z}/2)^{r'} \times H_1(S; \mathbb{Z})^r$ (see Lemma 4.3), while, when the partition $\lambda \vdash k$ is such that $\lambda_l \ge 3$ for all $1 \le l \le r$, the transformation groups $\mathcal{Q}_{(\lambda,3)}(\Sigma_{g,1})$, $\mathcal{Q}_{(\lambda,3)}^u(\Sigma_{g,1})$, $\mathcal{Q}_{(\lambda,3)}(\mathcal{N}_{h,1})$ and $\mathcal{Q}_{(\lambda,3)}^u(\mathcal{N}_{h,1})$ are calculated in Proposition 4.5 (see (4.3) and (4.4)). Apart from one specific case when $\ell = 3$ detailed in Example 3.6 below, there are no representations of surface braid groups in the literature whose ground rings coincide with those of the functors (3.3) for $\ell \in \{2,3\}$, which thus appear to be new.

**Example 3.6** (The An-Ko representations.) Let us focus on the closed variant $\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,3})}(\Sigma_{g,1})$ for orientable surfaces. Since it takes values in $\text{Cov}_{\bullet}^{\text{pr}}$ by Lemma 2.47, we may use Borel-Moore homology in the construction of Definition 2.20 and we denote by $L_k^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,3})}(\Sigma_{g,1}))$ the corresponding homological representation functor. This functor defines representations of $\mathbf{B}_n(\Sigma_{g,1})$ for any $k \ge 1$ and $n \ge 0$. The procedure to construct these representations may be seen as a reinterpretation following Bellingeri, Godelle and Guaschi [BGG17] of the work of An and Ko [AK10], who extend some homological representations from the classical braid groups to the surface braid groups. Namely, in [AK10, §3.A], the transformation group $\mathcal{Q}_{(k,3)}(\Sigma_{g,1})$ and the quotient $\mathbf{B}_{k,n}(\Sigma_{g,1})/\Gamma_3$ are introduced in a completely different way, as abstract groups satisfying certain technical homological constraints. Then, for $k \ge 3$, [BGG17, §4] redefine these groups via the third lower central quotient of $\mathbf{B}_{k,n}(\Sigma_{g,1})$ and prove that they are isomorphic to those of [AK10, §3.A].

For $k \ge 3$, one may straightforwardly check from the definitions that the $k$-th An-Ko representation [AK10, Th. 3.2] of $\mathbf{B}_n(\Sigma_{g,1})$ is the tensor product $L_k^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,3})}(\Sigma_{g,1}))(n) \otimes_{\mathbb{Z}[\mathcal{Q}_{(k,3)}(\Sigma_{g,1})]} \mathbb{Z}[\mathbf{B}_{k,n}(\Sigma_{g,1})/\Gamma_3]$. The case of $k \le 2$ is trickier: a careful analysis of the definitions of [AK10, §3.A] shows, using the canonical map $\mathcal{Q}_{(k,3)}(\Sigma_{g,1}) \to \mathcal{Q}_{(3,3)}(\Sigma_{g,1}) \to \mathbf{B}_{3,n}(\Sigma_{g,1})/\Gamma_3$, that for $k \le 2$ the $k$-th An-Ko representation [AK10, Th. 3.2] corresponds to $L_k^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,3})}(\Sigma_{g,1}))(n) \otimes_{\mathbb{Z}[\mathcal{Q}_{(k,3)}(\Sigma_{g,1})]} \mathbb{Z}[\mathbf{B}_{3,n}(\Sigma_{g,1})/\Gamma_3]$. The general method applied in this section thus elucidates the geometric origins of these groups, moreover proving that the use of the third lower central quotient is a key tool in defining the homological representations. Our framework also gives an alternative to the technical result [AK10, Lem. 3.1] to justify that the representations are well-defined. In contrast, the representations encoded by the functors $L_i^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,3})}(S))$ for any non-trivial partition $\lambda \vdash k$, as well as all of the untwisted versions $L_i^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,3})}^u(S))$, appear to be new.

In addition, if $\lambda$ is a partition of the form $\{1; \lambda'\}$ or $\{2; \lambda''\}$ where $\lambda'$ and $\lambda''$ are any partitions of $k-1$ and $k-2$ respectively, and for $S \in \{\Sigma_{g,1}, \mathcal{N}_{h,1} | g \ge 1, h \ge 2\}$, the sequel [PS22b] proves that $L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,\ell})(S)}) \ne L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{k,\ell+1})(S)})$ for each $\ell \ge 2$; see [PS22b, Prop. 5.2]. Therefore, for the aforementioned reasons, the functors (3.2) for $\ell \ge 2$ appear, in general, to define new representations of the braid groups on the surfaces $\Sigma_{g,1}$ and $\mathcal{N}_{h,1}$.

### 3.2.1.3. Loop braid groups

To apply the construction of §2.2 to extended loop braid groups, we consider restrictions of the continuous semifunctor $\mathcal{F}_{(Z,\mathcal{G},\ell)}: \mathcal{U}\mathcal{D}_3 \to \text{Cov}_{\bullet}$ of Theorem 2.28, with $Q := Q_{\Gamma_\ell}$ for some $\ell \ge 1$, to the subcategory $\mathcal{U}\mathcal{L}\mathcal{B}' \subset \mathcal{U}\mathcal{D}_3$ (see §3.1.3). For non-extended loop braid groups, we instead consider restrictions to $\mathcal{U}\mathcal{L}\mathcal{B} \subset \mathcal{U}\mathcal{D}_3^+$ (see §3.1.3) of the continuous semifunctor $\mathcal{F}_{(Z,\mathcal{G},\ell)}: \mathcal{U}\mathcal{D}_3^+ \to \text{Cov}_{\bullet}$: see Theorem 2.28. The automorphism groups of $\mathcal{L}\mathcal{B}'$ and $\mathcal{L}\mathcal{B}$ are the (braided) diffeomorphism groups $\text{Diff}_{\text{dec}}^{\text{br}}(\mathbb{D}_n^3)$ and $\text{Diff}_{\text{dec}}^{\text{br,+}}(\mathbb{D}_n^3)$ respectively; see §3.1.3. Hence $\mathcal{L}\mathcal{B}'$ and $\mathcal{L}\mathcal{B}$ are motion groupoids (see Definition 2.54). It also follows from the definitions that the posets $\mathcal{P}(\mathcal{L}\mathcal{B}', \mathcal{L}\mathcal{B}')$ and $\mathcal{P}(\mathcal{L}\mathcal{B}, \mathcal{L}\mathcal{B})$ (see Definition 2.49) are both isomorphic to the poset of natural numbers $\mathbb{N}$ with its usual order, so in particular they are directed sets. Now, two choices for the submanifold $Z \subset \mathbb{R}^3$ naturally arise as relevant inputs to construct homological representations: a set of points or an unlink.

**Using configurations of points.** We take $Z = \underline{k}$ a set of $k \ge 1$ points in $\mathbb{R}^3$ and $\mathcal{G} = \mathfrak{S}_\lambda$. We consider the restrictions of the continuous semifunctors of Theorem 2.28 to the subcategories $\mathcal{U}\mathcal{L}\mathcal{B} \subset \mathcal{U}\mathcal{D}_3^+$ and $\mathcal{U}\mathcal{L}\mathcal{B}' \subset \mathcal{U}\mathcal{D}_3$. Using the colimit coefficient systems (2.31) associated to these semifunctors, by Corollary 2.27, Proposition 2.58 and Proposition 2.59, we have the following homological representation functors for all $i \ge 0$:

$$ \begin{align} L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell})(\mathbb{D}^3)}): \mathcal{U}\mathcal{L}\beta &\to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(P,\lambda,\ell)}(\mathbb{D}^3)]}^{\text{tw}}; \quad L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell})(\mathbb{D}^3)}^u): \mathcal{U}\mathcal{L}\beta &\to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(P,\lambda,\ell)}^u(\mathbb{D}^3)]} \\ L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell})(\mathbb{D}^3)}': \mathcal{U}\mathcal{L}\beta') &\to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(P,\lambda,\ell)}(\mathbb{D}^3)]}^{\text{tw}}; \quad L_i(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell})(\mathbb{D}^3)}^{\text{u}}): \mathcal{U}\mathcal{L}\beta' &\to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(P,\lambda,\ell)}^{\text{u}}(\mathbb{D}^3)]} \tag{3.4} \end{align} $$
Here, $\mathcal{Q}_{(P,\lambda,\ell)}(\mathbb{D}^3)$ and $\mathcal{Q}_{(P,\lambda,\ell)}^u(\mathbb{D}^3)$ denote the colimit group $\mathcal{Q}_{\text{col}}(\mathbb{R} \circ \mathcal{F})$ (Notation 2.53) associated to $\mathcal{F} = \mathcal{F}_{(\underline{k},\mathfrak{S},\lambda,\ell)}(\mathbb{D}^3)$ and $\mathcal{F} = \mathcal{F}_{(\underline{k},\mathfrak{S},\lambda,\ell)}^u(\mathbb{D}^3)$ respectively, and similarly for the versions with $'$.
The symbol 'P' here simply indicates that we are in the setting where $Z$ is a set of points. For $\ell = 2$, we compute that $\mathcal{Q}_{(P,\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^r \times (\mathbb{Z}/2)^{r'}$ and $\mathcal{Q}_{(P,\lambda,2)}'(\mathbb{D}^3) \cong (\mathbb{Z}/2)^{r+r'}$; see Lemma 4.7. Among the representations introduced by (3.4), the only ones that we currently understand in detail are those defined with the parameters $i = k = 1$ and $\ell = 2$:

**Example 3.7 (The loop Burau representations.)** In [PS22a], we explicitly compute the matrices of the representations encoded by the functors $L_1(\mathcal{F}_{(\underline{1},0,2)}(\mathbb{D}^3))$ and $L_1(\mathcal{F}_{(\underline{1},0,2)}'(\mathbb{D}^3))$: these extend the Burau representations of the classical braid groups to $\mathbf{LB}_n$ and $\mathbf{LB}_n'$ respectively. To describe these calculations, we recall that the loop braid group $\mathbf{LB}_n$ admits a presentation given by generators $\{\sigma_i, \tau_i | 1 \le i \le n-1\}$, where $\{\sigma_1, \dots, \sigma_{n-1}\}$ satisfy the relations of the classical braid group $\mathfrak{B}_n$ and $\{\tau_1, \dots, \tau_{n-1}\}$ satisfy those of the symmetric group $\mathfrak{S}_n$, together with three additional mixed relations; see [Dam17, Prop. 3.14 and 3.16] for details. We show in [PS22a] that the matrices of the representations $L_1(\mathcal{F}_{(\underline{1},0,2)}(\mathbb{D}^3))(n): \mathbf{LB}_n \to \text{Aut}_{\mathbb{Z}[\mathbb{Z}]}(\mathbb{Z}[\mathbb{Z}]^{\oplus n-1})$ are those of:

* the Burau representation $\mathfrak{L}\mathfrak{B}_1(n)$ (see §3.2.1.1) for the generators $\{\sigma_1, \dots, \sigma_{n-1}\}$;

* the standard representation of the symmetric group $\mathfrak{S}_n$ for the generators $\{\tau_1, \dots, \tau_{n-1}\}$.

The matrices for the representations $L_1(\mathcal{F}_{(\underline{1},0,2)}'(\mathbb{D}^3))(n)$ of the extended loop braid groups $\mathbf{LB}_n'$ over $R = \mathbb{Z}[\mathbb{Z}/2] \cong \mathbb{Z}[t^{\pm 1}]/(t^2 - 1)$ are more subtle, since the underlying $R$-module is not free in this case: it is $R^{\oplus n-1} \oplus R/(t-1)$. They can however be computed; see [PS22a, Table 1].

On the other hand, apart from the setting of Example 3.7, the representations of the (extended and non-extended) loop braid groups encoded by the functors (3.4) appear to be new because of their ground rings. Furthermore, for any partition $\lambda = \{2; \lambda'\}$, we prove in [PS22b, Prop. 6.2] that $L_i(\mathcal{F}_{(\underline{k},\mathfrak{S},\lambda,\ell)}(\mathbb{D}^3)) \neq L_i(\mathcal{F}_{(\underline{k},\mathfrak{S},\lambda,\ell+1)}(\mathbb{D}^3))$ for each $\ell \ge 2$.

**Using configurations of unlinks.** We now set $Z = \underline{k}\mathfrak{S}^1$ a $k$-component unlink in $\mathbb{R}^3$. For each partition $\lambda \vdash k$, we focus on two choices for the group $\mathfrak{G}$: the group $\text{Diff}(\underline{\lambda}\mathfrak{S}^1)$ of diffeomorphisms of $\underline{k}\mathfrak{S}^1$ preserving the partition $\lambda$ or its subgroup $\text{Diff}^+(\underline{\lambda}\mathfrak{S}^1)$ of orientation-preserving diffeomorphisms. Then we consider the restrictions of the continuous semifunctors of Theorem 2.28 to the full subcategories $\mathfrak{ULB} \subset \mathfrak{UD}_3^+$ and $\mathfrak{ULB}' \subset \mathfrak{UD}_3$ along with their associated colimit coefficient systems (2.31).

**The oriented version.** We take $\mathfrak{G} = \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1)$. By Corollary 2.27, Proposition 2.58 and Proposition 2.59, we have the following homological representation functors for all $i \ge 0$:

$$ 
\begin{align}
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1), \ell)}(\mathbb{D}^3)) & : \mathfrak{ULB} \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}(\mathbb{D}^3)]}^{\text{tw}}; \nonumber \\
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1), \ell)}^u(\mathbb{D}^3)) & : \mathfrak{ULB} \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}^u(\mathbb{D}^3)]}^{\text{tw}}; \nonumber \\
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1), \ell)}'(\mathbb{D}^3)) & : \mathfrak{ULB}' \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}(\mathbb{D}^3)]}^{\text{tw}}; \tag{3.5} \\
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1), \ell)}'^u(\mathbb{D}^3)) & : \mathfrak{ULB}' \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}'^u(\mathbb{D}^3)]}^{\text{tw}}. \nonumber
\end{align}
$$

The group $\mathcal{Q}_{(\underline{S},\lambda,\ell)}(\mathbb{D}^3)$ and its variants with $^u$ or $'$ are defined similarly to above, except that the symbol 'S' indicates that we are now in the setting where $Z$ is an *oriented unlink*. For $\ell = 2$, we compute that $\mathcal{Q}_{(\underline{S},\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^{r^2+r+r'} \times (\mathbb{Z}/2)^{r'}$ and $\mathcal{Q}_{(\underline{S},\lambda,2)}'(\mathbb{D}^3) \cong \mathbb{Z}^{r^2+r'} \times (\mathbb{Z}/2)^{r+r'}$; see Lemma 4.7. Moreover, for partitions of the form $\lambda = \{b; \lambda'\}$ with $b \in \{2, 3, \{1, 1\}\}$ and generically denoting both $\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1), \ell)}$ and $\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}^+(\underline{\lambda}\mathfrak{S}^1), \ell)}'$ by $\mathcal{F}_{(\lambda,\ell)}$, we prove in [PS22b, Prop. 6.2] that $L_i(\mathcal{F}_{(\lambda,\ell)}(\mathbb{D}^3)) \neq L_i(\mathcal{F}_{(\lambda,\ell+1)}(\mathbb{D}^3))$ for each $\ell \ge 2$.

**The unoriented version.** We take $\mathfrak{G} = \text{Diff}(\underline{\lambda}\mathfrak{S}^1)$. By Corollary 2.27, Proposition 2.58 and Proposition 2.59, we have the following homological representation functors for all $i \ge 0$:

$$ 
\begin{align}
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}(\underline{\lambda}\mathfrak{S}^1), \ell)}(\mathbb{D}^3)) & : \mathfrak{ULB} \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}(\mathbb{D}^3)]}^{\text{tw}}; \nonumber \\
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}(\underline{\lambda}\mathfrak{S}^1), \ell)}^u(\mathbb{D}^3)) & : \mathfrak{ULB} \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}^u(\mathbb{D}^3)]}^{\text{tw}}; \nonumber \\
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}(\underline{\lambda}\mathfrak{S}^1), \ell)}'(\mathbb{D}^3)) & : \mathfrak{ULB}' \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}(\mathbb{D}^3)]}^{\text{tw}}; \nonumber \\
L_i(\mathcal{F}_{(\underline{k}\mathfrak{S}^1, \text{Diff}(\underline{\lambda}\mathfrak{S}^1), \ell)}'^u(\mathbb{D}^3)) & : \mathfrak{ULB}' \to \text{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\underline{S},\lambda,\ell)}'^u(\mathbb{D}^3)]}^{\text{tw}}. \nonumber
\end{align}
$$
The group $\mathcal{Q}_{(\mathfrak{S},\lambda,\ell)}(\mathbb{D}^3)$ and its variants with $^u$ or $'$ are defined similarly to above, except that the symbol '$\mathfrak{S}'$ indicates that we are now in the setting where $Z$ is an *unoriented unlink*. For $\ell = 2$, we compute that $\mathcal{Q}_{(\mathfrak{S},\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^r \times (\mathbb{Z}/2)^{2r'+r(r+1)}$ and $\mathcal{Q}'_{(\mathfrak{S},\lambda,2)}(\mathbb{D}^3) \cong (\mathbb{Z}/2)^{2r'+r(r+2)}$; see Lemma 4.7. Moreover, for partitions of the form $\lambda = \{b; \lambda'\}$ with $b \in \{1,2,3\}$ and generically denoting both $\tilde{\mathcal{F}}_{(\mathfrak{k}\mathbb{S}^1, \text{Diff}(\mathfrak{S}^1), \ell)}$ and $\tilde{\mathcal{F}}'_{(\mathfrak{k}\mathbb{S}^1, \text{Diff}(\mathfrak{S}^1), \ell)}$ by $\mathcal{F}_{(\lambda,\ell)}$, we prove in [PS22b, Prop. 6.2] that $L_i(\mathcal{F}_{(\lambda,\ell)}(\mathbb{D}^3)) \neq L_i(\mathcal{F}_{(\lambda,\ell+1)}(\mathbb{D}^3))$ for each $\ell \ge 2$.

Therefore, because of the computations of their ground rings, all of the representations encoded by the functors (3.5) and (3.6) appear to be new.

### 3.2.2. Mapping class groups of surfaces

Although it is best adapted to motion groups, the general construction of §2.2 may also be used to construct representations of mapping class groups, and recovers several classical constructions. This is detailed in §3.2.2.1. We then apply in §3.2.2.2 the method of §2.3 to define other representations for mapping class groups of surfaces.

#### 3.2.2.1. Homological representations from the construction of §2.2

Let $\mathcal{U}\mathcal{D}_2^\emptyset$ denote the full subcategory of $\mathcal{U}\mathcal{D}_2$ on the decorated manifolds $(M, A)$ where $A = \emptyset$. We consider the continuous semifunctor $\tilde{\mathcal{F}}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}$ of Theorem 2.28 where $Z := \underline{k}$ is a finite set of $k \ge 1$ points, $G := \mathfrak{S}_\lambda$ and $Q := Q_{\Gamma_\ell}$ for some $\ell \ge 1$, which we first restrict to $\mathcal{U}\mathcal{D}_2^\emptyset$. We note that, on this subcategory, the short exact sequences of the diagram (2.24) degenerate in the sense that the right-hand side is the trivial group. Restricting further, we then denote by $\tilde{\mathcal{F}}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}(\Gamma)$ and $\tilde{\mathcal{F}}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}(\mathcal{N})$ the restrictions of this semifunctor to the subcategories $\mathcal{U}\mathcal{M}_2^{+\mathrm{,t}}$ and $\mathcal{U}\mathcal{M}_2^{-\mathrm{,t}}$ of $\mathcal{U}\mathcal{D}_2^\emptyset$ (see §3.1.1). Note that, by definition of the groupoids $\mathcal{M}_2^+$ and $\mathcal{M}_2^-$, the posets $\mathcal{P}(\mathcal{M}_2^+, \mathcal{M}_2^+)$ and $\mathcal{P}(\mathcal{M}_2^-, \mathcal{M}_2^-)$ (see Definition 2.49) are both isomorphic to the poset of natural numbers $\mathbb{N}$ with its usual order, so in particular they are directed sets. Hence, using the colimit coefficient systems (2.31) associated to the semifunctors under consideration, we obtain from Corollary 2.27 and Proposition 2.58 the following homological representation functors for all $i \ge 0$:

$$ L_i(\tilde{\mathcal{F}}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}(\Gamma)) : \mathcal{U}\mathcal{M}_2^+ \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\lambda,\ell)}^{\mathcal{F}}(\Gamma)]}^{\mathrm{tw}} \quad \text{and} \quad L_i(\tilde{\mathcal{F}}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}(\mathcal{N})) : \mathcal{U}\mathcal{M}_2^- \to \mathrm{Mod}_{\mathbb{Z}[\mathcal{Q}_{(\lambda,\ell)}^{\mathcal{F}}(\mathcal{N})]}^{\mathrm{tw}} \quad (3.7) $$

**Remark 3.8** Since the groupoids $\mathcal{M}_2^+$ and $\mathcal{M}_2^-$ are not motion groupoids, Proposition 2.59 does not apply in this setting, so we cannot deduce that the $\tilde{\mathcal{F}}^u$ variants of Theorem 2.28 lead to untwisted versions of (3.7); thus we do not consider these variants here. This issue is precisely what is solved in §3.2.2.2 by applying the construction of §2.3 instead of that of §2.2.

The transformation groups $\mathcal{Q}_{(\lambda,\ell)}^{\mathcal{F}}(\Gamma)$ and $\mathcal{Q}_{(\lambda,\ell)}^{\mathcal{F}}(\mathcal{N})$ in (3.7) are defined (via taking a colimit) from the $\ell$-th lower central quotients $\mathbf{B}_\lambda(\Sigma_{g,1})/\Gamma_\ell$ or $\mathbf{B}_\lambda(\mathcal{N}_{h,1})/\Gamma_\ell$ for all $g,h \ge 1$ respectively. These lower central quotients are computed in [DPS22, §6.5–§6.6], from which one may deduce complete calculations of all of these transformation groups.

A simple modification of the above construction of (3.7) consists in replacing the lower central quotient $Q = Q_{\Gamma_\ell}$ with the identity quotient $Q = \mathrm{id}_{\mathrm{Grp}}$. The corresponding functors encode the representations given by the natural actions on the homology of the universal covers of $\lambda$-partitioned configuration spaces on the surfaces. In particular, it is easy to check from the definitions that the functor corresponding to $\lambda = \{1\}$ and $Q = \mathrm{id}_{\mathrm{Grp}}$ encodes the (reduced) *Magnus representations* of the mapping class groups, whose topological interpretation was introduced in [Suz05].

Another interesting modification of the construction of (3.7) consists in removing the basepoint $p_0 \in \partial S$ from the surface and allowing the configuration points to lie in the boundary of the surface. Namely, we replace each surface $S$ by the surface $S \setminus \{p_0\}$ (formally, this means slightly changing the domain category of the homological representation functors, but its automorphism groups are the same up to isomorphism) and we use the closed variant $\mathcal{F}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}$ of §2.2.1.3 instead of $\tilde{\mathcal{F}}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}$, so that Borel-Moore homology may be applied to this variant by Lemma 2.47. In this way we define homological representation functors $L_i^{\mathrm{BM}}(\mathcal{F}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}(\Gamma))$ and $L_i^{\mathrm{BM}}(\mathcal{F}_{(\underline{k},\mathfrak{S}_\lambda,\ell)}(\mathcal{N}))$ analogous to those of (3.7). These alternatives have the advantage of encoding representations endowed with natural free generating sets; see [PS23, §2.2]. In particular, we make here the connection between
the representations arising from the functor $L_k^{\text{BM}}(\mathcal{F}_{(\underline{k},0,1)}(\Gamma))$ (i.e. $i = k$, $\lambda = \{1^k\}$ and $\ell = 1$) with those introduced by Moriyama in [Mor07], whose kernels are the $k$-th terms of the Johnson filtration. More precisely, for each $g$, the $\Gamma_{g,1}$-representation introduced by [Mor07] is given by the $\Gamma_{g,1}$-action on the relative homology group $H_k(\Sigma_{g,1}^{\times k}, \Delta \cup A_g; \mathbb{Z})$, where $\Delta$ denotes the “fat diagonal” of $\Sigma_{g,1}^{\times k}$ where at least two points coincide and $A_g$ denotes the subspace of $\Sigma_{g,1}^{\times k}$ where at least one point is equal to $p_0$, a chosen basepoint on $\partial\Sigma_{g,1}$.

**Proposition 3.9** *The restriction of the functor $L_k^{\text{BM}}(\mathcal{F}_{(\underline{k},0,1)}(\Gamma))$ to the $g$-th mapping class group $\Gamma_{g,1}$ is the $k$-th Moriyama representation $\Gamma_{g,1} \to \text{Aut}_{\mathbb{Z}}(H_k(\Sigma_{g,1}^{\times k}, \Delta \cup A_g; \mathbb{Z}))$.*

*Proof.* We write $\Sigma'_{g,1} = \Sigma_{g,1} \setminus \{p_0\}$ and $F_k(\Sigma'_{g,1})$ for $C_{\{1^k\}}(\Sigma'_{g,1})$. Since $\Sigma_{g,1}^{\times k}$ is a compactification of $F_k(\Sigma'_{g,1}) = \Sigma_{g,1}^{\times k} \setminus (\Delta \cup A_g)$, the Borel-Moore homology of $F_k(\Sigma'_{g,1})$ is isomorphic to the relative homology $H_*(\Sigma_{g,1}^{\times k}, \Delta \cup A_g; \mathbb{Z})$. That this isomorphism is $\Gamma_{g,1}$-equivariant follows from the fact that the $\Gamma_{g,1}$-actions on $H_*^{\text{BM}}(F_k(\Sigma'_{g,1}); \mathbb{Z})$ and $H_*(\Sigma_{g,1}^{\times k}, \Delta \cup A_g; \mathbb{Z})$ are both induced from the diagonal action of $\Gamma_{g,1}$ on $\Sigma_{g,1}^{\times k}$. $\square$

On the other hand, the other representations encoded by the homological representation func-
tors (3.7), as well as their variants $L_i^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(\Gamma))$ and $L_i^{\text{BM}}(\mathcal{F}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(\mathcal{N}))$, do not appear in
the literature and thus seem to be new.

**3.2.2.2. Homological representations from the construction of §2.3**

As pointed out in Remark 3.8, the construction of §2.2 typically does not lead to untwisted
representations when applied to (full) mapping class groups, since Proposition 2.59 requires the
groupoid $\mathcal{M}$ to be a motion groupoid (its automorphism groups must be braided mapping class
groups). The purpose of the alternative construction in §2.3 is that it solves this problem: the
analogue of Proposition 2.59 in this setting is Proposition 2.65, which does not assume that $\mathcal{M}$ is
a motion groupoid. In this section we illustrate this alternative construction with some examples
in the context of mapping class groups of surfaces.

Let us consider the continuous semifunctors $\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}$ and $\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\text{u}}$ of Theorem 2.62, where $Z := \underline{k}$ is a finite set of $k \ge 1$ points, $G := \mathfrak{S}_{\lambda}$ and $Q := Q_{\Gamma_{\ell}}$ for some $\ell \ge 1$. We denote the restrictions of these semifunctors to the subcategories $\mathcal{U}\mathcal{M}_2^{+ ,t}$ and $\mathcal{U}\mathcal{M}_2^{- ,t}$ of $\mathcal{U}\mathcal{D}_2$ (see §3.1.1) by $\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(\Gamma)$, $\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\text{u}}(\Gamma)$, $\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(\mathcal{N})$ and $\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\text{u}}(\mathcal{N})$ respectively. Recall from §3.2.2.1 that the posets $\mathcal{P}(\mathcal{M}_2^{+}, \mathcal{M}_2^{+})$ and $\mathcal{P}(\mathcal{M}_2^{-}, \mathcal{M}_2^{-})$ are both directed sets. Using the appropriate colimit coefficient system $V_{\text{col}}(\mathcal{F})$ of §2.3.3 in each case, we therefore obtain from Corollary 2.64, Proposition 2.58 and Proposition 2.65 the following homological representation functors for all $i \ge 0$:

$$
\begin{equation} \tag{3.8}
\begin{aligned}
L_i(\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(\Gamma)) : \mathcal{U}\mathcal{M}_2^+ &\to \operatorname{Mod}_{\mathbb{Z}[\mathcal{Q}^{\check{\mathfrak{S}}_{(\lambda, \ell)}}(\Gamma)]}^{\text{tw}}; &
L_i(\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\text{u}}(\Gamma)) : \mathcal{U}\mathcal{M}_2^+ &\to \operatorname{Mod}_{\mathbb{Z}[\mathcal{Q}^{u, \check{\mathfrak{S}}_{(\lambda, \ell)}}(\Gamma)]} \\
L_i(\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}(\mathcal{N})) : \mathcal{U}\mathcal{M}_2^- &\to \operatorname{Mod}_{\mathbb{Z}[\mathcal{Q}^{\check{\mathfrak{S}}_{(\lambda, \ell)}}(\mathcal{N})]}^{\text{tw}}; &
L_i(\check{\mathfrak{S}}_{(\underline{k}, \mathfrak{S}_{\lambda}, \ell)}^{\text{u}}(\mathcal{N})) : \mathcal{U}\mathcal{M}_2^- &\to \operatorname{Mod}_{\mathbb{Z}[\mathcal{Q}^{u, \check{\mathfrak{S}}_{(\lambda, \ell)}}(\mathcal{N})]}.
\end{aligned}
\end{equation}
$$

In contrast with §3.2.2.1, the representations encoded by the functors (3.8) for $\ell \ge 2$ are more
novel. For instance, for $\ell = 2$, we have $\mathcal{Q}^{\check{\mathfrak{S}}_{(\lambda,2)}}(\Gamma) \cong (\mathbb{Z}/2)^{r'}$ and $\mathcal{Q}^{\check{\mathfrak{S}}_{(\lambda,2)}}(\mathcal{N}) \cong (\mathbb{Z}/2)^{r'} \times (\mathbb{Z}/2)^r$;
see Corollary 4.9. As far as the authors know, there are no representations of the mapping class
groups $\Gamma_{g,1}$ and $\mathcal{N}_{h,1}$ in the literature whose ground rings are of the above form. Therefore, many
new representations of the mapping class groups arise from the functors (3.8).

**4. Appendix: computations of transformation groups**

This appendix aims to study some of the transformation groups (i.e. the groups whose group
rings are the ground rings) of the homological representation functors that we construct in §3, in
order to offer a more concrete understanding of these representations. We first deal with some
necessary recollections of presentations of surface braid groups in §4.1. Then we compute and
present some properties of the transformation groups of the homological representation functors
of §3 in §4.2.
## 4.1. Presentations of surface braid groups

Presentations of braid groups on surfaces with one boundary component may be found in [HL02, §4] and in [Bel04, Th. 1.1 and A.2]; see also [DPS22, §6.3]. We fix three integers $k \ge 0$, $g \ge 0$ and $h \ge 1$. We shall use the same notation $\Sigma_{g,1}^k$ and $\mathcal{N}_{h,1}^k$ for surfaces as in §3.1.1. In the following presentations, we write $x \rightsquigarrow y$ to denote the relation saying that $x$ and $y$ commute.

**Proposition 4.1** The braid group on $n$ strands on the orientable surface $\Sigma_{g,1}^k$, denoted by $\mathbf{B}_n(\Sigma_{g,1}^k)$, admits the presentation with generators $S = \{\sigma_i\}_{1 \le i \le n-1}$, $A = \{a_i\}_{1 \le i \le g}$, $B = \{b_i\}_{1 \le i \le g}$ and $X = \{\xi_i\}_{1 \le i \le k}$ and relations given by the braid relations for the elements of $S$, to which are added the following families of relations (where $x$ and $y$ denote either a or b, and $1 \le r, s \le g$):

$$ \left\{ \begin{array}{ll} (BS1) & \sigma_i \rightsquigarrow x_r \quad \text{for all } r \text{ and all } 1 \le i \le n-2; \\ (BS2) & x_r \rightsquigarrow \sigma_{n-1} y_s \sigma_{n-1}^{-1} \quad \text{for } s < r; \\ (BS3) & (\sigma_{n-1} x_r)^2 = (x_r \sigma_{n-1})^2 \quad \text{for all } r; \\ (BS4) & [\sigma_{n-1} b_r \sigma_{n-1}^{-1}, a_r^{-1}] = \sigma_{n-1}^2 \quad \text{for all } r; \\ (BS5) & \xi_j \rightsquigarrow \sigma_i \quad \text{for all } 1 \le j \le k \text{ and all } 1 \le i \le n-2; \\ (BS6) & x_r \rightsquigarrow \sigma_{n-1} \xi_j \sigma_{n-1}^{-1} \quad \text{for all } 1 \le j \le k \text{ and all } 1 \le r \le g; \\ (BS7) & \xi_i \rightsquigarrow \sigma_{n-1} \xi_j \sigma_{n-1}^{-1} \quad \text{for } i < j. \end{array} \right. \qquad (4.1) $$

The braid group on $n$ strands on the non-orientable surface $\mathcal{N}_{h,1}^k$, denoted by $\mathbf{B}_n(\mathcal{N}_{h,1}^k)$, admits the presentation with generators $S = \{\sigma_i\}_{1 \le i \le n-1}$, $C = \{c_i\}_{1 \le i \le h}$ and $X = \{\xi_i\}_{1 \le i \le k}$ and relations given by the braid relations for the elements of $S$, to which are added the following families of relations (where $1 \le r, s \le h$):

$$ \left\{ \begin{array}{ll} (BN1) & \sigma_i \leftrightarrow c_r \quad \text{for all } r \text{ and all } 1 \le i \le n-2; \\ (BN2) & c_r \leftrightarrow \sigma_{n-1} c_s \sigma_{n-1}^{-1} \quad \text{for } s < r; \\ (BN3) & [\sigma_{n-1} c_r \sigma_{n-1}^{-1}, c_r^{-1}] = \sigma_{n-1}^2 \quad \text{for all } r; \\ (BN4) & \xi_j \leftrightarrow \sigma_i \quad \text{for all } 1 \le j \le k \text{ and all } 1 \le i \le n-2; \\ (BN5) & c_r \leftrightarrow \sigma_{n-1} \xi_j \sigma_{n-1}^{-1} \quad \text{for all } 1 \le j \le k \text{ and all } 1 \le r \le h; \\ (BN6) & \xi_i \leftrightarrow \sigma_{n-1} \xi_j \sigma_{n-1}^{-1} \quad \text{for } i < j; \\ (BN7) & (\sigma_{n-1} \xi_j)^2 = (\xi_j \sigma_{n-1})^2 \quad \text{for all } 1 \le j \le k. \end{array} \right. \qquad (4.2) $$

Now we consider a partition $\lambda = \{\lambda_1; \dots; \lambda_r\} \vdash k$. There is an isomorphism $\mathbf{B}_\lambda(S) \cong \mathbf{B}_{\{\lambda_1; \dots; \lambda_{r-1}\}}(\mathbb{D}_{\lambda_r}^2 \otimes S) \times \mathbf{B}_{\lambda_r}(S)$, which may be deduced from the split short exact sequence (1.7). There is a classical method of constructing a presentation of a group extension from a presentation of the quotient and a presentation of the kernel; see [HEO05, §2.4.3] and [DPS22, Appendix B]. For instance, a presentation of the group $\mathbf{B}_{k,n}(\Sigma_{g,1})$ is detailed in [BGG17, Prop. 3.2] following this method, while a presentation of $\mathbf{B}_{k,n}(\mathcal{N}_{h,1})$ is given in [DPS22, Prop. 6.58]. It is routine to generalise this to give full presentations for any partition $\lambda$. We thus obtain from Proposition 4.1 the following result for the partitioned surface braid groups.

**Proposition 4.2** Let $\lambda = \{\lambda_1; \dots; \lambda_r\}$ be a partition of $k \ge 1$. The surface braid group $\mathbf{B}_\lambda(S)$ admits a presentation whose generating sets are:

* $X^{(\rho)} = \{\xi_i^{(\rho)} | 1 \le i \le \Sigma_{\rho}\}$ with $\Sigma_{\rho} := \sum_{\rho+1 \le l \le r} \lambda_l$, for each block $1 \le \rho \le r-1$;

* $S^{(\rho')} = \{\sigma_i^{(\rho')}\}_{1 \le i \le \lambda_{\rho'} - 1}$ for each block $1 \le \rho' \le r$ such that $r_{\rho'} \ge 2$;

* if $S = \Sigma_{g,1}$: $A^{(\rho)} = \{a_i^{(\rho)}\}_{1 \le i \le g}$ and $B^{(\rho)} = \{b_i^{(\rho)}\}_{1 \le i \le g}$ for each block $1 \le \rho \le r$;

* if $S = \mathcal{N}_{h,1}$: $C^{(\rho)} = \{c_i^{(\rho)}\}_{1 \le i \le h}$ for each block $1 \le \rho \le r$.

The relations between generators of the same blocks are those of (4.1) and (4.2), while the relations between generators of different blocks are analogous to those of (c.1)-(c.8) in [BGG17, Prop. 3.2].

## 4.2. Properties of the transformation groups

This section deals with some properties of the homological representation functors of §3. Throughout §4.2, we consider an integer $\ell \ge 1$ corresponding to a lower central series index, and
an integer $k \ge 1$ and a partition $\lambda = \{\lambda_1; \dots; \lambda_r\} \vdash k$. We also denote by $r'$ the number of indices $1 \le i \le r$ such that $\lambda_i \ge 2$. For simplicity, we denote the decorated surface $(\mathbb{D}^2, \underline{n})$ for each non-negative integer $n$ by $\mathbb{D}_n$ and call it the $n$-th marked 2-disc, and we denote the decorated surface $(S, \underline{n})$ by $S^{(n)}$. If $n=0$, we abbreviate $(S, \emptyset) = S^{(0)}$ to $S$.

### 4.2.1. Surface braid groups

We continue to follow the notation of §4.1 and consider a compact, connected, smooth surface $S$ with one boundary component. We deal here with the computation of the transformation groups of the homological representation functors of the form $L_i(\tilde{\mathcal{F}}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell})(S)})$ for surface braid groups defined in §3.2.1.1–§3.2.1.2. We denote by $\mathcal{Q}_{(\lambda, \ell)}(S^{(n)})$ the group of the form $\mathcal{Q}(S, \underline{n})$ induced by diagram (2.24) for each integer $n \ge 0$ and by $\mathcal{Q}_{(\lambda, \ell)}(S)$ the colimit of these groups as $n \to \infty$, which is the colimit transformation group $\mathcal{Q}_{\text{col}}(T)$ of Notation 2.53 in this setting.

**Abelian quotients.** We start with the homological representation functor defined using the $\Gamma_2$ term of the lower central series. The abelianisations of the groups $\mathbf{B}_{\lambda,n}(S)$ are explicitly computed in [DPS22, Prop. 3.5 and 6.47], and may be described via the corresponding generating set of Proposition 4.2. In particular, a generating set for $\mathbf{B}_{\lambda,n}(S)^{ab}$ is given by:

* the common image $t_{i'}$ of the generators of $S^{(i')}$ for each $1 \le i' \le r$ such that $r_{i'} \ge 2$;

* if $S = \mathbb{D}$: the common image $q_i$ of all $\xi_j^{(i)} \in X^{(i)}$ with $j \ge 1 + \sum_{i+1 \le l \le r} \lambda_l$ for each $1 \le i \le r$, and the common image $s_{i_1, i_2}$ of all $\xi_j^{(i_1)}$ with $j \in \{j' + \sum_{i_1+1 \le l \le i_2-1} \lambda_l \mid 1 \le j' \le i_2\}$ for each pair $1 \le i_1 < i_2 \le r$;

* if $S = \Sigma_{g,1}$: the images $\{A_j^{(i)}, B_j^{(i)}\}_{1 \le j \le g}$ of the sets $A^{(\rho)}$ and $B^{(\rho)}$ for each $1 \le i \le r$;

* if $S = \mathcal{N}_{h,1}$: the images $\{C_j^{(i)}\}_{1 \le j \le h}$ of the set $C^{(\rho)}$ for each $1 \le i \le r$.

**Lemma 4.3** *We have $\mathcal{Q}_{(\lambda,2)}(\mathbb{D}) \cong \mathbb{Z}^{r'} \times \mathbb{Z}^{r(r-1)/2} \times \mathbb{Z}^r$ and $\mathcal{Q}_{(\lambda,2)}(S) \cong (\mathbb{Z}/2)^{r'} \times H_1(S; \mathbb{Z})^r$ if $S \not\cong \mathbb{D}$.*

*Proof.* We compute the group $\mathcal{Q}_{(\lambda,2)}(S^{(n)})$ for $n \ge 3$ as the kernel of the map $\mathbf{B}_{\lambda,n}(S)^{ab} \to \mathbf{B}_n(S)^{ab}$ given by diagram (2.24), the abelianisations $\mathbf{B}_{\lambda,n}(S)^{ab}$ and $\mathbf{B}_n(S)^{ab}$ being calculated in [DPS22, Prop. 3.5 and 6.47]. We deduce from these computations that $\mathcal{Q}_{(\lambda,2)}(S^{(n)}) \cong \mathcal{Q}_{(\lambda,2)}(S^{(n+1)})$ for all $n \ge 3$ and a fortiori that the colimit $\mathcal{Q}_{(\lambda,2)}(S)$ is isomorphic to $\mathcal{Q}_{(\lambda,2)}(S^{(3)})$, thus giving the result. $\square$

**Further $\Gamma_\ell$-quotients.** We now consider more generally the surface braid group homological representation $L_i(\tilde{\mathcal{F}}_{(\underline{k}, \mathfrak{S}_{\lambda, \ell})(S)})$ defined using any parameter $\ell \ge 2$.

**Proposition 4.4** *For any integers $\ell \ge 2$ and $n \ge 4$, we have $\mathcal{Q}_{(\lambda,\ell)}(S^{(n)}) \cong \mathcal{Q}_{(\lambda,\ell)}(S^{(n+1)}$).*

*Proof.* We recall that $\mathcal{Q}_{(\lambda,\ell)}(S^{(n)})$ is defined as the kernel of the surjection of $\mathbf{B}_{\lambda,n}(S)/\Gamma_\ell$ onto $\mathbf{B}_n(S)/\Gamma_\ell$. The data which depends on $n$ in the presentation of $\mathbf{B}_{\lambda,n}(S)$ are the set of braid generators $S^{(n)}$ of the $n$-th block, and, for each block $1 \le \rho \le r$, the subset of $X^{(\rho)}$ of the pure braid generators $\{\chi_i^{(\rho)} := \xi_{\Sigma_\rho+i}^{(\rho)} \mid 1 \le i \le n\}$ where $\Sigma_\rho$ denotes the sum $\sum_{\rho+1 \le l \le r} \lambda_l$. For a group $G$, we generically denote by $\gamma_\ell$ the projection onto the $\ell$-nilpotent quotient $G/\Gamma_\ell$.

Since the assignment $S \mapsto \mathbf{B}_n(S)$ is functorial with respect to embeddings of surfaces, we have a canonical injection $\mathbf{B}_n \hookrightarrow \mathbf{B}_n(S) \hookrightarrow \mathbf{B}_{\lambda,n}(S)$. In particular, this morphism sends $\Gamma_\infty(\mathbf{B}_n)$ to $\Gamma_\infty(\mathbf{B}_{\lambda,n}(S))$. Since $\Gamma_\infty(\mathbf{B}_n) = \Gamma_2(\mathbf{B}_n)$ (see for instance [DPS22, Ex. 2.3]), we know that $\sigma_i\sigma_j^{-1} \in \Gamma_\infty(\mathbf{B}_n)$ for all $1 \le i,j \le n-1$. A fortiori, we deduce that $\sigma_i^{(n)} \equiv \sigma_j^{(n)} \pmod{\Gamma_\infty(\mathbf{B}_n(S))}$ and we denote by $\sigma_i^{(n)} \in \mathbf{B}_n(S)/\Gamma_\ell$ the common image of all the $\sigma_i^{(n)}$ under $\gamma_\ell$.

Furthermore, for each $1 \le \rho \le r$, we have the relations $\sigma_i^{(n)}\chi_i^{(\rho)}(\sigma_i^{(n)})^{-1} = (\chi_i^{(\rho)})^{-1}\chi_{i+1}^{(\rho)}\chi_i^{(\rho)}$, $\sigma_i^{(n)}\chi_{i+1}^{(\rho)}(\sigma_i^{(n)})^{-1} = \chi_i^{(\rho)}$ and $\sigma_i^{(n)}\chi_j^{(\rho)}(\sigma_i^{(n)})^{-1} = \chi_j^{(\rho)}$ if $j \notin \{i, i+1\}$ by Proposition 4.2. These are the typical relations between pure braids and Artin generators of a given block induced by the injection $\mathbf{B}_{\lambda,n} \hookrightarrow \mathbf{B}_{\lambda,n}(S)$; see [BGG17, Prop. 3.2] for the case of $S = \Sigma_{g,1}$ and $\lambda = \{k\}$. Then, we deduce from these relations that:

* $\gamma_\ell(\chi_{i+1}^{(\rho)}) = \gamma_\ell((\sigma_i^{(n)})^{-1}\chi_i^{(\rho)}\sigma_i^{(n)}) = \gamma_\ell((\sigma_{i-1}^{(n)})^{-1}\chi_i^{(\rho)}\sigma_{i-1}^{(n)}) = \gamma_\ell(\chi_i^{(\rho)})$ for all $2 \le i \le n-1$;
$$ \bullet \gamma_{\ell}(\chi_2^{(\rho)}) = \gamma_{\ell}((\sigma_1^{(n)})^{-1}\chi_1^{(\rho)}\sigma_1^{(n)}) = \gamma_{\ell}((\sigma_3^{(n)})^{-1}\chi_1^{(\rho)}\sigma_3^{(n)}) = \gamma_{\ell}(\chi_1^{(\rho)}). $$

We denote by $\chi^{(\rho)} \in \mathbf{B}_{\lambda,n}(S)/\Gamma_{\ell}$ the common image of all the $\chi_i^{(\rho)}$ under $\gamma_{\ell}$.

Therefore, the presentation of $\mathbf{B}_{\lambda,n}(S)/\Gamma_{\ell}$ is independent of $n$. In particular, it is routine to check that there is a well-defined map $\gamma'_{\ell}: \mathbf{B}_{\lambda,n+1}(S) \to \mathbf{B}_{\lambda,n}(S)/\Gamma_{\ell}$ defined by $\sigma_i^{(n+1)} \mapsto \sigma^{(n)}$, $\chi_j^{(\rho)} \mapsto \chi^{(\rho)}$ (with $1 \le j \le n+1$) and the assignment of $\gamma_{\ell}$ for the other generators. Then $\gamma'_{\ell}$ induces an inverse to the canonical map $\mathbf{B}_{\lambda,n}(S)/\Gamma_{\ell} \to \mathbf{B}_{\lambda,n+1}(S)/\Gamma_{\ell}$, which is thus an isomorphism. We also know the analogous result for $\mathbf{B}_n(S)/\Gamma_{\ell}$ by [BGG17, Prop. 3.13] (see also [DPS22, Prop. 6.43]), whence the result. $\square$

It follows from Proposition 4.4 that $Q_{(\lambda,\ell)}(S) \cong Q_{(\lambda,\ell)}(S^{(n)})$ for each $\ell \ge 2$ and $n \ge 4$. Thus, for each $\ell \ge 2$, we may compute the colimit transformation group from a computation of $Q_{(\lambda,\ell)}(S^{(n)})$ for any $n \ge 4$. For $\ell = 2$, this was done in Lemma 4.3 above; the following result gathers further results for $\ell \ge 2$.

**Proposition 4.5** We assume that the partition is such that $\lambda_l \ge 3$ for all $1 \le l \le r$. For the classical braid groups, we have for all $\ell \ge 2$:

$$ Q_{\{2;\lambda\},\ell}(\mathbb{D}) \cong \mathbb{Z}^{\binom{r+2}{2}-1} \times ((\mathbb{Z}^2/2^{\ell-2}\bar{\Delta})^{r+1} \rtimes \mathbb{Z}), $$

where $\bar{\Delta} = (1, -1) \in \mathbb{Z}^2$ and $1 \in \mathbb{Z}$ acts on each copy of $\mathbb{Z}^2/2^{\ell-2}\bar{\Delta}$ by swapping coordinates.

For the surfaces different from the disc, we have the following computations for $\ell = 3$. For orientable surfaces, for all $g \ge 1$:

$$ Q_{(\lambda,3)}(\Sigma_{g,1}) \cong ((\mathbb{Z}^{r(r-1)/2} \times \mathbb{Z}^r \times \mathbb{Z}^{rg}) \rtimes \mathbb{Z}^{rg}) \times \mathbb{Z}^r. \quad (4.3) $$

In more detail, the right-hand side of (4.3) may be written as

$$ \left( \langle s_{r_1,r_2} \rangle_{1 \le r_1 < r_2 \le r} \times \langle t_1, \dots, t_r \rangle \times \prod_{1 \le \rho \le r} \langle A_1^{(\rho)}, \dots, A_g^{(\rho)} \rangle \right) \times \prod_{1 \le \rho \le r} \langle B_1^{(\rho)}, \dots, B_g^{(\rho)} \rangle \times \langle q_1, \dots, q_r \rangle $$

where the action defining the semi-direct product structure is determined by

$$ \bullet \ [A_i^{(\rho)}, B_i^{(\rho)}] = t_{\rho}^2 \text{ for all } 1 \le \rho \le r; $$

$$ \bullet \ [A_i^{(r_1)}, B_i^{(r_2)}] = [A_i^{(r_2)}, B_i^{(r_1)}] = s_{r_1, r_2} \text{ for all } 1 \le r_1 < r_2 \le r; $$

• all other pairs of generators commute.

We deduce that $Q_{(\lambda,3)}^u(\Sigma_{g,1}) \cong Q_{(\lambda,3)}(\Sigma_{g,1})/\langle q_1, \dots, q_r \rangle$. For non-orientable surfaces, for all $h \ge 1$:

$$ Q_{(\lambda,3)}(\mathcal{N}_{h,1}) \cong ((\mathbb{Z}^{r-1} \times (\mathbb{Z}^{r-2} \times \dots \times (\mathbb{Z}^2 \times (\mathbb{Z} \times \mathbb{Z}^h) \rtimes \mathbb{Z}^h) \rtimes \dots \rtimes \mathbb{Z}^h) \rtimes \mathbb{Z}^h) \times (\mathbb{Z}/2)^r \times \mathbb{Z}^r. \quad (4.4) $$

In more detail, the right-hand side of (4.4) may be written as

$$ \left( \langle s_{1,r_2} \rangle_{r_2 \ge 2} \times \dots \times \langle s_{r-1,r} \rangle \times \langle C_1^{(r)}, \dots, C_h^{(r)} \rangle \right) \rtimes \dots \rtimes \langle C_1^{(2)}, \dots, C_h^{(2)} \rangle \rtimes \langle C_1^{(1)}, \dots, C_h^{(1)} \rangle \times \langle t_1, \dots, t_r \rangle \times \langle q_1, \dots, q_r \rangle $$

where the action defining the semi-direct product structure is determined by

$$ \bullet \ [C_i^{(r_1)}, C_i^{(r_2)}] = [C_i^{(r_2)}, C_i^{(r_1)}] = s_{r_1, r_2} \text{ for all } 1 \le r_1 < r_2 \le r; $$

• all other pairs of generators commute.

We deduce that $Q_{(\lambda,3)}^u(\mathcal{N}_{h,1}) \cong Q_{(\lambda,3)}(\mathcal{N}_{h,1})/\langle q_1, \dots, q_r \rangle$.

**Proof.** That $Q_{\{2;\lambda\},\ell}(\mathbb{D}) = Q_{\{2;\lambda\},\ell}^u(\mathbb{D})$ and its explicit computation for each $\ell \ge 3$ is done in [PS22b, §4, Cor. 5.4].

For the surfaces different from the disc, we first recall that the quotient $\mathbf{B}_n(\Sigma_{g,1})/\Gamma_3$ is computed by [BGG17, Prop. 3.13], while $\mathbf{B}_n(\mathcal{N}_{h,1})/\Gamma_3 = \mathbf{B}_n(\mathcal{N}_{h,1})^{ab}$ by [DPS22, Th. 6.42]. Now the computation of $\mathbf{B}_{\lambda,n}(S)/\Gamma_3$ follows the same steps as the proof of [DPS22, Prop. 6.58], which computes $\mathbf{B}_{k,n}(\mathcal{N}_{h,1})/\Gamma_3$ and generalises mutatis mutandis as follows.
* Using the presentation from Proposition 4.2, let $N$ be the normal closure of the $\sigma_i^{(\rho)}(\sigma_{i+1}^{(\rho)})^{-1}$ for $i < \lambda_{\rho}$ together with the $\xi_j^{(\rho)}(\xi_{j+1}^{(\rho)})^{-1}$ for $j < \Sigma_{\rho}$, for each $1 \le \rho \le n$. Therefore, $N \subseteq \Gamma_3(\mathbf{B}_{\lambda,n}(S))$ because its generators are in $\Gamma_3(\mathbf{B}_{\lambda,n}(S))$; see for instance [DPS22, Lem. 6.49] and its proof. We claim that, in fact, $N = \Gamma_3(\mathbf{B}_{\lambda,n}(S))$.

* Consider the partition $\{\lambda, n\}$ as a partition $\lambda'$ of $k+n$ with length $r+1$ and we make the identification $s_{i,n} := q_i$ for each $1 \le i \le r$. It is routine, although lengthy (and an inductive generalisation of the analogous point in the proof of [DPS22, Prop. 6.58]), to check from the presentation of Proposition 4.2 that the quotient $\mathbf{B}_{\lambda,n}(S)/N$ is

$$ \left( \langle s_{r_1, r_2} \rangle_{1 \le r_1 < r_2 \le r+1} \times \langle t_1, \dots, t_{r+1} \rangle \times \prod_{1 \le \rho \le r+1} \langle A_1^{(\rho)}, \dots, A_g^{(\rho)} \rangle \right) \times \prod_{1 \le \rho \le r+1} \langle B_1^{(\rho)}, \dots, B_g^{(\rho)} \rangle \quad (4.5) $$

if $S = \Sigma_{g,1}$, and

$$ \left( \langle s_{1,r_2} \rangle_{2 \le r_2 \le r+1} \times \dots \times (\langle s_{r,r+1} \rangle \times \langle C_1^{(r+1)}, \dots, C_h^{(r+1)} \rangle) \times \dots \times \langle C_1^{(1)}, \dots, C_h^{(1)} \rangle \right) \times \langle t_1, \dots, t_{r+1} \rangle \quad (4.6) $$

if $S = \mathcal{N}_{h,1}$.

* The proof of the claim in the first point thus follows from the observation that, using the second point, $\mathbf{B}_{\lambda,n}(S)/N$ is a 2-nilpotent group. Indeed, in both cases, the commutator subgroup is generated by the elements $s_{r_1,r_2}$ and $t_i^2$ (if $S = \Sigma_{g,1}$ for the latter): all these generators are also clearly central in $\mathbf{B}_{\lambda,n}(S)/N$, which proves our claim.

The computations of $Q_{(\lambda,3)}(S)$ then directly follow from the above descriptions of $\mathbf{B}_{\lambda,n}(S)/\Gamma_3$ and $\mathbf{B}_n(S)/\Gamma_3$.

Finally, we compute the untwisted quotients $Q_{(\lambda,3)}^u(\Sigma_{g,1})$ and $Q_{(\lambda,3)}^u(\mathcal{N}_{h,1})$ as follows. First we know from the presentation of $\mathbf{B}_{\lambda,n}(S)/\Gamma_3$ (see (4.5) and (4.6)) that the only generators of $Q_{(\lambda,3)}(S)$ on which the action of $\mathbf{B}_n(S)$ (given by conjugation) is not trivial are $A_i^{(\rho)}$ and $B_i^{(\rho)}$ for all $i$ and each $1 \le \rho \le r$ if $S = \Sigma_{g,1}$, or the $C_j^{(\rho)}$ for all $j$ and each $1 \le \rho \le r$ if $S = \mathcal{N}_{h,1}$. Then, we deduce from the presentation of Proposition 4.2 that for all $1 \le \rho \le r$:

$$ q_{\rho} B_i^{(\rho)} = A_i^{(r+1)} B_i^{(\rho)} (A_i^{(r+1)})^{-1}, $$

and the analogous relation swapping $A$ and $B$, for all $1 \le i \le g$, and

$$ q_{\rho} C_j^{(\rho)} = C_j^{(r+1)} C_j^{(\rho)} (C_j^{(r+1)})^{-1} $$

for all $1 \le j \le h$. This proves that the quotienting submodule defining the coinvariants is $\langle q_1, \dots, q_r \rangle$ in each case. $\square$

**Remark 4.6** If $\lambda_i \le 2$ for some $1 \le i \le r$ and $\ell \ge 3$, we do not know whether the quotient $Q_{(\lambda,\ell)}^u(S)$ of $Q_{(\lambda,\ell)}(S)$ is a proper quotient.

### 4.2.2. Loop braid groups

We now compute the transformation groups of the homological representation functors defined in §3.2.1.3 for loop braid groups with the parameter $\ell = 2$.

**Lemma 4.7** We have the following descriptions of the colimit transformation groups:

* for (3.4): $Q_{(P,\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^r \times (\mathbb{Z}/2)^{r'}$ and $Q'_{(P,\lambda,2)}(\mathbb{D}^3) \cong (\mathbb{Z}/2)^{r+r'}$;

* for (3.5): $Q_{(\mathbb{S}_+,\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^{r^2+r+r'} \times (\mathbb{Z}/2)^{r'}$ and $Q'_{(\mathbb{S}_+,\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^{r^2+r'} \times (\mathbb{Z}/2)^{r+r'}$;

* for (3.6): $Q_{(\mathbb{S},\lambda,2)}(\mathbb{D}^3) \cong \mathbb{Z}^r \times (\mathbb{Z}/2)^{2r'+r(r+1)}$ and $Q'_{(\mathbb{S},\lambda,2)}(\mathbb{D}^3) \cong (\mathbb{Z}/2)^{2r'+r(r+2)}$.
Proof. Recall the quotients of embedding spaces $C_\lambda(\mathbb{D}_n^3)$, $U_\lambda^+(\mathbb{D}_n^3)$ and $U_\lambda(\mathbb{D}_n^3)$ introduced in (3.1); henceforth we denote by $C_{\lambda,n}$ any one of these spaces. The abelianisations of all of the groups $\pi_1(C_{\lambda,n}) \rtimes \mathbf{L}\mathbf{B}_n$, $\mathbf{L}\mathbf{B}_n$, $\pi_1(C_{\lambda,n}) \rtimes \mathbf{L}\mathbf{B}'_n$ and $\mathbf{L}\mathbf{B}'_n$ are computed in [DPS22, Props. 4.46 and 5.10]. In particular, via these computations, one sees that $(\pi_1(C_{\lambda,n}) \rtimes \mathbf{L}\mathbf{B}_n)^{ab} \cong (\pi_1(C_{\lambda,n+1}) \rtimes \mathbf{L}\mathbf{B}_{n+1})^{ab}$, $\mathbf{L}\mathbf{B}_n^{ab} \cong \mathbf{L}\mathbf{B}_{n+1}^{ab}$, $(\pi_1(C_{\lambda,n}) \rtimes \mathbf{L}\mathbf{B}'_n)^{ab} \cong (\pi_1(C_{\lambda,n+1}) \rtimes \mathbf{L}\mathbf{B}'_{n+1})^{ab}$ and $(\mathbf{L}\mathbf{B}'_n)^{ab} \cong (\mathbf{L}\mathbf{B}'_{n+1})^{ab}$ for all $n \ge 4$. Therefore, the groups of the statement of the form $\mathcal{Q}$ given by diagram (2.24) are computed as the kernels of the maps $(\pi_1(C_{\lambda,n}) \rtimes \mathbf{L}\mathbf{B}_n)^{ab} \to \mathbf{L}\mathbf{B}_n^{ab}$ and $(\pi_1(C_{\lambda,n}) \rtimes \mathbf{L}\mathbf{B}'_n)^{ab} \to (\mathbf{L}\mathbf{B}'_n)^{ab}$ for some fixed $n \ge 4$. $\square$

### 4.2.3. Mapping class groups

We finally study the transformation groups of the homological representation functors for mapping class group from §3.2.2 with the parameter $\ell = 2$. For the sake of completeness, we recall that the Lickorish generators together with the Dehn twist along a simple closed curve encircling the boundary component generate the mapping class group $\Gamma_{g,1}$ with $g \ge 1$ (see for instance [FM12, §4.4]), and that generating sets for the mapping class group $\mathcal{N}_{h,1}$ have been worked out by Stukow in [Stu06, Th. A.7] for $h = 2$ and in [Stu10, Th. 5.2] for $h \ge 3$, while $\mathcal{N}_{0,1} = \mathcal{N}_{1,1}$ are trivial by [Eps66]. For brevity, we shall typically use the notation $\mathrm{MCG}(S, \lambda)$ for the mapping class group $\pi_0(\mathrm{Diff}_{\mathrm{dec}}(S, \lambda))$, where $S$ is a compact, connected, smooth surface with one boundary component equipped with $k$ marked points that are fixed setwise, respecting the partition $\lambda \vdash k$. We first establish the following general decomposition for abelianisations of mapping class groups:

**Proposition 4.8** For a compact, connected, smooth, non-planar surface $S$ with one boundary component, we have:

$$ \mathrm{MCG}(\mathbb{D}_k \# S, \lambda)^{ab} \cong (\mathbb{Z}/2)^{r'} \times (H_1(S; \mathbb{Z})^r)_{\mathrm{MCG}(S)} \times \mathrm{MCG}(S)^{ab}, \quad (4.7) $$

where each of the first $r'$ $\mathbb{Z}/2$-summands is generated by the image in the abelianisation $\sigma^{(\rho')}$ (with $1 \le \rho' \le r$ such that $\lambda_{\rho'} \ge 2$) of a standard braid generator (considered as a mapping class) interchanging two points in the corresponding $\rho'$-th block of the partition.

**Proof.** Considering the split short exact sequence (1.9) and using the computation of $\mathbf{B}_\lambda(S)^{ab}$ (see for instance [DPS22, Prop. 6.47]), it follows from the general formula for the calculation of the abelianisation of a semi-direct product that $\mathrm{MCG}(\mathbb{D}_k \# S, \lambda)^{ab} \cong ((\mathbb{Z}/2)^{r'} \times H_1(S; \mathbb{Z})^r)_{\mathrm{MCG}(S)} \times \mathrm{MCG}(S)^{ab}$. We note that the splitting of (1.9) is induced by the embedding of surfaces $S \hookrightarrow \mathbb{D}_k \# S$. Therefore, each $\sigma^{(\rho')}$ may be represented as a mapping class supported in the subsurface $\mathbb{D}_k$ on which $\mathrm{MCG}(S)$ acts trivially, whence the result. $\square$

Using this, we deduce the following calculations of the transformation groups of the homological representation functors $L_i(\mathfrak{F}_{(\underline{k},\mathfrak{S}_{\lambda,\ell})(\Gamma)})$ and $L_i(\mathfrak{F}_{(\underline{k},\mathfrak{S}_{\lambda,\ell})(\mathcal{N})})$ of (3.8) for $\ell = 2$. Let us write $\mathfrak{G} \in \{\Gamma, \mathcal{N}\}$ and correspondingly $S \in \{\Sigma_{1,1}, \mathcal{N}_{1,1}\}$. Denote by $\mathcal{Q}^{\mathfrak{G}}_{(\lambda,2,n)}(\mathfrak{G})$ the group of the form $\mathcal{Q}(S^{\#n}, \emptyset)$ induced by diagram (2.32) for each integer $n \ge 0$ and by $\mathcal{Q}^{\mathfrak{G}}_{(\lambda,2)}(\mathfrak{G})$ the colimit of these groups as $n \to \infty$, which is the colimit transformation group $\mathcal{Q}_{\mathrm{col}}(T)$ of Notation 2.53 in the setting of (3.8).

**Corollary 4.9** We have $\mathcal{Q}^{\mathfrak{G}}_{(\lambda,2)}(\Gamma) \cong (\mathbb{Z}/2)^{r'}$ and $\mathcal{Q}^{\mathfrak{G}}_{(\lambda,2)}(\mathcal{N}) \cong (\mathbb{Z}/2)^{r'} \times (\mathbb{Z}/2)^r$.

**Proof.** Considering the discrete partition $\{1^k\}$, we know from [Kor02, Th. 5.1] that the abelianisations of $\Gamma_{g,1}$ and of $\Gamma_{g,1}^{\{1^k\}}$ are trivial for $g \ge 3$, while it follows from [Stu10, Th. 6.21] that $(\mathcal{N}_{h,1}^{\{1^k\}})^{ab} \cong (\mathbb{Z}/2)^r \times (\mathcal{N}_{h,1})^{ab}$ for $h \ge 7$. Using Proposition 4.8 for $\lambda = \{1^k\}$, we deduce that $(H_1(\Sigma_{g,1}; \mathbb{Z})^r)_{\Gamma_{g,1}} = 0$ and that $(H_1(\mathcal{N}_{h,1}; \mathbb{Z})^r)_{\mathcal{N}_{h,1}} \cong (\mathbb{Z}/2)^r$. Applying Proposition 4.8 again, now for arbitrary $\lambda$, we then deduce that $\mathcal{Q}^{\mathfrak{G}}_{(\lambda,2,g)}(\Gamma) \cong (\mathbb{Z}/2)^{r'}$ and $\mathcal{Q}^{\mathfrak{G}}_{(\lambda,2,h)}(\mathcal{N}) \cong (\mathbb{Z}/2)^{r'} \times (\mathbb{Z}/2)^r$ for all $g \ge 3$ and $h \ge 7$. In particular, these are independent of $g$ and $h$ respectively, so the result follows by taking the colimits as $g \to \infty$ and $h \to \infty$. $\square$
References

[AK10] B. H. An and K. H. Ko. *A family of representations of braid groups on surfaces*. Pacific J. Math. 247.2 (2010), pp. 257-282 (↑ 6, 67).

[Bas23] L. Basualdo Bonatto. *Decoupling decorations on moduli spaces of manifolds*. Math. Proc. Cambridge Philos. Soc. 174.1 (2023), pp. 163-198 (↑ 11).

[BB01] S. J. Bigelow and R. D. Budney. *The mapping class group of a genus two surface is linear*. Algebr. Geom. Topol. 1 (2001), pp. 699-708 (↑ 1, 2).

[BB05] J. S. Birman and T. E. Brendle. *Braids: a survey*. Handbook of knot theory (2005), pp. 19-103 (↑ 1).

[Bel04] P. Bellingeri. *On presentations of surface braid groups*. J. Algebra 274.2 (2004), pp. 543-563 (↑ 71).

[BGG17] P. Bellingeri, E. Godelle and J. Guaschi. *Abelian and metabelian quotient groups of surface braid groups*. Glasg. Math. J. 59.1 (2017), pp. 119-142 (↑ 67, 71-73).

[BH01] T. E. Brendle and H. Hamidi-Tehrani. *On the linearity problem for mapping class groups*. Algebr. Geom. Topol. 1 (2001), pp. 445-468 (↑ 2).

[Big01] S. J. Bigelow. *Braid groups are linear*. J. Amer. Math. Soc. 14.2 (2001), pp. 471-486 (↑ 1, 2, 6, 66).

[Bol12] S. K. Boldsen. *Improved homological stability for the mapping class group with integral or twisted coefficients*. Math. Z. 270.1-2 (2012), pp. 297-329 (↑ 3).

[BPS21] C. Blanchet, M. Palmer and A. Shaukat. *Heisenberg homology on surface configurations*. ArXiv: 2109.00515. 2021 (↑ 3).

[Bre97] G. E. Bredon. *Sheaf theory*. Second. Vol. 170. Graduate Texts in Mathematics. Springer-Verlag, New York, 1997, pp. xii+502 (↑ 43).

[Bro60] M. Brown. *A proof of the generalized Schoenflies theorem*. Bull. Amer. Math. Soc. 66 (1960), pp. 74-76 (↑ 24).

[Bro62] M. Brown. *Locally flat imbeddings of topological manifolds*. Ann. of Math. (2) 75 (1962), pp. 331-341 (↑ 13).

[BT01] C.-F. Bödigheimer and U. Tillmann. *Stripping and splitting decorated mapping class groups*. Cohomological methods in homotopy theory (Bellaterra, 1998). Vol. 196. Progr. Math. Birkhäuser, Basel, 2001, pp. 47-57 (↑ 11).

[Bur35] W. Burau. *Über Zopfgruppen und gleichsinnig verdrillte Verkettungen. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*. Vol. 11. 1. Springer. 1935, pp. 179-186 (↑ 2, 66).

[But16] J. O. Button. *Mapping class groups are not linear in positive characteristic*. ArXiv: 1610.08464. 2016 (↑ 1).

[Cer61] J. Cerf. *Topologie de certains espaces de plongements*. Bull. Soc. Math. France 89 (1961), pp. 227-380 (↑ 24, 27, 50, 52, 63, 64).

[Cer68] J. Cerf. *Sur les difféomorphismes de la sphère de dimension trois* (Γ₄ = 0). Vol. No. 53. Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, 1968, pp. xii+133 (↑ 64).

[Dam17] C. Damiani. *A journey through loop braid groups*. Expo. Math. 35.3 (2017), pp. 252-285 (↑ 63, 68).

[DK01] J. F. Davis and P. Kirk. *Lecture notes in algebraic topology*. Vol. 35. Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 2001, pp. xvi+367 (↑ 43).

[DPS22] J. Darné, M. Palmer and A. Soulié. *When the lower central series stops: a comprehensive study for braid groups and their relatives*. To appear in the Memoirs of the American Mathematical Society; ArXiv: 2201.03542. 2022 (↑ 65, 69, 71-75).

[DV19] A. Djament and C. Vespa. *Foncteurs faiblement polynomiaux*. Int. Math. Res. Not. IMRN 2 (2019), pp. 321-391 (↑ 3).

[EN02] K. Erdmann and D. K. Nakano. *Representation type of Hecke algebras of type A*. Trans. Amer. Math. Soc. 354.1 (2002), pp. 275-285 (↑ 3, 8).

[Eps66] D. B. A. Epstein. *Curves on 2-manifolds and isotopies*. Acta Math. 115 (1966), pp. 83-107 (↑ 75).

[FM12] B. Farb and D. Margalit. *A primer on mapping class groups*. Vol. 49. Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 2012, pp. xiv+472 (↑ 1, 2, 75).

[FP92] E. Formanek and C. Procesi. *The automorphism group of a free group is not linear*. J. Algebra 149.2 (1992), pp. 494-499 (↑ 1).

[Gal21] S. Galatius. *Lecture on invertible field theories*. Quantum field theory and manifold invariants. Vol. 28. IAS/Park City Math. Ser. Amer. Math. Soc., Providence, RI, 2021, pp. 347-402 (↑ 10).
[GK95] E. Getzler and M. M. Kapranov. *Cyclic operads and cyclic homology. Geometry, topology, & physics*. Vol. IV. Conf. Proc. Lecture Notes Geom. Topology. Int. Press, Cambridge, MA, 1995, pp. 167–201 (↑ 14).

[Gra73] A. Gramain. *Le type d'homotopie du groupe des difféomorphismes d'une surface compacte. Annales scientifiques de l'École Normale Supérieure*. Vol. 6. 1. Elsevier. 1973, pp. 53–66 (↑ 64).

[Gra76] D. Grayson. *Higher algebraic K-theory. II (after Daniel Quillen)* (1976), 217–240. Lecture Notes in Math., Vol. 551 (↑ 3, 5, 28).

[Hat02] A. Hatcher. *Algebraic topology*. Cambridge University Press, Cambridge, 2002, pp. xii+544 (↑ 50).

[HEO05] D. F. Holt, B. Eick and E. A. O'Brien. *Handbook of computational group theory*. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton, FL, 2005, pp. xvi+514 (↑ 71).

[Hir76] M. W. Hirsch. *Differential topology*. Graduate Texts in Mathematics, No. 33. New York: Springer-Verlag, 1976, pp. x+221 (↑ 12, 26).

[HL02] R. Häring-Oldenburg and S. Lambropoulou. *Knot theory in handlebodies*. Vol. 11. 6. Knots 2000 Korea, Vol. 3 (Yongpyong). 2002, pp. 921–943 (↑ 71).

[Iva93] N. V. Ivanov. *On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients*. *Mapping class groups and moduli spaces of Riemann surfaces (Göttingen, 1991/Seattle, WA, 1991)*. Vol. 150. Contemp. Math. Amer. Math. Soc., Providence, RI, 1993, pp. 149–194 (↑ 3).

[KLS19] T. Koberda, F. Luo and H. Sun. *An effective Lie-Kolchin theorem for quasi-unipotent matrices*. *Linear Algebra Appl.* 581 (2019), pp. 304–323 (↑ 1).

[Kor02] M. Korkmaz. *Low-dimensional homology groups of mapping class groups: a survey*. *Turkish J. Math.* 26.1 (2002), pp. 101–114 (↑ 75).

[Kra02] D. Krammer. *Braid groups are linear*. *Ann. of Math.* (2) 155.1 (2002), pp. 131–156 (↑ 1, 2, 66).

[Kra19] M. Krannich. *Homological stability of topological moduli spaces*. *Geom. Topol.* 23.5 (2019), pp. 2397–2474 (↑ 30).

[Law90] R. J. Lawrence. *Homological representations of the Hecke algebra*. *Comm. Math. Phys.* 135.1 (1990), pp. 141–191 (↑ 2, 6, 66).

[Lim63] E. L. Lima. *On the local triviality of the restriction map for embeddings*. *Comment. Math. Helv.* 38 (1963), pp. 163–164 (↑ 24, 63).

[Mac98] S. Mac Lane. *Categories for the working mathematician*. Second. Vol. 5. Graduate Texts in Mathematics. Springer-Verlag, New York, 1998, pp. xii+314 (↑ 3, 9, 36, 63).

[Mar19] D. Margalit. *Problems, questions, and conjectures about mapping class groups*. *Breadth in contemporary topology*. Vol. 102. Proc. Sympos. Pure Math. Amer. Math. Soc., Providence, RI, 2019, pp. 157–186 (↑ 1, 2).

[Mat69] J. N. Mather. *Stability of $C^\infty$ mappings. II. Infinitesimal stability implies stability*. *Ann. of Math.* (2) 89 (1969), pp. 254–291 (↑ 12, 17, 23, 52).

[Maz59] B. Mazur. *On embeddings of spheres*. *Bull. Amer. Math. Soc.* 65 (1959), pp. 59–65 (↑ 24).

[Mil65] J. Milnor. *Lectures on the h-cobordism theorem*. Notes by L. Siebenmann and J. Sondow. Princeton University Press, Princeton, NJ, 1965, pp. v+116 (↑ 24).

[MMSS01] M. A. Mandell, J. P. May, S. Schwede and B. Shipley. *Model categories of diagram spectra*. *Proc. London Math. Soc.* (3) 82.2 (2001), pp. 441–512 (↑ 25).

[Mor07] T. Moriyama. *The mapping class group action on the homology of the configuration spaces of surfaces*. *J. Lond. Math. Soc.* (2) 76.2 (2007), pp. 451–466 (↑ 6, 70).

[Mor60] M. Morse. *A reduction of the Schoenflies extension problem*. *Bull. Amer. Math. Soc.* 66 (1960), pp. 113–115 (↑ 24).

[Pal18] M. Palmer. *Twisted homological stability for configuration spaces*. *Homology, Homotopy and Applications* 20.2 (2018), pp. 145–178. {arxiv:1308.4397} (↑ 43).

[Pal21] M. Palmer. *Homological stability for moduli spaces of disconnected submanifolds, I*. *Algebr. Geom. Topol.* 21.3 (2021), pp. 1371–1444 (↑ 16–18, 50, 52).

[Pal60a] R. S. Palais. *Extending diffeomorphisms*. *Proc. Amer. Math. Soc.* 11 (1960), pp. 274–277 (↑ 13, 24).

[Pal60b] R. S. Palais. *Local triviality of the restriction map for embeddings*. *Comment. Math. Helv.* 34 (1960), pp. 305–312 (↑ 16–18, 24, 26, 50, 63).

[PS22a] M. Palmer and A. Soulié. *The Burau representations of loop braid groups*. *C. R. Math. Acad. Sci. Paris* 360 (2022), pp. 781–797 (↑ 6, 68).

[PS22b] M. Palmer and A. Soulié. *The pro-nilpotent Lawrence-Krammer-Bigelow representation*. *ArXiv: 2211.01855*. 2022 (↑ 7, 66–69, 73).
[PS23] M. Palmer and A. Soulié. *Polynomiality of surface braid and mapping class group representations*. ArXiv: 2302.08827. 2023 (↑ 3, 69).

[RW17] O. Randal-Williams and N. Wahl. *Homological stability for automorphism groups*. Adv. Math. 318 (2017), pp. 534–626. {arxiv:1409.3541} (↑ 3, 28).

[Sch] C. Schommer-Pries (https://mathoverflow.net/users/184/chris-schommer-pries). *Skeleton of a braided monoidal category*. MathOverflow. URL: https://mathoverflow.net/q/31599 (↑ 36).

[Sch06] A. Schoenflies. *Beiträge zur Theorie der Punktmengen. III*. Math. Ann. 62.2 (1906), pp. 286–328 (↑ 24).

[Sma59] S. Smale. *Diffeomorphisms of the 2-sphere*. Proc. Amer. Math. Soc. 10 (1959), pp. 621–626 (↑ 63).

[Sta23] T. Stacks project authors. *The Stacks project*. https://stacks.math.columbia.edu. 2023 (↑ 56).

[Ste51] N. Steenrod. *The Topology of Fibre Bundles*. Princeton Mathematical Series, vol. 14. Princeton University Press, Princeton, N. J., 1951, pp. viii+224 (↑ 39).

[Stu06] M. Stukow. *Dehn twists on nonorientable surfaces*. Fund. Math. 189.2 (2006), pp. 117–147 (↑ 75).

[Stu10] M. Stukow. *Generating mapping class groups of nonorientable surfaces with boundary*. Adv. Geom. 10.2 (2010), pp. 249–273 (↑ 75).

[Sup76] D. A. Suprunenko. *Matrix groups*. Ed. by K. A. Hirsch. Vol. Vol. 45. Translations of Mathematical Monographs. Translated from the Russian. American Mathematical Society, Providence, RI, 1976, pp. viii+252 (↑ 1).

[Suz05] M. Suzuki. *Geometric interpretation of the Magnus representation of the mapping class group*. Kobe J. Math. 22.1-2 (2005), pp. 39–47 (↑ 69).

[TV08] B. Toën and G. Vezzosi. *Homotopical algebraic geometry. II*. Geometric stacks and applications. Mem. Amer. Math. Soc. 193.902 (2008), pp. x+224 (↑ 25).

Martin Palmer, *Institutul de Matematică Simion Stoilow al Academiei Române, 21 Calea Griviţei, 010702 Bucharest, Romania*. Email address: mpanghel@imar.ro

Arthur Soulié, *Normandie Univ., UNICAEN, CNRS, LMNO, 14000 Caen, France*. Email address: artsou@hotmail.fr, arthur.soulie@unicaen.fr