isomorphism

$$|\Delta| \cong |\nu\Delta|.$$

Explicitly, this isomorphism associates to the equivalence class of a basis $((v_1, v_1), \dots, (v_n, v_n))$, $v_i \in T_q M$ of $T_{(q,q)}\Delta$ the equivalence class of the basis $([[(0, v_1)], \dots, [(0, v_n)]]$ of $\nu_{(q,q)}\Delta$.

## A.2. Homology with local systems

By local system, we mean a local system of $\mathbb{Z}$-graded rank 1 free $\mathbb{Z}$-modules. On each path-connected component of the underlying space, we think of such a local system in one of the following three equivalent ways: either as the data of the parallel transport representation of the fundamental groupoid, or as the data of the monodromy representation from the fundamental group $\pi$ to the multiplicative group $\{\pm 1\}$ together with the data of an integer (the degree), or as the data of a $\mathbb{Z}$-graded $\mathbb{Z}[\pi]$-module which is free and of rank 1 as a $\mathbb{Z}$-module. Isomorphism classes of local systems on a path-connected space $X$ are thus in bijective correspondence with $H^1(X; \mathbb{Z}/2) \times \mathbb{Z}$, where the first factor corresponds to the monodromy representation and the second factor to the grading. Here, and in the sequel, we identify the multiplicative group $\{\pm 1\}$ with the additive group $\mathbb{Z}/2$. We refer to [9] for a comprehensive discussion with emphasis on local systems on free loop spaces. One other point of view on local systems describes these as locally constant sheaves, but we will only marginally touch upon it in §A.3.

Given a local system $\nu$, we can change the coefficients to any commutative ring $R$ by considering $\nu_R = \nu \otimes_{\mathbb{Z}} R$. The monodromy of such a local system still takes values in $\{\pm 1\}$, and this property characterises local systems of rank 1 free $R$-modules which are obtained from local systems of rank 1 free $\mathbb{Z}$-modules by tensoring with $R$.

Let $X$ be a path-connected space admitting a universal cover $\tilde{X}$. Denote its fundamental group at some fixed basepoint $\pi = \pi_1(X)$. Interpreting a local system $\nu$ on $X$ as a $\mathbb{Z}[\pi]$-module, one defines singular homology/cohomology with coefficients in $\nu$ in terms of singular chains on $\tilde{X}$ as

$$H_*(X; \nu) = H_*(C_*(\tilde{X}; \mathbb{Z}) \otimes_{\mathbb{Z}[\pi]} \nu),$$

$$H^*(X; \nu) = H_*(\operatorname{Hom}_{\mathbb{Z}[\pi]}(C_*(\tilde{X}; \mathbb{Z}), \nu)).$$

The homology/cohomology with local coefficients extended to a commutative ring $R$ are the $R$-modules

$$H_*(X; \nu_R) = H_*(C_*(\tilde{X}; \mathbb{Z}) \otimes_{\mathbb{Z}[\pi]} \nu_R),$$

$$H^*(X; \nu_R) = H_*(\operatorname{Hom}_{\mathbb{Z}[\pi]}(C_*(\tilde{X}; \mathbb{Z}), \nu_R)).$$

In our grading convention, the cohomology with constant coefficients is supported in nonpositive degrees and equals the usual cohomology in the opposite degree. The induced differential on the dual group $\operatorname{Hom}_{\mathbb{Z}[\pi]}(C_*(\tilde{X}; \mathbb{Z}), \nu_R)$ has degree $-1$.

The tensor product $\nu_1 \otimes \nu_2$ of two local systems is again a local system. Its $\mathbb{Z}[\pi]$-module structure is the diagonal one and its degree is the sum of the degrees of the factors. Note that viewing $\nu_1, \nu_2$ as elements in $H^1(X; \mathbb{Z}/2) \times \mathbb{Z}$,
their tensor product is given by their sum $\nu_1 + \nu_2$. Operations like cap or cup product naturally land in homology/cohomology with coefficients in the tensor product of the coefficients of the factors.

Homology/cohomology with local coefficients behave functorially in the following sense. Given a continuous map $f: X \to Y$ and a local system $\nu$ on $Y$ described as a $\mathbb{Z}[\pi_1(Y)]$-module, the pullback local system $f^*\nu$ on $X$ is defined by inducing a $\mathbb{Z}[\pi_1(X)]$-module structure via $f_*$. We then have canonical maps

$$f_*: H_*(X; f^*\nu) \to H_*(Y; \nu), \quad f^*: H^*(Y; \nu) \to H^*(X; f^*\nu).$$

The *algebraic duality isomorphism* with coefficients in a field $K$ takes the form

$$H^{-k}(X; \nu_{\mathbb{K}}^{-1}) \stackrel{\sim}{\longrightarrow} H_k(X; \nu_{\mathbb{K}})^{\vee}, \quad k \in \mathbb{Z}.$$

The map is induced by the canonical evaluation of cochains on chains. We check that degrees fit in the case of graded local systems: given a local system $\nu_{\mathbb{K}}$ of degree $d$, and recalling our notation $\nu_{\mathbb{K}} = \nu_{\mathbb{K}}[d]$ and $\nu_{\mathbb{K}}^{-1} = \nu_{\mathbb{K}}^{-1}[-d]$, we have

$$H_k(X; \nu_{\mathbb{K}}) = H_{k-d}(X; \nu_{\mathbb{K}}), \quad H^{-k}(X; \nu_{\mathbb{K}}^{-1}) = H^{-k+d}(X; \nu_{\mathbb{K}}^{-1}),$$

so $H_k(X; \nu_{\mathbb{K}})^{\vee}$ and $H^{-k}(X; \nu_{\mathbb{K}}^{-1})$ both live in degree $d-k$.

### A.3. Poincaré duality

Consider a manifold $M$ of dimension $n$. We denote by $|M|$ the local system on $M$ whose fibre at any point $q \in M$ is the orientation line $|T_q M|$, supported by definition in degree $n$. We refer to $|M|$ as the *orientation local system of* $M$. The monodromy along a loop $\gamma$ is $+1$ if the loop preserves the orientation (i.e. the pullback bundle $\gamma^*TM$ is orientable), and $-1$ if the loop reverses it. The local system $|M|$ is trivial if and only if the manifold $M$ is orientable. A choice of orientation is equivalent to the choice of one of the two possible isomorphisms $|M| \simeq \mathbb{Z}$. The local system $|M| \otimes \mathbb{Z}/2$ is trivial, and this reflects the fact that any manifold is $\mathbb{Z}/2$-orientable.

Suppose now that $M$ is closed. Then, it carries a *fundamental class* $[M] \in H_n(M; |M|) = H_0(M; |M|^{-1})$.

For any local system $\nu$ on $M$, the cap product with a fundamental class defines a *Poincaré duality isomorphism*

$$H^*(M; \nu) \stackrel{\sim}{\longrightarrow} H_*(M; \nu \otimes |M|^{-1}), \quad \alpha \mapsto [M] \cap \alpha.$$

*Remark A.2.* Here is a description of the fundamental class using the interpretation of local systems as locally constant sheaves (see for example [28], Lemma 3.27 and Example 3 H.3). Let $M' \xrightarrow{p} M$ be the orientation double cover. Given the constant local system $\mathbb{Z}$ on $M'$, the pushforward $p_*\mathbb{Z}$ to $M$ has rank 2 and can be decomposed as $|M| \oplus \mathbb{Z}$ (the map $\mathbb{Z} \oplus \mathbb{Z} \to \mathbb{Z} \oplus \mathbb{Z}$, $(x, y) \mapsto (y, x)$ fixes the diagonal and acts by $-\mathrm{Id}$ on the anti-diagonal). The composition $H_*(M'; \mathbb{Z}) \xrightarrow{p_*} H_*(M; p_*\mathbb{Z}) \stackrel{\sim}{\longrightarrow} H_*(M; |M|) \oplus H_*(M; \mathbb{Z})$ is an isomorphism because $p_*$ is an isomorphism. Since $H_n(M; \mathbb{Z}) = 0$ if $M$
is nonorientable, we obtain that $H_n(M; |M|)$ has rank 1. A generator is the image of a generator in $H_n(M'; \mathbb{Z})$ via the above composition.

### A.4. Thom isomorphism and Gysin sequence

Let $E \xrightarrow{p} X$ be a real vector bundle of rank $r$, and denote $\dot{E}$ the complement of the zero section. Let $|E|$ be the local system on $X$ whose fibre at a point $x \in X$ is the orientation line $|E_x|$ of the fibre of $E$ at $x$. The local system $|E|$ is called the orientation local system of $E$ and is supported in degree $r$. The *Thom class* is a generator

$$ \tau \in H^{-r}(E, \dot{E}; p^*|E|) = H^0(E, \dot{E}; p^*|E|). $$

The *Thom isomorphism* takes the form

$$ H_k(E, \dot{E}) \xrightarrow{\simeq} H_{k-r}(X; |E|) = H_k(X; |E|), \quad k \in \mathbb{Z} $$

(cap product with $\tau$), respectively

$$ H^k(X; |E|^{-1}) = H^{k+r}(X; |E|^{-1}) \xrightarrow{\simeq} H^k(E, \dot{E}), \quad k \in \mathbb{Z} $$

(cup product with $\tau$). More generally, for any local system $\nu$ on $X$, we have isomorphisms

$$ H_k(E, \dot{E}; p^*\nu) \xrightarrow{\simeq} H_{k-r}(X; \nu \otimes |E|) = H_k(X; \nu \otimes |E|), $$

$$ H^k(X; \nu \otimes |E|^{-1}) = H^{k+r}(X; \nu \otimes |E|^{-1}) \xrightarrow{\simeq} H^k(E, \dot{E}; p^*\nu). $$

Pulling back the Thom class under the inclusion $i: X \to E$ of the zero section yields the *Euler class*

$$ e = i^* \tau \in H^{-r}(X; |E|) = H^0(X; |E|). $$

Denote by $S \subset \dot{E}$ the sphere bundle with projection $\pi = p|_S: S \to X$. Then, the long exact sequence of the pair $(E, \dot{E})$ fits into the commuting diagram

$$ \begin{tikzcd} \cdots H^k(E, \dot{E}) \arrow[r] & H^k(E) \arrow[r] & H^k(\dot{E}) \arrow[r] & H^{k-1}(E; \dot{E}) \arrow[r, "\cong"] & \cdots \\ \bigcup_{\tau} \arrow[r, "\cong"'] & i^* \arrow[r, "\cong"'] & \arrow[r, "\cong"'] & \bigcup_{\tau} \arrow[r, "\cong"'] & \cdots \\ \cdots H^k(X; |E|^{-1}) \arrow[r, "\cong"'] & H^k(X) \arrow[r, "\cong"'] & H^k(S) \arrow[r, "\cong"'] & H^{k-1}(X; |E|^{-1}) \arrow[r, "\cong"'] & \cdots \end{tikzcd} $$

where the lower sequence is the *Gysin sequence*. More generally, for each local system $\nu$ on $X$, we get a Gysin sequence

$$ \begin{aligned} \cdots H^k(X; \nu \otimes |E|^{-1}) &\xrightarrow{\cup e} H^k(X; \nu) \\ &\xrightarrow{\pi^*} H^k(S; \pi^*\nu) \xrightarrow{\pi_*} H^{k-1}(X; \nu \otimes |E|^{-1}) \cdots \end{aligned} $$

### A.5. Spaces of loops with self-intersection

Let $M$ be a manifold of dimension $n$, $\Lambda = \Lambda M$ its space of free loops of Sobolev class $W^{1,2}$, and $\text{ev}_s: \Lambda \to M$ the evaluation of loops at time $s$. We define

$$ \mathcal{F} = \{(\gamma, \delta) \in \Lambda \times \Lambda \mid \gamma(0) = \delta(0)\} \subset \Lambda \times \Lambda $$
(pairs of loops with the same basepoint), and

$$ \mathcal{F}_s = \{\gamma \in \Lambda \mid \gamma(s) = \gamma(0)\} \subset \Lambda, \quad s \in (0, 1) $$

(loops with a self-intersection at time s). Denoting $f: \Lambda \times \Lambda \to M \times M$, $f = \text{ev}_0 \times \text{ev}_0$ and $f_s: \Lambda \to M \times M$, $f_s = (\text{ev}_0, \text{ev}_s)$, we can equivalently write

$$ \mathcal{F} = f^{-1}(\Delta), \quad \mathcal{F}_s = f_s^{-1}(\Delta). $$

The maps $f$ and $f_s$ are smooth and transverse to the diagonal $\Delta$, so that $\mathcal{F}$ and $\mathcal{F}_s$ are Hilbert submanifolds of codimension $n$. Denoting $\nu\mathcal{F}$ and $\nu\mathcal{F}_s$ their normal bundles, we obtain canonical isomorphisms

$$ \nu\mathcal{F} \cong f^*\nu\Delta, \quad \nu\mathcal{F}_s \cong f_s^*\nu\Delta. $$

In view of Example A.1, we infer canonical isomorphisms

$$ |\nu\mathcal{F}| \cong f^*|\Delta| \cong \text{ev}_0^*|M|, \quad |\nu\mathcal{F}_s| \cong f_s^*|\Delta| \cong \text{ev}_0^*|M|, \quad (36) $$

where, in the first formula, $\text{ev}_0: \mathcal{F} \to M$ is the evaluation of pairs of loops at their common origin.

Denote $i: \mathcal{F} \hookrightarrow \Lambda \times \Lambda$ and $i_s: \mathcal{F}_s \hookrightarrow \Lambda$ the inclusions. Recall the restriction maps (7). Define the cutting map at time $s$

$$ c_s: \mathcal{F}_s \to \mathcal{F}, \quad c_s(\gamma) = (\gamma|_{[0,s]}, \gamma|_{[s,1]}) $$

and the concatenation map at time $s$

$$ g_s: \mathcal{F} \to \mathcal{F}_s, \quad g_s(\gamma_1, \gamma_2)(t) = \begin{cases} \gamma_1(\frac{t}{s}), & t \in [0, s], \\ \gamma_2(\frac{t-s}{1-s}), & t \in [s, 1]. \end{cases} $$

The maps $c_s$ and $g_s$ are smooth diffeomorphisms inverse to each other. The situation is summarised in the diagram

$$ \Lambda \times \Lambda \leftarrow_{i} \mathcal{F} \xrightarrow{\substack{g_s \\ \sim \\ c_s}} \mathcal{F}_s \rightarrow_{i_s} \Lambda. $$

**Lemma A.3.** Let $\nu$ be a local system (of rank 1 free abelian groups) on $\Lambda$ supported in degree 0. Denote $p_{1,2}: \Lambda \times \Lambda \to \Lambda$ the projections on the two factors. The following two conditions are equivalent:

$$ c_s^*(p_1^*\nu \otimes p_2^*\nu)|_{\mathcal{F}} \simeq \nu|_{\mathcal{F}_s}, \quad (37) $$

and

$$ (p_1^*\nu \otimes p_2^*\nu)|_{\mathcal{F}} \simeq g_s^*(\nu|_{\mathcal{F}_s}). \quad (38) $$

*Proof.* The first condition is $c_s^*i^*(p_1^*\nu \otimes p_2^*\nu) \simeq i_s^*\nu$. Since $g_s$ is a homeomorphism, this is equivalent to

$$ g_s^*c_s^*i^*(p_1^*\nu \otimes p_2^*\nu) \simeq g_s^*i_s^*\nu. $$

In view of $c_s g_s = \text{Id}_{\mathcal{F}_s}$, this is the same as the second condition. $\square$

**Definition A.4.** A degree 0 local system $\nu$ on $\Lambda$ is compatible with products if it satisfies the equivalent conditions of Lemma A.3.
A local system $\nu$ which is compatible with products must necessarily have degree 0 (and rank 1). Also, $\nu|M$ must be trivial: restricting both sides of (37) or (38) to the constant loops yields $\nu|M \otimes \nu|M \simeq \nu|M$.

**Remark A.5.** Local systems which are compatible with products play a key role in the sequel definition of the loop product and loop coproduct with local coefficients. Condition (37) is the one that ensures the coproduct is defined with coefficients twisted by $\nu$, whereas condition (38) is the one that ensures the product is defined with coefficients twisted by $\nu$. That the two conditions are equivalent can be seen as yet another instance of Poincaré duality for free loops.

We refer to Remark A.11 for an additional condition on the isomorphisms (38) which is needed for the associativity of the product and co associativity of the coproduct.

**Example A.6.** (Transgressive local systems) Let $\Lambda = \sqcup_{\alpha} \Lambda_{\alpha} M$ be the decomposition of the free loop space into connected components, indexed by conjugacy classes $\alpha$ in the fundamental group. We view loops $\gamma : S^1 \to \Lambda$ as maps $\gamma \times S^1 : S^1 \times S^1 \to M$, $(u, t) \mapsto \gamma(u)(t)$. This induces a map $\pi_1(\Lambda_{\alpha} M) \to H_1(\Lambda_{\alpha} M; \mathbb{Z}) \to H_2(M; \mathbb{Z})$, $[\gamma] \mapsto [\gamma \times S^1]$. Dually, and specialising to $\mathbb{Z}/2$-coefficients, any cohomology class $c \in H^2(M; \mathbb{Z}/2)$ determines a cohomology class $\tau_c \in H^1(\Lambda; \mathbb{Z}/2) = \prod_{\alpha} \mathrm{Hom}(\pi_1(\Lambda_{\alpha} M); \mathbb{Z}/2)$ via

$$ \langle \tau_c, [\gamma] \rangle = \langle c, [\gamma \times S^1] \rangle. $$

We denote the corresponding local system on $\Lambda$ also by $\tau_c$. Degree 0 local systems obtained in this way are called transgressive [7].

Transgressive local systems are compatible with products. Indeed, the identity (38) is a direct consequence of the equality $[g_s(\gamma_1, \gamma_2) \times S^1] = [\gamma_1 \times S^1] + [\gamma_2 \times S^1]$, which holds in $H_2(M; \mathbb{Z})$ for all $(\gamma_1, \gamma_2) : S^1 \to \mathcal{F}$.

The transgressive local system

$$ \sigma = \tau_{w_2} \qquad (39) $$

defined by the second Stiefel–Whitney class $w_2 \in H^2(M; \mathbb{Z}/2)$ will play a special role in the sequel.

**Example A.7.** Following Abouzaid [7], define for each loop $\gamma \in \Lambda$ the shift

$$ w(\gamma) = \begin{cases} 0, & \text{if } \gamma \text{ preserves the orientation,} \\ -1, & \text{if } \gamma \text{ reverses the orientation.} \end{cases} $$

Define the local system

$$ \tilde{o} = \mathrm{ev}_0^* |M|^{-w} \qquad (40) $$

to be trivial on the components where $\gamma$ preserves the orientation, and equal to $\mathrm{ev}_0^*|M|$ on components where $\gamma$ reverses the orientation.

The local system $\tilde{o}$ is compatible with products: the equality $w(\gamma_1) + w(\gamma_2) = w(g_s(\gamma_1, \gamma_2))$ holds in $\mathbb{Z}/2$ for all $(\gamma_1, \gamma_2) \in \mathcal{F}$. Note that the local system $\tilde{o}$ is not transgressive and, in case $M$ is nonorientable, it is nontrivial on all connected components $\Lambda_{\alpha} M$ whose elements reverse orientation.
**Question A.8.** Characterise in cohomological terms the local systems on $\Lambda$ which are compatible with products. For example, it follows from [9, Lemma 1] that, on a simply connected manifold, a local system $\nu$ is compatible with products if and only if $\nu|_M$ is trivial. A mild generalisation is given by [9, Proposition 10].

**A.6. Loop product with local coefficients**

Following [27], we view the loop product as being defined by going from left to right in the diagram

$$ \Lambda \times \Lambda \leftrightarrow \mathcal{F} \xrightarrow{g} \Lambda, $$

where $g = i_s g_s$ for some fixed $s \in (0, 1)$. More precisely, the loop product with integer coefficients is defined as the composition

$$ \begin{align*} H_i(\Lambda; \mathbb{Z}) \otimes H_j(\Lambda; \mathbb{Z}) &\xrightarrow{\epsilon \times} H_{i+j}(\Lambda \times \Lambda; \mathbb{Z}) \\ &\longrightarrow H_{i+j}(\nu\mathcal{F}, \dot{\nu}\mathcal{F}; \mathbb{Z}) \\ &\stackrel{\sim}{\longrightarrow} H_{i+j}(\mathcal{F}; \mathrm{ev}_0^*|M|) \\ &\xrightarrow{g_*} H_{i+j}(\Lambda; \mathrm{ev}_0^*|M|). \end{align*} $$

The first map is the homology cross-product corrected by a sign $\epsilon = (-1)^{n(i+n)}$ ([29, Appendix B]), the second map is the composition of the map induced by inclusion $\Lambda \times \Lambda \hookrightarrow (\Lambda \times \Lambda, \Lambda \times \Lambda \setminus \mathcal{F})$ with excision and the tubular neighbourhood isomorphism, and the third map is the Thom isomorphism. In case $M$ is not orientable the loop product does not land in homology with integer coefficients and thus fails to define an algebra structure on $H_*(\Lambda; \mathbb{Z})$. This can be corrected by using at the source homology with local coefficients.

**Definition A.9.** Define on $\Lambda$ the local system

$$ \mu := \mathrm{ev}_0^* |M|^{-1}. $$

The *archetypal loop product* is the bilinear map

$$ \bullet : H_i(\Lambda; \mu) \otimes H_j(\Lambda; \mu) \to H_{i+j}(\Lambda; \mu) $$

defined as the composition

$$ \begin{align*} H_i(\Lambda; \mu) \otimes H_j(\Lambda; \mu) &\xrightarrow{\epsilon \times} H_{i+j}(\Lambda \times \Lambda; p_1^* \mu \otimes p_2^* \mu) \\ &\longrightarrow H_{i+j}(\nu \mathcal{F}, \dot{\nu} \mathcal{F}; p_1^* \mu \otimes p_2^* \mu|_{\nu \mathcal{F}}) \\ &\stackrel{\sim}{\longrightarrow} H_{i+j}(\mathcal{F}; (p_1^* \mu \otimes p_2^* \mu)|_{\mathcal{F}} \otimes \mathrm{ev}_0^*|M|) \\ &\xrightarrow{g_*} H_{i+j}(\Lambda; \mu). \end{align*} $$

The description of the maps is the same as above, with $\epsilon = (-1)^{ni}$ because of the shift $H_i(\Lambda; \mu) = H_{i+n}(\Lambda; \mu)$. However, one still needs to check that the local systems of coefficients are indeed as written. For the first, second and third map the behaviour of the coefficients follows general patterns. For the last map, we use that

$$ (p_1^* \mu \otimes p_2^* \mu)|_{\mathcal{F}} \otimes \mathrm{ev}_0^*|M| \simeq g^* \mu, $$

which is true for our specific $\mu = \mathrm{ev}_0^*|M|^{-1}$.
The archetypal loop product is associative, graded commutative, and it has a unit represented by the fundamental class

$$[M] \in H_0(M; |M|^{-1}) = H_n(M; |\underline{M}|)$$

from §A.3. With our grading conventions, the archetypal loop product has degree 0 and the local system $\mu$ is supported in degree $-n$. In the case where $M$ is oriented, we recover the usual loop product.

More generally, the loop product can be defined with further twisted coefficients.

**Definition A.10.** Let $\nu$ be a degree 0 local system (of rank 1 free $\mathbb{Z}$-modules) on $\Lambda$ which is compatible with products. The loop product with coefficients twisted by $\nu$ is the bilinear map

$$\bullet : H_i(\Lambda; \nu \otimes \mu) \otimes H_j(\Lambda; \nu \otimes \mu) \to H_{i+j}(\Lambda; \nu \otimes \mu)$$

(with $\mu = \text{ev}_0^*|M|^{-1}$ as above) defined as the composition

$$\begin{align*}
& H_i(\Lambda; \nu \otimes \mu) \otimes H_j(\Lambda; \nu \otimes \mu) \\
\xrightarrow{\text{e} \times} & H_{i+j}(\Lambda \times \Lambda; (p_1^* \nu \otimes p_2^* \nu) \otimes (p_1^* \mu \otimes p_2^* \mu)) \\
\longrightarrow & H_{i+j}(\nu \mathcal{F}, \dot{\nu} \mathcal{F}; (p_1^* \nu \otimes p_2^* \nu) \otimes (p_1^* \mu \otimes p_2^* \mu)|_{\nu \mathcal{F}}) \\
\xrightarrow{\sim} & H_{i+j}(\mathcal{F}; (p_1^* \nu \otimes p_2^* \nu) \otimes (p_1^* \mu \otimes p_2^* \mu)|_{\mathcal{F}} \otimes \text{ev}_0^*|M|) \\
\xrightarrow{g_*} & H_{i+j}(\Lambda; \nu \otimes \mu).
\end{align*}$$

As before, we have $\epsilon = ni$. For the last map, we use the isomorphism $(p_1^*\mu \otimes p_2^*\mu)|_{\mathcal{F}} \otimes \text{ev}_0^*|M| \simeq g^*\mu$, and the isomorphism $(p_1^*\nu \otimes p_2^*\nu)|_{\mathcal{F}} \simeq g^*\nu$ which expresses the compatibility with products for $\nu$.

The loop product with twisted coefficients is graded commutative and unital. Recalling that the compatibility with products for $\nu$ forces its restriction to $M$ to be trivial, the unit is again represented by the fundamental class

$$[M] \in H_0(M; \nu|_M \otimes |M|^{-1}) = H_0(M; |M|^{-1}) = H_n(M; |\underline{M}|).$$

**Remark A.11.** Associativity of the loop product with twisted coefficients depends on the following associativity condition on the isomorphisms (37) and (38). Given $s, s' \in (0, 1)$ denote $s'' = (s' - ss')/(1 - ss')$, so that $g_{s'} \circ (g_s \times \text{id}) = g_{ss'} \circ (\text{id} \times g_{s''})$. Denoting $\Phi_s : (p_1^*\nu \otimes p_2^*\nu)|_{\mathcal{F}} \xrightarrow{\sim} g_s^*\nu|_{\mathcal{F}_s}$ the isomorphism from (38), we require the associativity condition

$$\Phi_{ss'} \circ (\text{Id} \otimes \Phi_{s''}) = \Phi_{s'} \circ (\Phi_s \otimes \text{Id}).$$

This holds for the transgressive local systems from Example A.6 and for the local system in Example A.7.

Also, because (37) and (38) are equivalent, this condition on (38) will guarantee coassociativity of the coproduct, see below.
**A.7. Loop coproduct with coefficients**

Again following [27], we view the primary coproduct on loop homology as being defined by going from left to right in the diagram

$$ \Lambda \leftrightarrow \mathcal{F}_s \xrightarrow{c_s} \Lambda \times \Lambda $$

for some fixed $s \in (0, 1)$, where $c_s$ stands for $ic_s$ in the notation of §A.5. We restrict in this section to coefficients in a field $\mathbb{K}$ and all local systems are accordingly understood in this category. The reason for this restriction is explained below. The *primary coproduct with constant coefficients* is defined as the composition

$$ \begin{aligned} H_k(\Lambda; \mathbb{K}) &\longrightarrow H_k(\nu\mathcal{F}_s, \dot{\nu}\mathcal{F}_s; \mathbb{K}) \stackrel{\sim}{\longrightarrow} H_k(\mathcal{F}_s; \mathrm{ev}_0^*|M|) \\ \xrightarrow{c_s} H_k(\Lambda \times \Lambda; p_1^*\mathrm{ev}_0^*|M|) &\longrightarrow \bigoplus_{i+j=k} H_i(\Lambda; \mathrm{ev}_0^*|M|) \otimes H_j(\Lambda; \mathbb{K}). \end{aligned} $$

The first map is the composition of the map induced by inclusion $\Lambda \to (\Lambda, \Lambda \setminus \mathcal{F}_s)$ with the excision isomorphism towards the homology rel boundary of a tubular neighbourhood of $\mathcal{F}_s$. The second map is the Thom isomorphism. For the third map, we use that $c_s^*p_1^*\mathrm{ev}_0^* = \mathrm{ev}_0^*$. The fourth map is the Alexander-Whitney diagonal map followed by the Künneth isomorphism.<fnref n="7" /> Just like for the loop product, we see that if $M$ is nonorientable the primary coproduct fails to define a coalgebra structure on $H_*(\Lambda; \mathbb{K})$. This is corrected using homology with local coefficients as follows.

**Definition A.12.** Define on $\Lambda$ the local system

$$ o := \mathrm{ev}_0^* |M| = \mu^{-1}. $$

The *archetypal primary coproduct* is the bilinear map

$$ \forall_s : H_k(\Lambda; o) \to \bigoplus_{i+j=k} H_i(\Lambda; o) \otimes H_j(\Lambda; o) $$

(for some fixed $s \in [0, 1]$) defined as the composition

$$ \begin{aligned} H_k(\Lambda; o) &\longrightarrow H_k(\nu\mathcal{F}_s, \dot{\nu}\mathcal{F}_s; o|_{\nu\mathcal{F}_s}) \\ &\stackrel{\sim}{\longrightarrow} H_k(\mathcal{F}_s; o \otimes \mathrm{ev}_0^*|M|) \\ &\xrightarrow{c_s} H_k(\Lambda \times \Lambda; p_1^*o \otimes p_2^*o) \\ &\stackrel{AW}{\longrightarrow} \bigoplus_{i+j=k} H_i(\Lambda; o) \otimes H_j(\Lambda; o). \end{aligned} $$

With our grading conventions, this coproduct has degree 0. Taking into account that $o = \mathrm{ev}_0^*|M|$ is supported in degree $n$, this results in the coproduct having the usual degree $-n$ in ungraded notation. In the orientable case it recovers the usual primary coproduct.

Just like the product, the primary coproduct can be defined with further twisted coefficients.

<footnote n="7">⁷The Alexander-Whitney diagonal map [25, VI.12.26] takes values in $H_*(C_*(\Lambda; \mathrm{ev}_0^*|M|) \otimes C_*(\Lambda))$ with arbitrary coefficients. In order to further land in $H_*(\Lambda; \mathrm{ev}_0^*|M|) \otimes H_*(\Lambda)$, we need to restrict to field coefficients so the Künneth isomorphism holds. Alternatively, one needs to modify the target of the coproduct to be $H_*(\Lambda \times \Lambda; p_1^*\mathrm{ev}_0^*|M|)$, see [29].</footnote>
**Definition A.13.** Let $\nu$ be a degree 0 local system (of rank one $\mathbb{K}$-vector spaces) on $\Lambda$ which is compatible with products. The *primary coproduct with twisted coefficients* is the bilinear map

$$ \vee_s : H_k(\Lambda; \nu \otimes o) \to \bigoplus_{i+j=k} H_i(\Lambda; \nu \otimes o) \otimes H_j(\Lambda; \nu \otimes o) $$

(with $o = \text{ev}_0^*|M|$ as above and some fixed $s \in [0, 1]$) defined as the composition

$$ \begin{align*} H_k(\Lambda; \nu \otimes o) &\to H_k(\nu \mathcal{F}_s, \dot{\nu} \mathcal{F}_s; \nu \otimes o|_{\nu \mathcal{F}_s}) \\ &\stackrel{c_{s*}}{\rightleftharpoons} H_k(\mathcal{F}_s; \nu \otimes o^{\otimes 2}) \\ &\stackrel{c_{s*}}{\to} H_k(\Lambda \times \Lambda; p_1^*(\nu \otimes o) \otimes p_2^*(\nu \otimes o)) \\ &\stackrel{AW}{\to} \bigoplus_{i+j=k} H_i(\Lambda; \nu \otimes o) \otimes H_j(\Lambda; \nu \otimes o). \end{align*} $$

In the definition, we use $c_s^*(p_1^*o \otimes p_2^*o) \simeq o \otimes o$, and the condition $c_s^*(p_1^*\nu \otimes p_2^*\nu) \simeq \nu|_{\mathcal{F}_s}$ which is part of the condition of being compatible with products for $\nu$.

The arguments of Goresky-Hingston [27, §8] which show that, in the orientable case, there is a secondary coproduct of degree $-n+1$ defined on relative homology $H_*(\Lambda, \Lambda_0; \mathbb{K})$, apply *verbatim* in the current setup involving local coefficients. As an outcome, we obtain the following.

**Definition-Proposition A.14.** For any degree 0 local system $\nu$ of rank one $\mathbb{K}$-vector spaces on $\Lambda$ which is compatible with products, there is a well-defined (secondary) loop coproduct with twisted coefficients (abbreviate $o = \text{ev}_0^*|M|$)

$$ \vee : H_k(\Lambda, \Lambda_0; \nu \otimes o) \to \bigoplus_{i+j=k+1} H_i(\Lambda, \Lambda_0; \nu \otimes o) \otimes H_j(\Lambda, \Lambda_0; \nu \otimes o). $$

□

As explained in [29], in order for this secondary coproduct to be coassociative in the case of a constant local system $\nu$, we need to correct the *ij*-component of the secondary product induced by the previously defined primary product by a sign $\epsilon = (-1)^{(n-1)(j-n)}$ (see [29, Definition 1.7] and note the shift in grading $H_j(\Lambda, \Lambda_0; \nu \otimes o) = H_{j-n}(\Lambda, \Lambda_0; \nu \otimes \mathfrak{o})$). With this correction, the coproduct with constant local system $\nu$ is also graded cocommutative if gradings are shifted so that it has degree 0. However, it has no counit (this would contradict the infinite dimensionality of the homology of $\Lambda$).

In the case of coefficients twisted by a local system $\nu$ which is nonconstant, coassociativity further requires that the isomorphisms $\Phi_s$ expressing compatibility with products for $\nu$ satisfy the condition from Remark A.11. The coproduct is then also cocommutative.

One obtains dually a cohomology product [27, 29]. Note that, in contrast to the loop coproduct, the dual loop cohomology product is defined with arbitrary coefficients because its definition does not require the Künneth isomorphism.
field coefficients as in §A.7). Interpreted dually as a product on cohomology, this is defined with arbitrary coefficients on $MH^*(\Lambda, \chi_{pt}; \nu \otimes \mu)$.

The comparison between reduced loop homology and loop homology relative to a point goes as follows. Recalling that $\mu = \underline{o}$, we have a commutative diagram

$$ \begin{tikzcd} Rq_0 \arrow[r, "\cdot \chi"] \arrow[d, "\cdot \chi"] & MC_*(\Lambda; \nu \otimes \mu) \arrow[d, "\parallel"'] \\ 0 \arrow[r, "\chi Rq_0"] & MC_*(\Lambda; \nu \otimes \underline{o}) \arrow[r, "\chi_{pt}; \nu \otimes \mu"] & 0 \arrow[d, "\cdot \chi"'] \arrow[r] & 0 \end{tikzcd} $$

which induces

$$ \begin{tikzcd} R \arrow[r, "\cdot \chi"] \arrow[d, "\cdot \chi"'] & MH_*(\Lambda; \nu \otimes \mu) \arrow[d, "\parallel"'] & \overline{MH}_*(\Lambda; \nu \otimes \mu) \arrow[d, "\parallel"'] \\ \dots \arrow[r, "\chi R"] & MH_*(\Lambda; \nu \otimes \underline{o}) \arrow[r, "\chi_{pt}; \nu \otimes \underline{o}"] & MH_*(\Lambda; \chi_{pt}; \nu \otimes \underline{o}) \arrow[d, "\cdot \chi"'] & \dots \end{tikzcd} $$

We thus get a canonical map

$$ \overline{MH}_*(\Lambda; \nu \otimes \mu) \to MH_*(\Lambda, \chi_{pt}; \nu \otimes \underline{o}) \quad (41) $$

which is an isomorphism if and only if the map $\chi R \to MH_0(\Lambda; \nu \otimes \underline{o})$ is injective. To study its injectivity, we can restrict without loss of generality to the component of contractible loops, in which case the target of this map is $R$ if $\nu \otimes \underline{o}$ is trivial and $R/2R$ if $\nu \otimes \underline{o}$ is nontrivial on that component. We thus obtain injectivity of this map, and an isomorphism in (41), under any of the following conditions:

(i) $\nu \otimes \underline{o}$ is trivial on the component of contractible loops.

(ii) $\chi = 0$.

(iii) $R$ is 2-torsion.

## A.9. Isomorphism between symplectic homology and loop homology

We spell out in this section the isomorphism between the symplectic homology of the cotangent bundle and the homology of the free loop space with twisted coefficients.

For the next definition, recall the local systems

$$ \sigma = \tau_{w_2}, \quad \tilde{\sigma} = \mathrm{ev}_0^*|M|^{-w} $$

from (39) and (40), as well as the orientation local systems

$$ \mu = \mathrm{ev}_0^*|M|^{-1} = o^{-1}. $$

**Definition A.16.** (Abouzaid [7]) The fundamental local system for symplectic homology of the cotangent bundle is the local system on $\Lambda$ given by

$$ \eta = \sigma \otimes \mu \otimes \tilde{\sigma}. $$
The fundamental local system $\eta$ is supported in degree $-n$. Our previous discussion shows that the loop product is defined and has degree 0 on $H_*(\Lambda; \eta)$, and the loop coproduct is defined and has degree +1 on $H_*(\Lambda, \Lambda_0; \eta^{-1})$. We can view the loop product as being defined on $H_*(\Lambda; \eta)$, where it has degree $-n$, and the loop coproduct as being defined (with field coefficients) on $H_*(\Lambda, \Lambda_0; \eta)$, where it has degree $1-n$. This point of view is useful when considering $\overline{H}_*(\Lambda; \eta)$, which is a common space of definition (to which the product descends and the coproduct extends).

As proved in [5,7], the chain map $\Psi = \Psi^{\text{quadratic}}$ discussed in Sect. 5.3 associated to a quadratic Hamiltonian acts as

$$ \Psi : FC_*(H) \to MC_*(S_L; \eta) $$

and induces an isomorphism $SH_*(D^*M) \xrightarrow{\simeq} H_*(\Lambda; \eta)$. Given any local system $\nu$, the same map acts as $\Psi : FC_*(H; \nu) \to MC_*(S_L; \nu \otimes \eta)$ and induces an isomorphism $SH_*(D^*M; \nu) \xrightarrow{\simeq} H_*(\Lambda; \nu \otimes \eta)$.

Our filtered chain map $\Psi = \Psi^{\text{linear}}$ from Sect. 5.5 associated to a linear Hamiltonian is a chain isomorphism $FC_*(H) \xrightarrow{\simeq} MC_*^{\leq \mu}(E^{1/2}; \eta)$, with $\mu$ the slope of the Hamiltonian. Given any local system $\nu$, we obtain a filtered chain isomorphism $FC_*(H; \nu) \xrightarrow{\simeq} MC_*^{\leq \mu}(E^{1/2}; \nu \otimes \eta)$.

In case the local system $\nu$ is compatible with products, the arguments of [1,3,7] adapt in order to show that the map $\Psi$ intertwines the pair-of-pants product on the symplectic homology side with the homology product on the Morse side. The arguments of Theorem 6.1 adapt in order to show that the map $\Psi$ descends on homology relative to the constant loops, where it intertwines the continuation coproduct with the loop coproduct (with field coefficients).

**Theorem A.17.** ([1,3,5,7], Theorem 5.3, Theorem 6.1) Given any local system $\nu$ compatible with products, the filtered chain level map $\Psi$ induces filtered isomorphisms

$$ \Psi_* : SH_*(D^*M; \nu) \xrightarrow{\simeq} H_*(\Lambda; \nu \otimes \eta), $$

$$ \Psi_*^{>0} : SH_*^{>0}(D^*M; \nu) \xrightarrow{\simeq} H_*(\Lambda, \Lambda_0; \nu \otimes \eta). $$

Moreover,

- $\Psi_*$ intertwines the pair-of-pants product with the Chas-Sullivan loop product,

- $\Psi_*^{>0}$ intertwines the continuation coproduct with the loop coproduct (with field coefficients).

□

There are also reduced versions of the map $\Psi$ which intertwine the product, and which also intertwine the coproducts provided both are defined using the same continuation data at the endpoints. The statements for the coproducts can, moreover, be interpreted as dual statements about products in cohomology.
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Kai Cieliebak

Universität Augsburg

Universitätsstrasse 14

86159 Augsburg

Germany

e-mail: kai.cieliebak@math.uni-augsburg.de

Nancy Hingston

Department of Mathematics and Statistics

College of New Jersey

Ewing New Jersey08628

USA

e-mail: hingston@tcnj.edu

Alexandru Oancea

Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG

Sorbonne Université and Université Paris Cité, CNRS

Paris

France

and

Institut de recherche mathématique avancée, IRMA

Université de Strasbourg and CNRS

Strasbourg

France

oancea@unistra.fr

Accepted: January 28, 2023.