2.1 Commutative algebras

Let $\mathsf {Com}$ denote the nonunital commutative operad in $\mathbb {Q}\text {-}\mathsf {mod}$ , having $\mathsf {Com}(r) = 0$ for $r=0$ and $\mathsf {Com}(r) = \mathbb {Q}$ as a trivial $\mathfrak {S}_r$ -representations otherwise. A commutative algebra in $\mathsf {Ch}$ is an algebra for the operad $\mathsf {Com}$ or, equivalently, the associated monad.

The (desuspended) Harrison complex of commutative algebra A is

where $[1]$ denotes suspension and $[-1]$ denotes desuspension of chain complexes, and the differential $d=d_A + d_{\mathrm {Harr}}$ is the sum of the differential $d_A$ induced by that on A and the differential $d_{\mathrm {Harr}}$ induced by the unique map of Lie coalgebras given on cogenerators by the commutative multiplication map $\mathrm {coLie}^2(A[1]) = \Lambda ^2(A[1])= \mathrm {Sym}^2(A)[2] \to A[1]$ . These satisfy $d_Ad_{\mathrm {Harr}}+d_{\mathrm {Harr}}d_A = 0$ , so $d^2 = 0$ .

If A is a nonunital commutative algebra object in the subcategory $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod}) \subset \mathsf {Ch}$ , then the objects $\mathrm {Harr}(A)$ inherit a further grading, as usual in homological algebra. We may think of this as follows: The grading of A determines a $\mathbb {Q}^{\times }$ -action on A, where $u \in \mathbb {Q}^{\times }$ acts as $u^q$ in degree q. Functoriality gives a $\mathbb {Q}^{\times }$ -action on $\mathrm {Harr}(A)$ , and we write $\mathrm {Harr}_{*}(A)_q$ for the subspace on which $u \in \mathbb {Q}^{\times }$ acts as $u^q$ . It is clear that such eigenspaces exhaust $\mathrm {Harr}(A)$ , and we set

and call p the Harrison degree, q the internal degree and $p+q-1$ the total degree.

If A also has a weight grading, that is, is a nonunital commutative algebra object in $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})^{\mathbb {N}}$ , then the Harrison complex and its homology obtains a further weight grading, and we write

for the piece of weight w.

2.2 Lie algebras

Let $\mathsf {Lie}$ denote the Lie operad, whose definition can be found in [Reference MillèsLV12, Section 13.2.3]. A Lie algebra in $\mathsf {Ch}$ is an algebra for the operad $\mathsf {Lie}$ , or equivalently the associated monad. The (desuspended) Chevalley–Eilenberg complex of a Lie algebra L is

$$\begin{align*}\mathrm{CE}(L)[-1] = \mathrm{coCom}(L[1])[-1],\end{align*}$$

where the differential $d=d_L + d_{\mathrm {CE}}$ is again the sum of the differential $d_L$ induced by that on L and the differential $d_{\mathrm { CE}}$ induced by the unique map of cocommutative coalgebras given on cogenerators by the Lie bracket map $\mathrm {coCom}^2(L[1]) = \mathrm { Sym}^2(L[1]) = \mathrm {Lie}^2(L)[2] \to L[1]$ . These satisfy $d_Ld_{\mathrm {CE}}+d_{\mathrm {CE}}d_L = 0$ , so $d^2 = 0$ .

If L is a Lie algebra object in $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ , then as above we can endow the Chevalley–Eilenberg complex with a further grading, and we set

If L also has a weight grading, then we write

for the piece of weight w.

2.3 Koszul duality

The operads $\mathsf {Com}$ and $\mathsf {Lie}$ are Koszul dual [Reference Garoufalidis and NakamuraGK94, Corollary 4.2.7], [Reference MillèsLV12, Proposition 13.1.5], and thus there is a Koszul duality between nonunital commutative algebras and Lie algebras.

A quadratic datum for a nonunital commutative algebra is a pair $(V,S)$ of a finite-dimensional graded vector space V and a subspace $S \subset \mathrm {Com}(2) \otimes _{\mathfrak {S}_2} V^{\otimes 2}$ . From this, we may form the quadratic nonunital commutative algebra $A(V,S)$ as the quotient of the free graded nonunital commutative algebra $\mathrm {Com}(V)$ by the ideal generated by S. As the relations are homogeneous in V (namely purely quadratic), we can endow this with a weight grading by declaring V to have weight 1. Similarly, a quadratic datum for a Lie algebra, given by a pair $(W,R)$ of a finite-dimensional graded vector space W and a subspace $R \subset \mathrm { Lie}(2) \otimes _{\mathfrak {S}_2} W^{\otimes 2}$ , yields a quadratic Lie algebra $L(W,R)$ ; it has a weight grading by declaring W to have weight 1.

Given a quadratically presented nonunital commutative algebra $A=A(V,S)$ , its quadratic dual Lie algebra is .Footnote ¹ Similarly, given a quadratically presented Lie algebra $L = L(W,R)$ , its quadratic dual nonunital commutative algebra is . These constructions are natural in the quadratic data, and are dualities in the sense that there are natural isomorphisms $A \cong (A^!)^!$ and $L \cong (L^!)^!$ . In particular, the quadratic dual preserves any group actions on the objects in question.

The definitions of Koszulness given in Definition 2.2 and Definition 2.3 are the diagonal criteria; a reference is [Reference MoritaMil12, Theorem 4.9 (ii)]. By [Reference MoritaMil12, Theorem 4.11], A is Koszul if and only if $A^!$ is Koszul. The same is true for the notions of Koszulness in a range by the same argument:

2.4 Orthogonal and symplectic representation theory

We recall some facts from [Reference Koike and TeradaKRW20c, Section 2.1]. Let $H(g)$ be a $2g$ -dimensional rational vector space with nonsingular $\epsilon $ -symmetric pairing $\lambda \colon H(g) \otimes H(g) \to \mathbb {Q}$ given by the hyperbolic form. Its dual $\epsilon $ -symmetric form $\omega \colon \mathbb {Q} \to H(g) \otimes H(g)$ characteristic by $(\lambda \otimes \mathrm {id})(- \otimes \omega ) = \mathrm {id}(-)$ . The automorphisms of $H(g)$ preserving $\lambda $ are denoted $\mathrm {O}_{\epsilon }(H(g))$ ; these are the $\mathbb {Q}$ -points of an algebraic group $\mathbf {O}_{g,g}$ for $\epsilon =1$ and $\mathbf {Sp}_{2g}$ for $\epsilon =-1$ . Observe that $\mathrm {O}_{\epsilon }(H(g))$ is a subgroup of $\mathrm {GL}_{2g}(\mathbb {Q})$ and the intersection $\mathrm {O}_{\epsilon }(H(g)) \cap \mathrm {GL}_{2g}(\mathbb {Z})$ is $\mathrm {O}_{g,g}(\mathbb {Z})$ if $\epsilon =1$ and $\mathrm {Sp}_{2g}(\mathbb {Z})$ if $\epsilon =-1$ .

For distinct i and j in $\{1,2,\ldots ,q\}$ , applying the pairing $\lambda $ to the ith and jth factors yields a map

$$\begin{align*}\lambda_{i,j} \colon H(g)^{\otimes q} \longrightarrow H(g)^{\otimes q-2}\end{align*}$$

and inserting the form $\omega $ in the ith and jth factors yields a map

$$\begin{align*}\omega_{i,j} \colon H(g)^{\otimes q-2} \longrightarrow H(g)^{\otimes q}.\end{align*}$$

We obtain representations of $\mathrm {O}_{\epsilon }(H(g))$ by

and the composition $H(g)^{[q]} \to H(g)^{\otimes q} \to H(g)_{[q]}$ is an isomorphism. The action of the symmetric group $\Sigma _q$ permuting the terms in the tensor product $H(g)^{\otimes q}$ descends to an action on $H(g)^{[q]}$ and $H(g)_{[q]}$ .

Recall that the irreducible rational representations of $\mathfrak {S}_q$ are in bijection with partitions $\lambda $ of q into positive integers; these are denoted $S^{\lambda }$ . Then we define

By construction, these are algebraic $\mathrm {O}_{\epsilon }(H(g))$ -representations. They are either zero or irreducible, and the irreducible ones are distinct and exhaust all isomorphism classes.

We will have a use for representations of arithmetic subgroups $G \subset \mathrm {O}_{\epsilon }(H(g))$ , that is, groups $G \subset \mathrm { O}_{\epsilon }(H(g))$ which are commensurable with $\mathrm {O}_{g,g}(\mathbb {Z})$ or $\mathrm {Sp}_{2g}(\mathbb {Z})$ . As long as $g \geq 2$ , the $V_{\lambda }$ restrict to irreducible algebraic representations of G and any algebraic representation of G is a direct sum of these [Reference Koike and TeradaKRW20c, Section 2.1].

3.1 Basic results

3.1.1 Definitions

Let $\Sigma ^r_{g,n}$ be a surface of genus g with n boundary components and r marked points, and $\Gamma ^r_{g,n}$ its mapping class group, consisting of isotopy classes of orientation-preserving diffeomorphism fixing pointwise the boundary components and the marked points. Its Torelli group is the subgroup $T_{g,n}^r \subset \Gamma _{g,n}^r$ of those isotopy classes which act as the identity on $H_1(\Sigma _g;\mathbb {Z})$ through the inclusion $\Sigma _{g,n}^r \hookrightarrow \Sigma _g$ . Its pro-unipotent, or Malcev, completion $\mathfrak {t}^r_{g,n}$ is the Lie algebra of the initial pro-unipotent algebraic group over $\mathbb {Q}$ under $T^r_{g,n}$ .

There is a tautological extension of groups $T_{g,n}^r \to \Gamma _{g,n}^r \to \mathrm {Sp}_{2g}(\mathbb {Z})$ , and the induced representation $\Gamma _{g,n}^r \to \mathrm {Sp}_{2g}(\mathbb {Q})$ into the $\mathbb {Q}$ -points of the algebraic group $\mathbf {Sp}_{2g}$ is Zariski dense. There is an initial pro-algebraic group $\mathcal {G}_{g,n}^r$ over $\mathbb {Q}$ with Zariski dense homomorphism to $\mathbf {Sp}_{2g}$ and pro-unipotent kernel [Reference HainHai93, §2]. The Lie algebra of $\mathcal {G}_{g,n}^r$ is $\mathfrak {g}_{g,n}^r$ and the pro-nilpotent Lie algebra of this kernel is the relative unipotent completion $\mathfrak {u}_{g,n}^r$ of $T_{g,n}^r$ .

Let $\mathfrak {p}^r_{g,n}$ denote the Lie algebra of the pro-unipotent completion of the fundamental group $\pi ^r_{g,n}$ of the ordered configuration space in $\Sigma _g$ of r points and n points with nonzero tangent vector (in the notation of [Reference HainHai97] we have $\mathfrak {p}_g = \mathfrak {p}^1_g$ ). The usual extensions of groups relating mapping class groups yield extensions of Lie algebras (Theorem 3.4 and Proposition 3.6 of [Reference HainHai97]).

Lemma 3.1. For $g \geq 3$ , there are extensions of Lie algebras

$$\begin{align*}\mathfrak{p}^r_{g,n} \longrightarrow \mathfrak{t}^r_{g,n} \longrightarrow \mathfrak{t}_g, \quad\quad\quad\quad \mathbb{Q} \longrightarrow \mathfrak{t}^r_{g,n} \longrightarrow \mathfrak{t}^{r+1}_{g,n-1},\end{align*}$$

$$ \begin{align*} \mathbb{Q} \longrightarrow \mathfrak{t}^r_{g,n} \longrightarrow \mathfrak{u}^r_{g,n}.\\[-20pt] \end{align*} $$

3.1.2 Mixed Hodge structures and lower central series

The associated graded of the lower central series of a Lie algebra $\mathfrak {g}$ is a Lie algebra ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {g}$ . We consider this as a Lie algebra in $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})^{\mathbb {N}}$ by giving it homological degree 0 and giving $\mathrm {Gr}^w_{\mathrm {LCS}}\, \mathfrak {g}$ weight w. In the following, there are unfortunately two things called weight: our additional grading and the Hodge theoretic weight filtration. We do not think any confusion is likely, but we shall refer to our weight grading as ‘the additional grading’ in places where confusion is possible.

As long as $g \geq 3$ , Theorem 4.10 of [Reference HainHai97] says that $\mathfrak {t}_{g,n}^r$ is equipped with a mixed Hodge structure whose weight filtration agrees, up to negation of the indexing, with the lower central filtration. By Corollary 4.8 and 4.9 of loc. cit., the same is true for $\mathfrak {u}_{g,n}^r$ . Corollary 5.3 of loc. cit. (stated over $\mathbb {C}$ , see [Reference HirschhornHai20, Section 7.4] for why it is also true over $\mathbb {Q}$ ) implies that for $g \geq 3$ there are isomorphisms of pro-nilpotent Lie algebras

$$\begin{align*}\mathfrak{t}^r_{g,n} \cong \prod_{s>0} \mathrm{Gr}^{s}_{\mathrm{LCS}}\, \mathfrak{t}^r_{g,n} \qquad \text{and} \quad \mathfrak{u}^r_{g,n} \cong \prod_{s>0} \mathrm{Gr}^{s}_{\mathrm{LCS}}\, \mathfrak{u}^r_{g,n}.\end{align*}$$

Similarly, $\mathfrak {p}^r_{g,n}$ comes with a mixed Hodge structure [Reference HainHai87, Theorem 1], and by Section 2 and Lemma 4.7 of [Reference HainHai97] its weight filtration agrees, up to negation of the indexing, with the lower central series filtration.

The existence of compatible mixed Hodge structures implies that taking the associated graded of the weight filtration is exact and hence so is taking the associated graded of the lower central series filtration. In Lemma 3.1, the two copies of $\mathbb {Q}$ are the Hodge structures $\mathbb {Q}(1)$ , so we obtain the following (using Section 13 and Theorem 4.10 of [Reference HainHai97]), where we clarify that the $\mathbb {Q}[2]$ are in weight $2$ and homological degree $0$ .

Lemma 3.2. For $g \geq 3$ , there are extensions of Lie algebras with additional grading

$$\begin{align*}{\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{p}^r_{g,n} \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}^r_{g,n} \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_g \quad\quad\quad \mathbb{Q}[2] \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}^r_{g,n} \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}^{r+1}_{g,n-1} ,\end{align*}$$

$$ \begin{align*} \mathbb{Q}[2] \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}^r_{g,n} \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{u}^r_{g,n}. \\[-39pt]\end{align*} $$

3.1.3 Koszulness

The general version of Theorem A is as follows.

Theorem 3.3. The Lie algebras with additional grading ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ , ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}^1$ , ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g$ , ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}_{g,1}$ , ${\mathrm { Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}^1_g$ and ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}_g$ are Koszul in weight $\leq \tfrac {g}{3}$ .

The following proposition shows that Theorem A implies Theorem 3.3, so we may focus on Theorem A for the remainder of this paper.

Proposition 3.4. Suppose $g \geq 3$ and that ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ is Koszul in weight $\leq W$ . Then the same is true for ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}^1$ , ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g$ , ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}_{g,1}$ , ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}^1_g$ and ${\mathrm { Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}_g$ .

Proof. Though we defined Koszulness in terms of Chevalley–Eilenberg homology, to verify its vanishing we may as well use Chevalley–Eilenberg cohomology, and take advantage of its multiplicative structure. We first prove that ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}^1_g$ is Koszul in additional gradings $ \leq W$ , by considering the spectral sequence in Lie algebra cohomology for the extension

$$\begin{align*}\mathbb{Q}[2] \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g,1} \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_g^1\end{align*}$$

of Lemma 3.2. This yields a Gysin sequence

$$\begin{align*}\cdots \longrightarrow H_{\mathrm{Lie}}^{p-2}({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g}^1)_{0,w-2} \overset{e \cdot -}\longrightarrow H_{\mathrm{Lie}}^{p}({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g}^1)_{0,w} \longrightarrow H_{\mathrm{Lie}}^{p}({\mathrm{ Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g,1})_{0,w} \longrightarrow \cdots\end{align*}$$

If $w \leq W$ , then the right-hand term is zero whenever $p \neq w$ , in which case $H_{\mathrm {Lie}}^{p}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}^1)_{0,w}$ is infinitely divisible by e and hence vanishes. Thus, ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}^1$ is also Koszul in additional gradings $ \leq W$ .

We next prove that ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,}\mathfrak {t}_g$ is Koszul in additional gradings $ \leq W$ , using the extension

$$\begin{align*}{\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{p}^1_g \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}^1_g \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_g\end{align*}$$

of Lemma 3.2. We will make use of continuous Lie algebra cohomology; see [Reference HainHai97, Section 5] for background. By Proposition 5.5 of [Reference HainHai97], we have an isomorphism

$$\begin{align*}H_{\mathrm{Lie}}^*({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{p}_g) = H_{\mathrm{Lie}}^*(\mathrm{Gr}^{-\bullet}_{\mathrm{W}} \mathfrak{p}_g) \cong \mathrm{Gr}^{-\bullet}_{\mathrm{W}} H_{\mathrm{Lie}, \text{cts}}^*(\mathfrak{p}_g)\end{align*}$$

and as surface groups are pseudo-nilpotent [Reference Kupers and Randal-WilliamsKO87] we have $H^*_{\mathrm {Lie}, \text {cts}}(\mathfrak {p}_g) = H^*(\Sigma _g; \mathbb {Q})$ . In particular, we have $H_{\mathrm {Lie}}^2({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {p}_g) = \mathbb {Q}$ and the cohomology in higher degrees vanishes. Thus, in the spectral sequence for this extension projection to the top row provides a fibre-integration map

$$\begin{align*}\pi_!\colon H_{\mathrm{Lie}}^*({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_g^1) \longrightarrow H_{\mathrm{Lie}}^{*-2}({\mathrm{ Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_g)\end{align*}$$

which as usual is a map of $H^{*}_{\mathrm {Lie}}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g)$ -modules and satisfies $\pi _!(e) = 2-2g \neq 0 \in H_{\mathrm {Lie}}^{0}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g)$ by comparison to the corresponding extension of groups $\pi _1(\Sigma _g) \to T_g^1 \to T_g$ . In particular, this implies that $\pi ^*\colon H_{\mathrm {Lie}}^{p}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g)_{0,w} \to H_{\mathrm {Lie}}^{p}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g^1)_{0,w}$ is injective. Assuming ${\mathrm { Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_g^1$ is Koszul the target of this map vanishes for $p \neq w$ and $w \leq W$ : Thus, the source does too.

For $\mathfrak {u}_{g,1}$ , we use the extension $\mathbb {Q}(1) \to \mathfrak {t}_{g,1} \to \mathfrak {u}_{g,1}$ of Lie algebras with mixed Hodge structure and proceed as we did above for $\mathfrak {t}^1_g$ , taking associated graded and using the Gysin sequence; $\mathfrak {u}_g^1$ and $\mathfrak {u}_g$ may be treated similarly.

Remark 3.5. The Lie algebra ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}$ for $g \geq 4$ can never actually be Koszul, and hence neither can ${\mathrm { Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}^1$ or ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ be by the proof of Proposition 3.4. If ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g}$ were Koszul, then ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}_{g}$ would be too by the Gysin sequence argument of Proposition 3.4 applied to

$$\begin{align*}\mathbb{Q}[2] \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g} \longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{u}_{g}.\end{align*}$$

This would imply that the cohomology algebras $H^*_{\mathrm {Lie}}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g})$ and $H^*_{\mathrm {Lie}}({\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {u}_{g})$ are both generated in tridegree $(1,0,1)$ by the same vector space $(\mathrm {Gr}_{\text {LCS}}^1\, \mathfrak {u}_{g})^{\vee } \overset {\sim } \to (\mathrm {Gr}_{\text {LCS}}^1\, \mathfrak {t}_{g})^{\vee }$ . This is finite-dimensional for $g \geq 3$ by a theorem of Johnson [Reference Kawazumi and MoritaJoh85], so the map

$$\begin{align*}H^*_{\mathrm{Lie}}({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{u}_{g}) \longrightarrow H^*_{\mathrm{Lie}}({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g})\end{align*}$$

is surjective and both cohomology groups vanish above some cohomological degree. Let the top degrees in which the cohomologies are nontrivial be called U and T, so this surjection shows that $U \geq T$ . On the other hand the Gysin sequence for the extension shows that $T = U+1$ , a contradiction.

As far as we can tell Koszulness of ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,}\mathfrak {u}_g$ cannot be ruled out in this way:

Question 3.6. Is it possible that ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,}\mathfrak {u}_g$ is Koszul?

3.2 Cohomology of $\mathfrak {t}_{g,1}$ in low degrees

Following Morita and Kawazumi [Reference Kohno and OdaKM96], in [Reference Koike and TeradaKRW20c, Section 5.1] we have constructed, for each $a,r \geq 0$ with $r+2a -2 \geq 0$ and tuple $v_1, v_2, \ldots , v_r \in H^1(\Sigma _g;\mathbb {Q})$ , a cohomology class

$$\begin{align*}\kappa_{e^a}(v_1 \otimes v_2 \otimes \cdots \otimes v_r) \in H^{r-2 + 2a}(T_{g,1};\mathbb{Q})\end{align*}$$

generalising the Miller–Morita–Mumford class $\kappa _{e^a}$ ; we refer to it as a twisted MMM-class. (We often write $1$ for $e^0$ .) This construction is linear in the $v_i$ , is alternating under permuting the $v_i$ and is $\mathrm {Sp}_{2g}(\mathbb {Z})$ -equivariant.

Lemma 3.8. If $\{a_i\}$ is a basis of $H^1(\Sigma _g;\mathbb {Q})$ and $\{a_i^{\#}\}$ is the dual basis characterised by $\langle a_i^{\#} \cdot a_j, [\Sigma _g] \rangle = \delta _{ij}$ , then there are relations

$$ \begin{align*} \sum_i \kappa_{e^a}(v_1 \otimes \cdots \otimes v_j \otimes a_i) \cdot \kappa_{e^b}(a_i^{\#} \otimes v_{j+1} \otimes \cdots \otimes v_r) &= \kappa_{e^{a+b}}(v_1 \otimes \cdots \otimes v_r)\\ \sum_i \kappa_{e^a}(v_1 \otimes \cdots \otimes v_r \otimes a_i \otimes a_i^{\#}) &=\kappa_{e^{a+1}}(v_1 \otimes \cdots \otimes v_r)\\ \sum_{i,j,k} \kappa_1(a_i \otimes a_j \otimes a_k) \cdot \kappa_1(a_i^{\#} \otimes a_j^{\#} \otimes a_k^{\#}) &=0. \end{align*} $$

This construction for $a=0$ and $r=3$ gives a map of $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representations

$$\begin{align*}\kappa_1\colon \Lambda^3 V_1 \longrightarrow H^{1}(T_{g,1};\mathbb{Q}).\end{align*}$$

Taking cup products of such classes determines an algebra homomorphism

$$\begin{align*}\phi \colon \Lambda^*[\Lambda^3 V_1[1]] \longrightarrow H^{*}(T_{g,1};\mathbb{Q}).\end{align*}$$

It follows from the discussion in [Reference Koike and TeradaKRW20c, Section 8.2] and the estimate in [Reference Koike and TeradaKRW20c, Section 9.3] that as long as $g \geq 4$ this map is an isomorphism in degree 1 and its kernel in degree 2 is spanned by the terms

(IH) $$ \begin{align} \sum_i \kappa_1(v_1 \otimes v_2 \otimes a_i) \cdot \kappa_1(a_i^{\#} \otimes v_5 \otimes v_6) - \sum_i \kappa_1(v_1 \otimes v_5 \otimes a_i) \cdot \kappa_1(a_i^{\#} \otimes v_6 \otimes v_2) \end{align} $$

for $v_1, v_2, v_5, v_6 \in H^1(\Sigma _{g,1};\mathbb {Q})$ , as well as the invariant vector

(Θ) $$ \begin{align} \sum_{i,j,k} \kappa_1(a_i \otimes a_j \otimes a_k) \cdot \kappa_1(a_i^{\#} \otimes a_j^{\#} \otimes a_k^{\#}) \end{align} $$

(which, as we showed in the proof of Lemma 3.8 above, represents $-3 \kappa _{\mathcal {L}_1}$ ). Furthermore, in degrees $\leq 2$ the image of $\phi $ is the maximal algebraic subrepresentation $H^{*}(T_{g,1};\mathbb {Q})^{\text {alg}}$ .

Theorem 3.10. There is a homomorphism of trigraded algebras

$$\begin{align*}\psi \colon \frac{\Lambda^*[\Lambda^3 V_1[1,0,1]]}{((\text{IH}), (\Theta))} \longrightarrow H_{\mathrm{Lie}}^{*}({\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g,1})_{*,*},\end{align*}$$

which as long as $g \geq 4$ is an isomorphism in cohomological degrees $* \leq 2$ .

Proof. We will work with the continuous Lie algebra cohomology $H_{\mathrm {Lie}, \text {cts}}^*(\mathfrak {t}_{g,1})$ , cf. [Reference HainHai97, Section 5]. By [Reference HainHai97, Proposition 5.5], there is an isomorphism

(1) $$ \begin{align} \mathrm{Gr}^W_{\bullet} H_{\mathrm{Lie}, \text{cts}}^*(\mathfrak{t}_{g,1}) \cong H_{\mathrm{Lie}}^*(\mathrm{Gr}^W_{\bullet} \mathfrak{t}_{g,1}), \end{align} $$

and by [Reference HainHai97, Theorem 4.10] there is an identification

(2) $$ \begin{align} \mathrm{Gr}^W_{\bullet} \mathfrak{t}_{g,1} = \mathrm{Gr}_{\mathrm{LCS}}^{-\bullet} \mathfrak{t}_{g,1}. \end{align} $$

By definition of the lower central series, this Lie algebra with additional grading is generated by its piece of grading 1. Its grading 1 piece is by definition the abelianisation of the Lie algebra $\mathfrak {t}_{g,1}$ , which is tautologically identified with $H_1(T_{g,1};\mathbb {Q})$ and so for $g \geq 3$ is the algebraic $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representation $\Lambda ^3 V_1$ by a theorem of Johnson [Reference Kawazumi and MoritaJoh85]. It follows that the Lie algebra with additional grading ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ is a finite-dimensional algebraic $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representation in each additional grading, so $\smash {\mathrm { Gr}^W_{\bullet }} H_{\mathrm {Lie}, \text {cts}}^*(\mathfrak {t}_{g,1})$ is a finite-dimensional algebraic $\mathrm { Sp}_{2g}(\mathbb {Z})$ -representation in each bigrading. As the weight filtration is finite in each cohomological degree, it follows that $H_{\mathrm {Lie}, \text {cts}}^*(\mathfrak {t}_{g,1})$ is a finite-dimensional algebraic $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representation in each degree.

Consider the natural map

(3) $$ \begin{align} H_{\mathrm{Lie}, \text{cts}}^*(\mathfrak{t}_{g,1}) \longrightarrow H^*(T_{g,1};\mathbb{Q}). \end{align} $$

By the discussion above, it has image in the maximal algebraic subrepresentation $ H^{*}(T_{g,1};\mathbb {Q})^{\text {alg}}$ . As $H^1(T_{g,1};\mathbb {Q}) \cong \Lambda ^3 V_1$ is finite-dimensional, [Reference HainHai97, Proposition 5.1] implies that (3) is an isomorphism in degree 1 and a monomorphism in degree 2. Thus, $\phi $ factors as

$$\begin{align*}\phi \colon \Lambda^*[\Lambda^3 V_1[1]] \overset{\psi"}\longrightarrow H_{\mathrm{Lie}, \text{cts}}^*(\mathfrak{t}_{g,1}) \longrightarrow H^{*}(T_{g,1};\mathbb{Q})^{\text{alg}}\end{align*}$$

and the relations (IH) and (Θ) hold in $H_{\mathrm {Lie}, \text {cts}}^2(\mathfrak {t}_{g,1})$ , so $\psi "$ descends to give

$$\begin{align*}\frac{\Lambda^*[\Lambda^3 V_1[1]]}{(({IH}), (\Theta))} \overset{\psi'}\longrightarrow H_{\mathrm{Lie}, \text{cts}}^{*}(\mathfrak{t}_{g,1}) \longrightarrow H^{*}(T_{g,1};\mathbb{Q})^{\text{alg}}\end{align*}$$

with $\psi '$ an isomorphism in degree 1 and a monomorphism in degree 2. By the discussion above the composition is an isomorphism in degrees $\leq 2$ , so it follows that $\psi '$ is too.

It follows from (1) and (2) that $H_{\mathrm {Lie}, \text {cts}}^{1}(\mathfrak {t}_{g,1})$ is a pure Hodge structure of weight $-1$ , so if we give the domain of $\psi '$ the weight filtration given by minus its degree, then $\psi '$ is an isomorphism of weight-filtered graded vector spaces in degrees $* \leq 2$ : passing to associated graded for the weight filtration, applying (1) and (2) again, and recalling that ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ is supported in homological degree 0, gives the map $\psi $ in the statement of the lemma and shows it is an isomorphism in cohomological degrees $* \leq 2$ .

Our next goal is to describe the quadratic presentation of ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ . This uses the following lemma.

Proof. As $\mathfrak {g}$ is quadratically presented it admits an additional grading by weight, where W is put in weight 1. This endows the Chevalley–Eilenberg chains $(\mathrm {coCom}(\mathfrak {g}[1])[-1], d_{CE})$ with an additional weight grading, with respect to which it takes the form (homological degree is displayed horizontally, weight vertically)

The maps from the second to the first column are given by $[-,-] \colon \Lambda ^2 \mathfrak {g} \to \mathfrak {g}$ so are surjective in weights $>1$ , as $\mathfrak {g}$ is generated in weight 1. Dualising this complex, it follows that $H^1_{\mathrm {Lie}}(\mathfrak {g})=W^{\vee }$ and that there is an exact sequence as claimed

$$\begin{align*}0 \longrightarrow (\mathrm{Lie}^2(W)/Q)^{\vee} \longrightarrow \Lambda^2 W^{\vee} = \Lambda^2 H^1_{\mathrm{Lie}}(\mathfrak{g}) \longrightarrow H^2_{\mathrm{Lie}}(\mathfrak{g}).\\[-36pt] \end{align*}$$

Let us write and $R^{\perp } \leq (\Lambda ^2 \Lambda ^3 V_1)^{\vee } \cong \Lambda ^2 \Lambda ^3 V_1$ for its annihilator.

Remark 3.13. As long as $g \geq 6$ we have the decomposition

$$\begin{align*}\Lambda^2 \Lambda^3 V_1 = 2 V_0 + 3 V_{1^2} + 2V_{1^4} + V_{1^6} + V_{2,1^2} + V_{2^2} + V_{2^2,1^2}\end{align*}$$

into irreducible $\mathrm {Sp}_{2g}(\mathbb {Z})$ -representations. The vectors (IH) generate the representation $V_0 + V_{1^2} + V_{2^2}$ (see [Reference Galatius and Randal-WilliamsGN98, Proof of Corollary 2.2 (ii)]Footnote ²), and the invariant vector ( $\Theta $ ) can be checked not to lie in (IH). Therefore, $R^{\perp } \cong 2 V_{1^2} + 2V_{1^4} + V_{1^6} + V_{2,1^2} + V_{2^2,1^2}$ . For $g < 6$ , the representation theory is degenerate but in principle a similar analysis can be made.

The following is essentially in [Reference HainHai97], though Hain does not give the final answer in the case of $\mathfrak {t}_{g,1}$ . The explicit presentation produced by Hain’s method in this case was given in [Reference IsaacsonHS00] (and agrees with the following).

Corollary 3.14 (Hain)

For $g \geq 4$ , there is an isomorphism

$$\begin{align*}\mathrm{Lie}(\Lambda^3 V_1 [0,1])/ ( R^{\perp} ) \overset{\sim}\longrightarrow {\mathrm{Gr}^{\bullet}_{\mathrm{LCS}}\,} \mathfrak{t}_{g,1}.\end{align*}$$

Using Lemma 2.4, it follows that for $g \geq 4$ the Lie algebra ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ is Koszul in weight $\leq W$ if and only if the commutative algebra object $\frac {\Lambda ^*[\Lambda ^3 V_1[1,1]]}{(({IH}), (\Theta ))}$ in $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})^{\mathbb {N}}$ is Koszul in weight $\leq W$ , allowing us to work with the latter.

4.1 The downward Brauer category

The following is identical to [Reference Koike and TeradaKRW20c, Definition 2.15]. One thinks of the morphisms of this category as in Figure 1.

Figure 1 A graphical representation of a morphism $(f,m_S)$ in $\mathsf {dBr}(S,T)$ from a 7-element set S to a 5-element set T. The order of crossings is irrelevant.

There is also the following signed version, identical to [Reference Koike and TeradaKRW20c, Definition 2.20].

We will also write $\mathsf {FB}$ for the category of finite sets and bijections and consider it (after $\mathbb {Q}$ -linearising) as a subcategory of both $\mathsf {dBr}$ and $\mathsf {dsBr}$ , consisting of all objects and morphisms spanned by those $(f,m_S)$ with f a bijection and $m_S=\varnothing $ . To treat the symmetric and skew-symmetric cases simultaneously, we will write $\mathsf {d(s)Br}$ for $\mathsf {dBr}$ if n is even and $\mathsf {dsBr}$ if n is odd.

Recall that $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ is the category of nonnegatively graded $\mathbb {Q}$ -vector spaces, considered as a $\mathbb {Q}\text {-}\mathsf {mod}$ -enriched category. We shall be interested in (graded) representations of the downward (signed) Brauer category, that is, enriched functors $F\colon \mathsf {d(s)Br} \to \mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ . The category of such functors $\mathsf {Fun}({\mathsf {d(s)Br}}, \mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod}))$ is again symmetric monoidal under Day convolution, using the symmetric monoidality on $\mathsf {d(s)Br}$ given by disjoint union and that on $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ given by tensor product of graded vector spaces and the Koszul sign rule. Using the tensoring of $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ over $\mathbb {Q}\text {-}\mathsf {mod}$ , it is given by the enriched coend

$$\begin{align*}(F \otimes G)(S) = \int^{S',S" \in \mathsf{d(s)Br}} \mathsf{d(s)Br}(S' \sqcup S",S) \otimes F(S') \otimes G(S"),\end{align*}$$

and the monoidal unit is the functor given by $\mathbb {Q}$ on the empty set and 0 on all other finite sets. One can compute the values of the Day convolution explicitly as

(4) $$ \begin{align}\bigoplus_{S',S"} \mathsf{Pair}(S' \sqcup S",S) \otimes_{\mathfrak{S}_{S'} \times \mathfrak{S}_{S"}} F(S') \otimes G(S") \overset{\cong}\longrightarrow (F \otimes G)(S),\end{align} $$

where the indexing set runs over all isomorphism classes of pairs of finite sets and $\mathsf {Pair}(S' \sqcup S",S) \subset \mathsf {d(s)Br}(S' \sqcup S",S)$ is the span of those $(f,m_{S' \sqcup S"})$ , where each pair in $m_{S' \sqcup S"}$ contains elements of both $S'$ and $S"$ . To see this, use that Day convolution preserves colimits in each variable to reduce to the case that F and G are representable and verify it by hand there.

4.1.1 Homotopy theory of representations of the downward Brauer category

We can do homotopy theory with these objects if we enlarge their target to $\mathsf {Ch}$ . One can use $\infty $ -categories or model structures; we opt for the latter because we need to do some explicit computations.

We endow $\mathsf {Ch}$ with the projective model structure [Reference HainGS07, Theorem 1.5]: the weak equivalences are quasi-isomorphisms, the cofibrations are degreewise monomorphisms and the fibrations are epimorphisms in positive degrees. Let us define chain complexes

(5)

Then this model structure is cofibrantly generated by the set I of generating cofibrations given by $0 \to S(0)$ and the inclusions $S(n-1) \hookrightarrow D(n)$ for $n \geq 1$ , and the set J of generating trivial cofibrations given by the inclusions $0 \hookrightarrow D(n)$ for $n \geq 1$ [Reference HainGS07, Examples 3.4(1)].

Lemma 4.3. The category $\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch})$ has a model structure whose weak equivalences and fibrations are objectwise. This is cofibrantly generated and monoidal with respect to Day convolution.

The proof proceeds as in the unenriched case [Reference Hain and LooijengaHir03, Theorem 11.6.1] by right transfer as in [Reference Hain and LooijengaHir03, Theorem 11.3.2]. The set of generating cofibrations $I'$ is given by the union over all $S \in \mathrm { ob}(\mathsf {d(s)Br})$ of the image of I under the left adjoint $S_*$ to the evaluation $ \mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch}) \to \mathsf {Ch}$ given by $F \mapsto F(S)$ , and the set of generating trivial cofibrations $J'$ is in the same manner with J in place of I. That this model structure is monoidal is [Reference JohnsonIsa09, Proposition 2.2.15].

By Lemma 4.3, the Day convolution tensor product preserves weak equivalences in each entry when all objects involved are cofibrant. The latter hypothesis is in fact unnecessary.

Lemma 4.4. The Day convolution tensor product on $\mathsf {d(s)Br}$ preserves weak equivalences in each entry.

Proof. Using (4), the result follows from the facts that for rational chain complexes tensor products and quotients by finite group actions preserve weak equivalences.

4.2 Commutative algebras on the downward Brauer category

Recall that $\mathsf {Com}$ denotes the nonunital commutative algebra operad. We write $\mathsf {Alg}_{\mathsf {Com}}(\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch}))$ for the category of $\mathsf {Com}$ -algebra objects in the symmetric monoidal category $\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch})$ ; these are nonunital commutative algebra objects.

4.2.1 Unital commutative algebras

Let $\mathsf {Com}^+$ denote the unital commutative algebra operad, having $\mathsf {Com}^+(r) = \mathbb {Q}$ with trivial $\mathfrak {S}_r$ -action for all $r \geq 0$ . We will write $\mathsf {Alg}_{\mathsf {Com}^+}(\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch}))$ for the category of $\mathsf {Com}^+$ -algebra objects in $\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch})$ ; these are unital commutative algebra objects.

The object

is canonically a unital commutative algebra object; an augmentation for a unital commutative algebra object R is a morphism

. Since the unit provides a section, there is a splitting

with $\overline {R} = \ker (\epsilon )$ the augmentation ideal. This is canonically a nonunital commutative algebra object and taking the augmentation ideal yields an equivalence of categories

(6) $$ \begin{align}\mathsf{Alg}^{\mathrm{augm}}_{\mathsf{Com}^+}(\mathsf{Fun}(\mathsf{d(s)Br}, \mathsf{Ch})) \overset{\cong}\longrightarrow \mathsf{Alg}_{\mathsf{Com}}(\mathsf{Fun}(\mathsf{d(s)Br}, \mathsf{Ch}))\end{align} $$

which has an inverse given by unitalisation.

4.2.2 Homotopy theory of commutative algebras and André–Quillen homology

To do homotopy theory with the nonunital commutative algebras in $\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch})$ , we transfer the model structure to these objects.

Lemma 4.5. The category $\mathsf {Alg}_{\mathsf {Com}}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch}))$ has a model structure whose weak equivalences and fibrations are objectwise. This is cofibrantly generated. $ \Box $

The proof proceeds as in the unenriched case by right transfer [Reference HainGS07, Example 3.7]. The set of generating cofibrations is $F^{\mathrm {Com}}I'$ and the set of generating trivial cofibrations is $F^{\mathrm {Com}} J'$ with $F^{\mathrm {Com}}$ the left adjoint in the free-forgetful adjunction

Remark 4.6. The same construction endows $\mathsf {Alg}_{\mathsf {Com}^+}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch}))$ with a model structure whose weak equivalences and fibrations are objectwise. Moreover, [Reference Hain and LooijengaHir03, Theorem 7.6.5] endows $\mathsf {Alg}^{\mathrm { augm}}_{\mathsf {Com}^+}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch}))$ with a model structure whose weak equivalences, cofibrations and fibrations are such when forgetting the augmentation. Then (6) is a Quillen equivalence.

The monad $\mathrm {Com}$ has an augmentation $\epsilon \colon \mathrm {Com} \to \mathrm {Id}$ by projection to the $n=1$ summand, and we define indecomposables $Q^{\mathrm {Com}}(\overline {R})$ of a nonunital commutative algebra object $\overline {R}$ as the following reflexive coequaliser in $\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch})$

This is the left adjoint in an adjunction

with right adjoint the trivial algebra functor $Z^{\mathrm {Com}}$ endowing F with the zero multiplication. With the model structures of Lemma 4.3 and Lemma 4.5 this is a Quillen adjunction, because the right adjoint preserves the underlying objects and hence (trivial) fibrations. We write $\mathbb {L} Q^{\mathrm {Com}}$ for the total left derived functor of $Q^{\mathrm {Com}}$ .

Definition 4.7. For a nonunital commutative algebra object $\overline {R} \colon \mathsf {d(s)Br} \to \mathsf {Ch}$ we define its André–Quillen homology

Notation 4.8. By a slight abuse of notation, for an augmented commutative algebra R with augmentation ideal $\overline {R}$ , we write $\mathbb {L} Q^{\mathrm {Com}^+}(R)$ for $\mathbb {L} Q^{\mathrm {Com}}(\overline {R})$ , and $\mathrm {AQ}_*(R)$ for $\mathrm { AQ}_*(\overline {R})$ .

4.2.3 Regular sequences

If $R \in \mathsf {Alg}^{\mathrm {augm}}_{\mathsf {Com}^+}(\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch}))$ and $z \in \overline {R}(\varnothing )$ is a cycle of degree $k \geq 0$ , then by adjunction there is a map $(\varnothing )_* S(k) \to R$ , where $(\varnothing )_*$ denotes the left Kan extension along the inclusion $\{\varnothing \} \to \mathsf {d(s)Br}$ and $S(k)$ is the chain complex of (5). Using that R is a unital commutative algebra object, this further extends to a morphism $F^{\mathrm { Com}^+}((\varnothing )_* S(k)) \to R$ , using which we may form the pushout

(7)

in the category $\mathsf {Alg}^{\mathrm {augm}}_{\mathsf {Com}^+}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch}))$ .

Definition 4.9. Let R and z be as above, then we say that z is not a zero divisor if the morphisms $z \cdot -\colon \mathbb {Q}[k] \otimes R(S) \to R(S)$ are injective for all finite sets S.

In particular, as $z^2 = (-1)^k z^2$ by graded-commutativity, k must be even.

Lemma 4.10. If $z \in \overline {R}(\varnothing )$ is not a zero divisor, then (7) is a homotopy pushout.

Proof. Consider the generating cofibration $S(k) \hookrightarrow D(k+1)$ of (5). Then

$$\begin{align*}F^{\mathrm{Com}^+}((\varnothing)_*S(k)) \longrightarrow F^{\mathrm{Com}^+}((\varnothing)_*D(k+1))\end{align*}$$

is a cofibration modelling the left vertical map of (7) (it is a generating cofibration). If $cR \overset {\sim }\to R$ is a cofibrant replacement (we may assume it is surjective) and $z' \in \overline {cR}(\varnothing )$ is a lift of z, the homotopy pushout is by definition given by

$$\begin{align*}cR \otimes_{F^{\mathrm{Com}^+}((\varnothing)_*S(k))} F^{\mathrm{Com}^+}((\varnothing)_*D(k+1))\end{align*}$$

which is $cR \otimes F^{\mathrm {Com}^+}((\varnothing )_*S(k+1))$ with differential determined by $d (x \otimes 1) = d_{cR}(x) \otimes 1$ , $d (1 \otimes z_{k+1}) = z' \otimes 1$ and the Leibniz rule.

We thus first prove that the map

$$\begin{align*}cR \otimes_{F^{\mathrm{Com}^+}((\varnothing)_*S(k))} F^{\mathrm{Com}^+}((\varnothing)_*D(k+1)) \longrightarrow R \otimes_{F^{\mathrm{ Com}^+}((\varnothing)_*S(k))} F^{\mathrm{Com}^+}((\varnothing)_*D(k+1))\end{align*}$$

induced by $cR \overset {\sim }\to R$ is a weak equivalence. To do so, we use that there is an isomorphism

(8) $$ \begin{align}F^{\mathrm{Com}^+}((\varnothing)_*S(k+1)) \cong (\varnothing)_*(\Lambda[z_{k+1}])\end{align} $$

and filter both sides by powers of $z_{k+1}$ (there will be only two nonzero powers, $1$ and $z_{k+1}$ , since $k+1$ is odd). On the associated gradeds, we get either a shift of the map $cR \to R$ or the map $0 \to 0$ , so the result follows.

Next, we observe that the isomorphism (8) together with the fact that $k+1$ is odd, identifies $R \otimes F^{\mathrm { Com}^+}((\varnothing )_*S(k+1))$ with the mapping cone of

$$\begin{align*}z \cdot -\colon \mathbb{Q}[k] \otimes R \longrightarrow R.\end{align*}$$

As we have assumed that z is not a zero divisor, this map is injective when evaluated on any finite set, so its cokernel and mapping cone are quasi-isomorphic: It follows that $R \otimes F^{\mathrm {Com}^+}((\varnothing )_*S(k+1)) \simeq R/(z)$ .

Corollary 4.11. If $z \in \overline {R}(\varnothing )$ is not a zero divisor, then there is a homotopy cofibre sequence

$$ \begin{align*} (\varnothing)_*(\mathbb{Q}\{z\}[k]) \longrightarrow \mathbb{L} Q^{\mathrm{Com}^+}(R) \longrightarrow \mathbb{L} Q^{\mathrm{ Com}^+}(R/(z)).\\[-20pt] \end{align*} $$

4.2.4 The Harrison complex

Koszul duality between the nonunital commutative and Lie operads [Reference MillèsLV12, Section 13.1.5] gives for $\overline {R} \in \mathsf {Alg}_{\mathsf {Com}}(\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch}))$ a weak equivalence of the form

$$\begin{align*}F^{\mathrm{Com}}(\mathrm{coLie}(\overline{R}[1])[-1]) \overset{\sim}\longrightarrow \overline{R},\end{align*}$$

where the differential of the domain is a certain functorial deformation of the differential induced by that on $\overline {R}$ [Reference MillèsLV12, Corollary 11.3.5], and the domain is in addition cofibrant [Reference MillèsLV12, Proof of Theorem 12.1.6]; the proofs in [Reference MillèsLV12] are given in $\mathsf {Ch}$ but go through for $\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Ch})$ . Taking indecomposables shows that the (shifted) Harrison complex (c.f. Section 2.1),

is a model for the derived indecomposables (cf. [Reference MillèsLV12, Proposition 13.1.4]).

Lemma 4.12. $\mathbb {L} Q^{\mathrm {Com}}(\overline {R}) \simeq \mathrm {Harr}(\overline {R})[-1]$ .

Proof. For $c\overline {R} \overset {\sim }\longrightarrow \overline {R}$ a cofibrant replacement, there is a commutative diagram

By definition $c\overline {R}$ is a cofibrant, and $F^{\mathrm {Com}}(\mathrm {coLie}(c\overline {R}[1])[-1])$ is too by the discussion above. The left horizontal map is thus a weak equivalence; it is a left Quillen functor applied to a weak equivalence between cofibrant objects. The right vertical map is a weak equivalence as $\mathrm {coLie}(-)$ preserves weak equivalences (as the tensor product does by Lemma 4.4 and coinvariants for a finite group action do).

4.2.5 Additional gradings and Koszulness

If a nonunital commutative algebra object $\overline {R}$ takes values in the subcategory $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod}) \subset \mathsf {Ch}$ , then the objects $\mathrm {AQ}_p(\overline {R})$ inherit a further grading, as in Section 2.1. We set

and call p the Harrison degree, q the internal degree and $p+q-1$ the total degree. If $\overline {R}$ also has a weight grading, that is, $\overline {R} \in \mathsf {Alg}_{\mathsf {Com}}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})^{\mathbb {N}}))$ , then its derived indecomposables can be formed in $\mathsf {Ch}^{\mathbb {N}}$ , giving them an additional weight grading (and similarly for the Harrison complex) and we write

for the piece of weight w.

Definition 4.13. For $\overline {R} \in \mathsf {Alg}_{\mathrm { Com}}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})^{\mathbb {N}}))$ , we say that $\overline {R}$ is Koszul in weight $ \leq W$ if $H^{\mathrm {Com}}_p(\overline {R})_{q,w}=0$ when $p \neq w$ and $w \leq W$ . If $W=\infty $ , then we simply say that R is Koszul.

The following lemma gives an easy criterion for half of this inequation:

Lemma 4.14. If each $\overline {R}(S)$ is supported in weight $\geq 1$ , then $H^{\mathrm {Com}}_p(\overline {R})_{q,w} = 0 \text { if } p> w$ .

Proof. Neglecting the differential, the Harrison complex splits as

$$\begin{align*}\bigoplus_{k \geq 0} \mathrm{coLie}(k) \otimes_{\mathfrak{S}_k} (\overline{R}[1])^{\otimes k}.\end{align*}$$

The weight of $\overline {R}[1]$ is $\geq 1$ , so the weight of $(\overline {R}[1])^{\otimes p}$ is $\geq p$ . As $H^{\mathrm {Com}}_p(\overline {R})$ is a subquotient of $\mathrm {coLie}(p) \otimes _{\mathfrak {S}_p} (\overline {R}[1])^{\otimes p}$ , it is supported in weight $\geq p$ .

4.3 Realisation

If H is a vector space with a (skew) symmetric form $\lambda $ , then we will now explain how a representation of the downward (signed) Brauer category yields a representation of $\mathrm {Aut}(H,\lambda )$ .

Definition 4.15. If $\lambda $ is a symmetric form on H, there is associated a functor

$$\begin{align*}K\colon \mathsf{dBr} \longrightarrow \mathsf{Gr}(\mathbb{Q}\text{-}\mathsf{mod})\end{align*}$$

given as follows: To the object S, we assign $K(S) = {H}^{\otimes S}$ , considered as a graded vector space in grading zero, and to the morphism $(f, m_S)\colon S \to T$ we assign the map ${H}^{\otimes S} \to {H}^{\otimes T}$ given by applying ${\lambda }$ to each of the pairs in $m_S$ and using the bijection $f(T) \overset {\sim }\to T$ to induce an isomorphism ${H}^{\otimes f(T)} \overset {\sim }\to {H}^{\otimes T}$ on the remaining factors.

Similarly, if $\lambda $ is a skew-symmetric form on H there is associated a functor

$$\begin{align*}K\colon \mathsf{dsBr} \longrightarrow \mathsf{Gr}(\mathbb{Q}\text{-}\mathsf{mod})\end{align*}$$

given in the same way on objects and given on a morphism $[(f,m_S)]\colon S \to T$ by applying $\lambda $ to each of the ordered pairs in $m_S$ ; as $\lambda $ is skew-symmetric this is well defined.

In either case, dualising K gives a functor $K^{\vee }\colon \mathsf {d(s)Br}^{\mathrm {op}} \to \mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ , and taking the coend with this defines a functor

$$\begin{align*}K^{\vee} \otimes^{\mathsf{d(s)Br}} -\colon \mathsf{Fun}(\mathsf{d(s)Br},\mathsf{Gr}(\mathbb{Q}\text{-}\mathsf{mod})) \longrightarrow \mathsf{Gr}(\mathbb{Q}\text{-}\mathsf{mod}).\end{align*}$$

As explained in [Reference Koike and TeradaKRW20c, Section 2.2.1] (following [Reference Turchin and WillwacherSS15]), this has a strong symmetric monoidality.

The data $({H},{\lambda })$ is – tautologically – equipped with an action of the group $\mathrm {Aut}({H},{\lambda })$ , and hence the functor K is too. Thus, the above construction may be promoted to a functor

$$\begin{align*}K^{\vee} \otimes^{\mathsf{d(s)Br}} -\colon \mathsf{Fun}(\mathsf{d(s)Br},\mathsf{Gr}(\mathbb{Q}\text{-}\mathsf{mod})) \longrightarrow \mathsf{Gr}(\mathsf{Rep}(\mathrm{Aut}({H},{\lambda}))),\end{align*}$$

that is, to take values in graded $\mathrm {Aut}({H},{\lambda })$ -representations.

From now on, we take $(H,\lambda )$ to be $(H(g),\lambda )$ as in Section 2.4 so that $\mathrm {Aut}(H,\lambda ) = \mathrm { O}_{\epsilon }(H(g))$ . This functor is a left adjoint so is right exact but is not left exact in general. However, it has the following stable left exactness property. Let us say that a functor $A\colon \mathsf {d(s)Br} \to \mathbb {Q}\text {-}\mathsf {mod}$ is supported on sets of size $\leq N$ if $A(S)=0$ whenever $|S|> N$ . (We take the target to be $\mathbb {Q}$ -modules rather than graded $\mathbb {Q}$ -modules as we will later wish to consider functors to graded $\mathbb {Q}$ -modules whose support depends on the grading.)

Lemma 4.17. For each $A \in \mathsf {Fun}(\mathsf {d(s)Br},\mathbb {Q}\text {-}\mathsf {mod})$ and finite set S there is a surjective map

$$\begin{align*}A(S) \longrightarrow [H(g)_{[S]} \otimes (K^{\vee} \otimes^{\mathsf{d(s)Br}} A)]^{\mathrm{O}_{\epsilon}(H(g))}\end{align*}$$

which is injective when $|S| \leq g$ .

Proof. Recall from Section 2.4 that $H(g)_{[S]}$ is a certain quotient of $H(g)^{\otimes S}$ , and consider

$$\begin{align*}[H(g)_{[S]} \otimes (K^{\vee} \otimes^{\mathsf{d(s)Br}} A)]^{\mathrm{O}_{\epsilon}(H(g))} = \int^{T \in \mathsf{d(s)Br}} [H(g)_{[S]} \otimes K(T)^{\vee}]^{\mathrm{O}_{\epsilon}(H(g))} \otimes A(T).\end{align*}$$

There is a natural transformation of two variables

$$\begin{align*}\mathsf{d(s)Br}(T, S) \longrightarrow [H(g)_{[S]} \otimes K(T)^{\vee}]^{\mathrm{O}_{\epsilon}(H(g))}\end{align*}$$

given by the functoriality of K, which is surjective, and is injective if $|S|+|T| \leq 2g$ , by [Reference Koike and TeradaKRW20c, Theorem 2.6]. By the co-Yoneda lemma it gives a map

$$\begin{align*}A(S) \longrightarrow [H(g)_{[S]} \otimes (K^{\vee} \otimes^{\mathsf{d(s)Br}} A)]^{\mathrm{O}_{\epsilon}(H(g))}\end{align*}$$

which is surjective, and injective for $|S| \leq g$ .

Lemma 4.18. If $0 \to A \to B \to C \to 0$ is a short exact sequence in the category $\mathsf {Fun}(\mathsf {d(s)Br},\mathbb {Q}\text {-}\mathsf {mod})$ such that B is supported on sets of size $\leq g$ , then

$$\begin{align*}0 \longrightarrow K^{\vee} \otimes^{\mathsf{d(s)Br}} A \longrightarrow K^{\vee} \otimes^{\mathsf{d(s)Br}} B \longrightarrow K^{\vee} \otimes^{\mathsf{d(s)Br}} C \longrightarrow 0\end{align*}$$

is again exact.

(The following argument provides the missing second half of the proof of [Reference Koike and TeradaKRW20c, Corollary 2.18], which is not as immediate as we had suggested.)

Proof. It is right exact as $K^{\vee } \otimes ^{\mathsf {d(s)Br}} -$ is a left adjoint, so we only need to show that the left-hand map is injective. As B is supported on sets of size $\leq g$ and A is a subobject of B, A also is supported on sets of size $\leq g$ .

The map

$$\begin{align*}A(S) \longrightarrow [H_{[S]} \otimes (K^{\vee} \otimes^{\mathsf{d(s)Br}} A)]^{\mathrm{O}_{\epsilon}(H(g))}\end{align*}$$

of Lemma 4.17 is an isomorphism for $|S| \leq g$ . As the domain vanishes if $|S|> g$ , this map is an isomorphism. The same argument holds with A replaced by B, and so we deduce that the map

$$\begin{align*}[H_{[S]} \otimes (K^{\vee} \otimes^{\mathsf{d(s)Br}} A)]^{\mathrm{O}_{\epsilon}(H(g))} \longrightarrow [H_{[S]} \otimes (K^{\vee} \otimes^{\mathsf{d(s)Br}} B)]^{\mathrm{O}_{\epsilon}(H(g))}\end{align*}$$

is injective for all finite sets S.

If $K^{\vee } \otimes ^{\mathsf {d(s)Br}} A \to K^{\vee } \otimes ^{\mathsf {d(s)Br}} B$ were not injective, then its kernel would contain an irreducible algebraic $\mathrm {O}_{\epsilon }(H(g))$ -representation, which would be detected by applying $[H_{[S]} \otimes -]^{\mathrm {O}_{\epsilon }(H(g))}$ for some finite set S. Thus, the map is injective as claimed.

4.3.1 Realisation and Koszulness

The above results imply that realisation preserves Koszulness in a range increasing with g.

Lemma 4.19. Suppose that the weight w piece of a nonunital commutative algebra $\overline {R} \in \mathsf {Alg}_{\mathrm { Com}}(\mathsf {Fun}(\mathsf {d(s)Br},\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})^{\mathbb {N}}))$ is supported on sets of size $\leq \lambda w$ for some $\lambda \in \mathbb {N}_{>0}$ . Then as long as $w \leq \tfrac {1}{\lambda }g$ we have

$$\begin{align*}K^{\vee} \otimes^{\mathsf{d(s)Br}} H^{\mathrm{Com}}_p(\overline{R})_{q,w} \cong H_p^{\mathrm{Com}}(K^{\vee} \otimes^{\mathsf{d(s)Br}} \overline{R})_{q,w}.\end{align*}$$

Proof. As an explicit model for the derived indecomposables of $\overline {R}$ we can take the Harrison complex (Lemma 4.12):

$$\begin{align*}\mathbb{L} Q^{\mathrm{Com}}(\overline{R}) \simeq \mathrm{coLie}(\overline{R}[1])[-1] = \mathrm{Harr}(\overline{R})[-1].\end{align*}$$

The weight w piece of $\overline {R}$ is supported on sets of size $\leq \lambda w$ , that is, $\overline {R}(S)_{q,w}=0$ if $w < \tfrac {1}{\lambda }|S|$ . It follows that tensor powers of $\overline {R}$ have the same property, and so Schur functors of $\overline {R}$ have the same property, and hence the chain complex with additional grading $\mathrm {Harr}(\overline {R})$ has this vanishing property too. In particular, for each $w \leq \tfrac {1}{\lambda } g$ , $\mathrm {Harr}(\overline {R})_{q,w}$ is a chain complex supported on sets of size $\leq g$ , so by Lemma 4.18 the operation $K^{\vee } \otimes ^{\mathsf {dsBr}} -$ is exact on it. Thus, as long as $w \leq \tfrac {1}{\lambda } g$ , we have

$$ \begin{align*} K^{\vee} \otimes^{\mathsf{d(s)Br}} H^{\mathrm{Com}}_{p}(\overline{R})_{q,w} &\cong H_{p+q}(K^{\vee} \otimes^{\mathsf{d(s)Br}} \mathrm{ Harr}(\overline{R}))_{q,w} \\ &\cong H_{p+q}( \mathrm{Harr}(K^{\vee} \otimes^{\mathsf{d(s)Br}} \overline{R}))_{q,w} \\ &\cong H^{\mathrm{Com}}_{p}(K^{\vee} \otimes^{\mathsf{d(s)Br}} \overline{R})_{q,w}, \end{align*} $$

where for the second step we have used that $K^{\vee } \otimes ^{\mathsf {d(s)Br}}-$ is an additive strong symmetric monoidal left adjoint and so commutes with the formation of the Harrison complex.

4.4 The main examples

There are two families of examples we shall work with, which are both special cases of the functors $\mathcal {P}(-; \mathcal {B})^{\prime }_{\geq 0} \otimes \det ^{\otimes n}$ from [Reference Koike and TeradaKRW20c, Definition 1.3], and we follow that definition.

Definition 4.20. A partition of a finite set S is a finite collection of possibly empty subsets $\{S_{\alpha }\}_{\alpha \in I}$ of S which are pairwise disjoint and whose union is S. A partition is admissible if each part has size $\geq 3$ .

For $n \in \mathbb {N}_{>0}$ , let $Z_n\colon \mathsf {d(s)Br} \to \mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ be the functor which to a finite set S assigns the vector space

$$\begin{align*}Z_n(S) = \mathbb{Q}\{\text{admissible partitions }\{S_{\alpha}\}_{\alpha \in I}\text{ of }S\} \otimes \det(\mathbb{Q}^S)^{\otimes n},\end{align*}$$

made into a graded vector space by declaring a part $S_{\alpha }$ to have degree $n(|S_{\alpha }|-2)$ , and a partition to have degree the sum of the degrees of its parts.

The linear map $Z_n(S) \to Z_n(T)$ induced by a bijection $(f, \varnothing )\colon S \to T$ is simply given by relabelling elements in a partition and by the induced map on determinants. The linear map induced by $(inc, (x,y))\colon S \to S \setminus \{x,y\}$ assigns to the element $[\{S_{\alpha }\}_{\alpha \in I}] \otimes (x \wedge y \wedge s_3 \wedge \cdots \wedge s_{|S|})^{\otimes n}$ the following:

1. (i) If some $S_{\beta }$ contains $\{x,y\}$ , then it assigns 0.

2. (ii) If x and y lie in different parts $S_{\beta }$ and $S_{\gamma }$ , then these are merged into a single new part $S_{\beta } \setminus \{x\} \cup S_{\gamma } \setminus \{y\}$ and it assigns

   $$\begin{align*}[\{S_{\alpha}\}_{\alpha \neq \beta, \gamma} \cup \{S_{\beta} \setminus\{x\} \cup S_{\gamma} \setminus\{y\}\}] \otimes (s_3 \wedge \cdots \wedge s_{|S|})^{\otimes n}.\end{align*}$$

On a more general morphism in $\mathsf {d(s)Br}$ , the effect of $Z_n$ is determined by the above and functoriality.

Figure 2 The effect of the morphism $(inc,\{1,2\}) \colon S = \underline {8} \to S' = \underline {8} \setminus \{1,2\}$ on two elements of $Z_n(S)$ : The first is the partition of S into parts $S_{ \alpha } = \{1,\ldots ,5\}$ and $S_{ \beta } = \{6,7,8\}$ and the second is the partition into parts $S^{\prime }_{ \alpha '} = \{2,\ldots ,6\}$ and $S^{\prime }_{ \beta '} = \{1,7,8\}$ . Similarly to [Reference Koike and TeradaKRW20c, Section 5.3], we represent each part in a partition by a corolla whose legs are labelled by the elements in that part; we have suppressed the orientations. In this graphical notation, morphisms in $\mathsf {d(s)Br}$ act by reordering labels and connecting legs, collapsing internal edges and mapping to zero whenever a loop appears.

Definition 4.21. A weighted partition of a set S is a partition $\{S_{\alpha }\}_{\alpha \in I}$ of S along with a weight $g_{\alpha } \in \{0,1,2,3,\ldots \}$ for each part. A weighted partition is admissible if

1. (i) Each part of size 0 has weight $\geq 2$ .

2. (ii) Each part of size 1 or 2 has weight $\geq 1$ .

For $n \in \mathbb {N}_{>0}$ , let $E_n\colon \mathsf {d(s)Br} \to \mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod})$ be the functor which to a finite set S assigns the vector space

$$\begin{align*}E_n(S) = \mathbb{Q}\{\text{admissible weighted partitions }\{(S_{\alpha}, g_{\alpha})\}_{\alpha \in I}\text{ of }S\} \otimes \det(\mathbb{Q}^S)^{\otimes n},\end{align*}$$

made into a graded vector space by declaring a weighted part $(S_{\alpha }, g_{\alpha })$ to have degree $n(2 g_{\alpha } + |S_{\alpha }|-2)$ , and a weighted partition to have degree the sum of the degrees of its weighted parts.

The linear map $E_n(S) \to E_n(T)$ induced by a bijection $(f, \varnothing )\colon S \to T$ is simply given by relabelling elements in a partition and by the induced map on determinants. The linear map induced by $(inc, (x,y))\colon S \to S \setminus \{x,y\}$ assigns to the element $[\{(S_{\alpha }, g_{\alpha })\}_{\alpha \in I}] \otimes (x \wedge y \wedge s_3 \wedge \cdots \wedge s_{|S|})^{\otimes n}$ the following:

1. (i) If some $S_{\beta }$ contains $\{x,y\}$ , then it assigns

   $$\begin{align*}[\{(S_{\alpha}, g_{\alpha})\}_{\alpha \neq \beta} \cup \{(S_{\beta} \setminus \{x,y\}, g_{\beta}+1)\}] \otimes ( s_3 \wedge \cdots \wedge s_{|S|})^{\otimes n},\end{align*}$$

2. (ii) If x and y lie in different parts $S_{\beta }$ and $S_{\gamma }$ , then it assigns

   $$\begin{align*}[\{(S_{\alpha}, g_{\alpha})\}_{\alpha \neq \beta, \gamma} \cup \{(S_{\beta} \setminus\{x\} \cup S_{\gamma} \setminus\{y\}, g_{\beta}+g_{\gamma})\}] \otimes ( s_3 \wedge \cdots \wedge s_{|S|})^{\otimes n}.\end{align*}$$

On a more general morphism in $\mathsf {d(s)Br}$ , the effect of $E_n$ is determined by the above and functoriality.

Figure 3 The effect of the morphism $(inc,\{1,2\}) \colon S = \underline {8} \to S' = \underline {8} \setminus \{1,2\}$ on two elements of $E_n(S)$ , once more representing each part in a partition by a corolla and suppressing the orientations. Each part $S_{ \alpha }$ in a partition – that is, each corolla – has a weight $g_{ \alpha } \geq 0$ indicated with a label at the vertex. Morphisms in $\mathsf {d(s)Br}$ act by reordering labels and connecting labels, collapsing internal edges (adding their weights) and removing loops (adding $1$ to the weight of the vertex it is attached to).

The functors $Z_n$ and $E_n$ have lax symmetric monoidalities given by disjoint union, making them into unital commutative algebra objects in $\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod}))$ . As they are concentrated in homological degrees which are multiples of n, we can give them an additional weight grading by declaring it to be the homological degree divided by n. Furthermore, both $Z_n$ and $E_n$ have unique augmentations, by sending all (weighted) partitions of nonempty sets to zero and the empty partition of the empty set to $1$ . Up to scaling the homological grading each of these augmented unital commutative algebra objects only depends on the parity of n.

For each n, there is a (lax symmetric monoidal) epimorphism $E_n \to Z_n$ given by sending a part $(S_{\alpha }, 0)$ to $S_{\alpha }$ , and parts with strictly positive weight to zero. It is compatible with the augmentations.

After having introduced $E_n$ and $Z_n$ we will now explain that, as commutative algebra objects, they differ only by the attachment of a single commutative algebra cell. As we mentioned above, there is a map of commutative algebra objects $E_n \to Z_n$ and hence maps $\mathbb {L} Q^{\mathrm { Com}^+}(E_n) \to \mathbb {L} Q^{\mathrm {Com}^+}(Z_n)$ . Taking the homotopy cofibre of this, we may define the relative homology groups $H_p^{\mathrm {Com}}(Z_n, E_n)_{q,w}$ which participate in the usual long exact sequence. We have the following description of this relative homology.

Theorem 4.22. We have

$$\begin{align*}H_p^{\mathrm{Com}}(Z_n, E_n)_{q,w}(S) \cong \begin{cases} \mathbb{Q} & \text{ if } |S|=1 \text{ and } (p,q,w)=(2,n,1),\\ 0 & \text{ otherwise.} \end{cases}\end{align*}$$

Proof. As the weight is given by the homological grading divided by n, we may neglect it. Let us write

and in particular $\underline {1} = \{1\}$ . There is a map $S(n) \to E_n(\underline {1})$ corresponding to the weighted partition $(\underline {1}, 1)$ of the set $\underline {1}$ , with $S(n)$ as in (5). This is adjoint to a map $\underline {1}_*S(n) \to E_n$ and hence, as the target is a unital commutative algebra object, to a map $\phi \colon F^{\mathrm {Com}^+}(\underline {1}_*S(n)) \longrightarrow E_n$ . Using this we may form the pushout

of unital commutative algebra objects. As $(\underline {1}, 1)$ maps to zero in $Z_n(\underline {1})$ , there is a factorisation

(9) $$ \begin{align} E_n \longrightarrow E^{\prime}_n \longrightarrow Z_n. \end{align} $$

Although $E^{\prime }_n : \mathsf {d(s)Br} \to \mathsf {Ch}$ does not take values in the subcategory $\mathsf {Gr}(\mathbb {Q}\text {-}\mathsf {mod}) \subset \mathsf {Ch}$ , we can nonetheless give it an ‘internal grading’ and ‘weight’ by declaring both $\underline {1}_* S(n)$ and $\underline {1}_* D(n+1)$ to have internal degree n and weight 1. Then the maps (9) both preserve these two additional gradings. We will show that the map $E^{\prime }_n \to Z_n$ is a weak equivalence: As the cofibre of the map $\mathbb {L} Q^{\mathrm {Com}^+}(E_n) \to \mathbb {L} Q^{\mathrm { Com}^+}(E^{\prime }_n)$ is $\underline {1}_* (D(n+1)/S(n)) = \underline {1}_* S(n+1)$ by construction (supported in total degree $n+1$ , internal degree n, and weight $1$ ) the formula in the statement of the theorem follows.

Pushouts of commutative algebra objects are given by the relative tensor product, and – neglecting the differential – we have

$$ \begin{align*} F^{\mathrm{Com}^+}(\underline{1}_*D(n+1)) &= F^{\mathrm{Com}^+}(\underline{1}_*S(n) \oplus \underline{1}_*S(n+1))\\ &= F^{\mathrm{Com}^+}(\underline{1}_*S(n)) \otimes F^{\mathrm{Com}^+}(\underline{1}_*S(n+1)) \end{align*} $$

so that – neglecting the differential – we have

$$\begin{align*}E^{\prime}_n = E_n \otimes F^{\mathrm{Com}^+}(\underline{1}_*S(n+1)).\end{align*}$$

Claim. We have $F^{\mathrm {Com}^+}(\underline {1}_*S(n+1))(T) \cong \det (\mathbb {Q}^T)^{\otimes n+1}[(n+1)|T|]$ , and the functoriality is such that a morphism $(f,m)$ in $\mathsf {d(s)Br}$ with $m \neq \varnothing $ acts as zero.

Proof of claim

By definition of Day convolution, we have

$$\begin{align*}(\underline{1}_*S(n+1))^{\otimes p}(T) = S(n+1)^{\otimes p} \otimes \mathsf{d(s)Br}(\{1,2,\ldots, p\}, T)\end{align*}$$

where $\mathfrak {S}_p$ acts diagonally, and so

$$\begin{align*}F^{\mathrm{Com}^+}(\underline{1}_*S(n+1))(T) = \bigoplus_{p \geq 0} S(n+1)^{\otimes p} \otimes_{\mathfrak{S}_p} \mathsf{d(s)Br}(\{1,2,\ldots, p\}, T).\end{align*}$$

Write $e_{n+1} \in S(n+1)$ for the basis element. If $(f, m) \in \mathsf {d(s)Br}(\{1,2,\ldots , p\}, T)$ has $m \neq \varnothing $ , say with $(x,y) \in m$ , then

$$ \begin{align*} (e_{n+1} \otimes \cdots \otimes e_{n+1}) \otimes_{\mathfrak{S}_p} (f, m) &= (x \, y) \cdot(e_{n+1} \otimes \cdots \otimes e_{n+1}) \otimes_{\mathfrak{S}_p} (x \, y) \cdot(f, m)\\ &= (-1)^{n+1} (e_{n+1} \otimes \cdots \otimes e_{n+1}) \otimes_{\mathfrak{S}_p} (-1)^n (f, m)\\ &= - (e_{n+1} \otimes \cdots \otimes e_{n+1}) \otimes_{\mathfrak{S}_p} (f, m) \end{align*} $$

and so this term vanishes. Hence, only the term with $p=|T|$ contributes, giving $F^{\mathrm {Com}^+}(\underline {1}_*S(n+1))(T) = S(n+1)^{\otimes |T|} \otimes _{\mathfrak {S}_{|T|}} \mathsf {d(s)Br}(\{1,2,\ldots , |T|\}, T)$ . The claim is simply an interpretation of this formula.

Using the expression (4) for Day convolution, neglecting the differential we have

(10) $$ \begin{align} E^{\prime}_n(S) = \bigoplus_{S', S"} \mathsf{Pair}(S' \sqcup S", S) \otimes_{\mathfrak{S}_{S'} \times \mathfrak{S}_{S"}} E_n(S') \otimes \det(\mathbb{Q}^{S"})^{\otimes n+1}[(n+1)|S"|]. \end{align} $$

We will give an interpretation of this in terms of decorated partitions. Define a biweighted partition of a set S to be a partition $\{S_{\alpha }\}_{\alpha \in I}$ of a subset of S along with a weight $g_{\alpha } \in \{0,1,2,3,\ldots \}$ and a further weight $h_{\alpha } \in \{0,1\}$ for each part. A biweighted partition is admissible if

1. (i) when $|S_{\alpha }|=0$ then $g_{\alpha }+h_{\alpha }\geq 2$ ,

2. (ii) when $|S_{\alpha }|=1,2$ then $g_{\alpha }+h_{\alpha }\geq 1$ .

For such a biweighted partition, we let , and . Define

$$\begin{align*}E^{\prime\prime}_n(S) = \bigoplus_{\substack{\text{admissible biweighted partitions} \\ \{(S_{\alpha}, g_{\alpha}, h_{\alpha})\}_{\alpha \in I} \text{ of } S}} \det(\mathbb{Q}^{J}) \otimes \det(\mathbb{Q}^{B}) \otimes \det(\mathbb{Q}^{S})^{\otimes n}\end{align*}$$

made into a graded vector space by declaring a biweighted partition to have degree $(n+1)|B| + \sum _{\alpha \in I} n(2 g_{\alpha } + |S_{\alpha }|-2) + (2n+1)h_{\alpha }$ .

Figure 4 Intuitive indication of the map $\chi $ . Weights $g_{\alpha }$ are not indicated, and weights $h_{\alpha }=1$ are indicated by a half-edge with a red end.

Define a morphism of graded vector spaces

$$\begin{align*}\chi \colon E^{\prime\prime}_n(S) \longrightarrow E^{\prime}_n(S)\end{align*}$$

on $(\{S_{\alpha }\}_{\alpha \in I}, g_{\alpha }, h_{\alpha }) \otimes (j_1 \wedge \cdots \wedge j_{|J|}) \otimes (b_1 \wedge \cdots \wedge b_{|B|}) \otimes (a_1 \wedge \cdots \wedge a_{|A|} \wedge b_1 \wedge \cdots \wedge b_{|B|})^{\otimes n}$ as follows. Let

and

, and let $\phi = (f, m) \colon S' \sqcup S" \to S$ be the morphism in $\mathsf {d(s)Br}$ with injection $f \colon S \to S' \sqcup S"$ given by sending $s \in S$ to $s \in S'$ if $s \in A$ , and to $s \in S"$ if $s \in B$ , and matching m given by pairing each element of $J \subset S'$ with the same element of $J \subset S"$ , putting that of $S'$ first. For $\alpha \in I$ , let

Then $\phi $ , $(\{S^{\prime }_{\alpha }\}_{\alpha \in I}, g_{\alpha }) \otimes (a_1 \wedge \cdots \wedge a_{|A|} \wedge j_1 \wedge \cdots \wedge j_{|J|})^{\otimes n}$ , $(b_1 \wedge \cdots \wedge b_{|B|} \wedge j_1 \wedge \cdots \wedge j_{|J|})^{\otimes n+1}$ represents an element of (10). Note that permuting the $j_i$ acts by the sign, permuting the $a_i$ acts by the nth power of the sign and permuting the $b_i$ acts via the $(n+1)$ st power of the sign, making this map well defined.

Claim. The map $\chi $ is an isomorphism.

Proof of claim

Suppose we are given a morphism $\phi = (f, m) \colon S' \sqcup S" \to S$ , $(\{S^{\prime }_{\alpha }\}_{\alpha \in I}, g_{\alpha }) \otimes (s^{\prime }_1 \wedge \cdots \wedge s^{\prime }_{|S'|})^{\otimes n} \in E_n(S')$ , and $(s^{\prime \prime }_1 \wedge \cdots \wedge s^{\prime \prime }_{|S"|})^{\otimes n+1} \in \det (\mathbb {Q}^{S"})^{\otimes n+1}[(n+1)|S"|]$ , representing an element of $E^{\prime }_n(S)$ in the description (10).

Suppose that this data is such that there are ordered pairs $(s^{\prime }_i, s^{\prime \prime }_i), (s^{\prime }_j, s^{\prime \prime }_j)$ with $i \neq j$ , but $s^{\prime }_i, s^{\prime }_j \in S^{\prime }_{\alpha }$ . The permutation $(s^{\prime }_i \, s^{\prime }_j)(s^{\prime \prime }_i \, s^{\prime \prime }_j)$ gives a morphism $\psi \colon S' \sqcup S" \to S' \sqcup S"$ such that $\phi \circ \psi = \psi $ . Thus, the data above is equivalent to the data $\phi $ , $(s^{\prime }_i \, s^{\prime }_j) \cdot ((\{S^{\prime }_{\alpha }\}_{\alpha \in I}, g_{\alpha }) \otimes (s^{\prime }_1 \wedge \cdots \wedge s^{\prime }_{|S'|})^{\otimes n})$ , $(s^{\prime \prime }_i \, s^{\prime \prime }_j) \cdot (s^{\prime \prime }_1 \wedge \cdots \wedge s^{\prime \prime }_{|S"|})^{\otimes n+1}$ . That is: The data $\phi $ , $(-1)^n (\{S^{\prime }_{\alpha }\}_{\alpha \in I}, g_{\alpha }) \otimes (s^{\prime }_1 \wedge \cdots \wedge s^{\prime }_{|S'|})$ , and $(-1)^{n+1}(s^{\prime \prime }_1 \wedge \cdots \wedge s^{\prime \prime }_{|S"|})^{\otimes n+1}$ and so minus the original element. Thus, such elements vanish, and we may suppose that for each $\alpha $ there is at most one pair $(s',s") \in m$ with $s' \in S^{\prime }_{\alpha }$ . Using this, it is easy to produce an inverse to the map $\chi $ : we set and $h_{\alpha }=1$ if $(s',s") \in m$ with $s' \in S^{\prime }_{\alpha }$ , and otherwise set and $h_{\alpha }=0$ .

The differential of $E^{\prime }_n(S)$ is given under the isomorphism $\chi $ by sending the element $(\{S_{\alpha }\}_{\alpha \in I}, g_{\alpha }, h_{\alpha }) \otimes (j_1 \wedge \cdots \wedge j_{|J|}) \otimes (b_1 \wedge \cdots \wedge b_{|B|}) \otimes (a_1 \wedge \cdots \wedge a_{|A|} \wedge b_1 \wedge \cdots \wedge b_{|B|})^{\otimes n}$ of $E^{\prime \prime }_n(S)$ to

$$ \begin{align*} &\sum_{\beta \in J \subset I} (\{S_{\alpha}\}_{\alpha \in I}, g_{\alpha} + \delta_{\alpha \beta}, h_{\alpha} - \delta_{\alpha \beta}) \otimes (j_1 \wedge \cdots \wedge j_{|J|})/\beta \\ &\quad\quad\quad\quad\quad \otimes (b_1 \wedge \cdots \wedge b_{|B|}) \otimes (a_1 \wedge \cdots \wedge a_{|A|} \wedge b_1 \wedge \cdots \wedge b_{|B|})^{\otimes n}\\ &+ \sum_{b \in B} (\{S_{\alpha}\}_{\alpha \in I \cup \{b\}}, g_{\alpha}, h_{\alpha}) \otimes (j_1 \wedge \cdots \wedge j_{|J|})\\ &\quad\quad\quad\quad\quad \otimes (b_1 \wedge \cdots \wedge b_{|B|})/b \otimes (a_1 \wedge \cdots \wedge a_{|A|} \wedge b_1 \wedge \cdots \wedge b_{|B|})^{\otimes n,} \end{align*} $$

where $S_b = \{b\}$ , $g_b = 1$ , $h_b=0$ . In particular, the partition $S = \bigcup _{\alpha } S_{\alpha } \cup \bigcup _{b \in B} \{b\}$ of S is preserved by the differential, so we recognise the chain complex $E^{\prime \prime }_n(S)$ as a direct sum of complexes, one for each partition of S. Furthermore, we recognise the chain complex for the partition $S = \bigcup _{\alpha } S_{\alpha } \cup \bigcup _{b \in B} \{b\}$ as the tensor product of chain complexes, one for each part of this partition.

1. (i) That corresponding to the part $S_{\alpha }$ is given by

   $$\begin{align*}\bigoplus_{g_{\alpha} + 1 \geq r_{\alpha}} (S_{\alpha}, g_{\alpha}, 1) \otimes \alpha \otimes 1 \overset{d}\longrightarrow \bigoplus_{g_{\alpha} \geq r_{\alpha}} (S_{\alpha}, g_{\alpha}, 0) \otimes 1 \otimes 1 \end{align*}$$
   (tensored with $\det (\mathbb {Q}^{S_{\alpha }})^{\otimes n}$ ) with $d((S_{\alpha }, g_{\alpha }, 1) \otimes \alpha \otimes 1) = (S_{\alpha }, g_{\alpha }+1, 0) \otimes 1 \otimes 1$ , where $r_{\alpha }$ is 2 if $|S_{\alpha }|$ is 0, is 1 if $|S_{\alpha }|$ is 1 or 2, and is 0 otherwise. The homology of this complex is given by $\bigoplus _{0 \geq r_{\alpha }} (S_{\alpha }, 0, 0) \otimes 1 \otimes 1$ , that is, is 1-dimensional as long as $|S_{\alpha }|> 2$ and zero otherwise, recalling the requirement $g_{\alpha } \geq 0$ .

2. (ii) That corresponding to the part $\{b\}$ with $b \in B$ is given by

   $$\begin{align*}(\varnothing, -, -) \otimes 1 \otimes s \underset{\sim}{\overset{d}\longrightarrow} (\{b\}, 1, 0) \otimes 1 \otimes 1\end{align*}$$
   (tensored with $\det (\mathbb {Q}^{b})^{\otimes n}$ ) so is acyclic.

Thus, the homology of $E_n"(S)$ is supported on those partitions with $B=\varnothing $ , $|S_{\alpha }|> 2$ and $g_{\alpha }=h_{\alpha }=0$ , and we recognise it as being isomorphic to $Z_n(S)$ . By considering the maps involved (or by counting dimensions) it follows that the map $E^{\prime }_n(S) \to Z_n(S)$ is an isomorphism on homology, so $E^{\prime }_n \to Z_n$ is a weak equivalence.

Proof. In the portion of the long exact sequence

we must show that the connecting map is injective. By the proof of the previous theorem, this connecting map sends the generator to the map

$$\begin{align*}\underline{1}_*S(n) \overset{\phi}\longrightarrow \overline{E_n} \longrightarrow \mathbb{L} Q^{\mathrm{Com}}(E_n),\end{align*}$$

a class in $\mathrm {AQ}_n(E_n)_{n,1}(\underline {1}) = H_1^{\mathrm {Com}}(E_n)_{n,1}(\underline {1})$ . As $1$ is the lowest weight in which $\overline {E_n}$ has nontrivial elements, elements of this weight cannot be decomposable: Thus, as $\phi $ is nontrivial the connecting map is too.

4.5 Relation to Torelli Lie algebras

The relation between $E_1$ and the Torelli Lie algebra was already indicated in [Reference Koike and TeradaKRW20c, Remark 8.4], but we explain it again here.

Theorem 4.25. The realisation

$$\begin{align*}K^{\vee} \otimes^{\mathsf{dsBr}} E_1/(\kappa_{e^2}) \in \mathsf{Gr}(\mathsf{Rep}(\mathrm{Sp}_{2g}(\mathbb{Z})))^{\mathbb{N}}\end{align*}$$

is a commutative algebra object, and as long as $g \geq 4$ it agrees in degrees $* \leq g$ with the quadratic dual of the Lie algebra object

$$\begin{align*}\mathrm{Gr}^{\bullet}_{\mathrm{LCS}} \mathfrak{t}_{g,1} \in \mathsf{Gr}(\mathsf{Rep}(\mathrm{Sp}_{2g}(\mathbb{Z})))^{\mathbb{N}}.\end{align*}$$

Proof. In the language of [Reference Koike and TeradaKRW20c, Section 5], we have $E_1 = \mathcal {P}(-;\mathcal {V})^{\prime }_{\geq 0} \otimes \det $ with $\mathcal {V}=\mathbb {Q}[e]$ having $|e|=2$ , and so $K^{\vee } \otimes ^{\mathsf {dsBr}} E_1$ is the object denoted $R^{\mathcal {V}}$ there. By the discussion of [Reference Koike and TeradaKRW20c, Section 8.2] (especially Remark 8.4), and using the notation of Section 3.2, there is a map of graded commutative algebras

$$\begin{align*}\frac{\Lambda^*[\Lambda^3 V_1[1]]}{(({IH}))} \longrightarrow R^{\mathcal{V}}\end{align*}$$

which is an isomorphism in a stable range of degrees. As in the proof of Lemma 3.8, the element $\Theta = \sum _{i,j,k} \kappa _1(a_i \otimes a_j \otimes a_k) \cdot \kappa _1(a_i^{\#} \otimes a_j^{\#} \otimes a_k^{\#})$ corresponds to $-\kappa _{e^2}$ . As the weight grading on $E_1$ coincides with its homological grading, there is a map

$$\begin{align*}\frac{\Lambda^*[\Lambda^3 V_1[1,1]]}{(({IH}), (\Theta))} \longrightarrow K^{\vee} \otimes^{\mathsf{dsBr}} E_1/(\kappa_{e^2})\end{align*}$$

which is an isomorphism in a stable range of degrees, and the domain is the quadratic dual of $\mathrm {Gr}^{\bullet }_{\mathrm {LCS}} \mathfrak {t}_{g,1}$ as long as $g \geq 4$ by the discussion in Section 3.2. Finally, the stability range is determined in [Reference Koike and TeradaKRW20c, Section 9.5]: This map is an isomorphism in degrees $* \leq g$ .

This yields a relation between Koszulness of ${\mathrm {Gr}^{\bullet }_{\mathrm {LCS}}\,} \mathfrak {t}_{g,1}$ and of $E_1$ .

Proof. Note that $E_1(\varnothing )$ is the polynomial algebra on the elements $\kappa _{e^j} = (\varnothing , j)$ for $j> 1$ , and each $E_1(S)$ is a free $E_1(\varnothing )$ -module, so the element $\kappa _{e^2} = (\varnothing ,2) \in E_1(\varnothing )_{2,2}$ is not a zero divisor in the sense of Section 4.2.3, and hence we may apply Corollary 4.11. Recalling the definition $\mathrm {AQ}_*({R}) = H_*(\mathbb {L} Q^{\mathrm {Com}}(\overline {R}))$ , there is a long exact sequence

$$\begin{align*}\cdots \longrightarrow (\varnothing)_*(\mathbb{Q}\{\kappa_{e^2}\}[2]) \longrightarrow \mathrm{AQ}_*(E_1) \longrightarrow \mathrm{ AQ}_*(E_1/(\kappa_{e^2})) \longrightarrow \cdots.\end{align*}$$

As $\kappa _{e^2}$ is decomposable in $E_1$ by the formula for (Θ), taking internal gradings and weight into account this gives short exact sequences

$$\begin{align*}0 \to H_p^{\mathrm{Com}}(E_1)_{q,w}(S) \to H_p^{\mathrm{Com}}(E_1/(\kappa_{e^2}))_{q,w}(S) \to \begin{cases} \mathbb{Q} & (p,q,w)=(2,2,2),\ S=\varnothing\\ 0 & \text{ otherwise} \end{cases} \to 0,\end{align*}$$

so $E_1$ is Koszul if and only if $E_1/(\kappa _{e^2})$ is.

Observe that the weight w part of $E_1/(\kappa _{e^2})$ is supported on sets of size $\leq 3w$ , so by Lemma 4.19 we have

$$\begin{align*}K^{\vee} \otimes^{\mathsf{dsBr}} H^{\mathrm{Com}}_p(E_1/(\kappa_{e^2}))_{q,w} \cong H^{\mathrm{Com}}_p(K^{\vee} \otimes^{\mathsf{dsBr}} E_1/(\kappa_{e^2}))_{q,w}\end{align*}$$

as long as $w \leq \tfrac {1}{3}g$ . Now, if $E_1/(\kappa _{e^2})$ is Koszul, then $H^{\mathrm {Com}}_{p}(E_1/(\kappa _{e^2}))_{q,w}=0$ for $p \neq w$ , and it follows that $H^{\mathrm {Com}}_{p}(K^{\vee } \otimes ^{\mathsf {dsBr}} E_1/(\kappa _{e^2}))_{q,w}=0$ for $p \neq w$ and $w \leq \tfrac {1}{3} g$ .

For any odd $n \geq 1$ , the commutative algebra object $E_n$ is isomorphic to $E_1$ with its homological grading scaled by n. Thus, $E_1$ is Koszul if and only if $E_n$ is Koszul, and using Corollary 4.27 and Proposition 3.4 this is what we shall show to prove Theorem A and hence Theorem 3.3. We will do so by showing that the $Z_n$ are Koszul and appealing to Corollary 4.23. Koszulness of the $Z_n$ has its own applications, to the rational homotopy Lie algebra of diffeomorphism groups of high-dimensional manifolds: We will explain these applications in Section 7.1.

5.1 Torelli groups of high-dimensional manifolds

For the remainder of this section, we suppose that $2n \geq 6$ . We will use the $2n$ -manifolds

A tangential structure for $2n$ -manifolds can equivalently be described by a map $\theta \colon B \to B\mathrm {O}(2n)$ , or by a $\mathrm { GL}_{2n}(\mathbb {R})$ -space $\Theta $ (see [Reference Goerss and SchemmerhornGRW19, Section 4.5]). Here, we use the former, as in [Reference Koike and TeradaKRW20c, Reference Kupers and Randal-WilliamsKRW20a], though we cite extensively from [Reference Kupers and Randal-WilliamsKRW20b], which uses the latter.

Fix a tangential structure $\theta \colon B \to B\mathrm {O}(2n)$ . A $\theta $ -structure on $W_{g,1}$ is a map of vector bundles $\ell \colon TW_{g,1} \to \theta ^* \gamma _{2n}$ , where $\gamma _{2n}$ is the universal bundle over $B\mathrm {O}(2n)$ . We will fix a boundary condition $\ell _{\partial } \colon TW_{g,1}|_{\partial W_{g,1}} \to \theta ^*\gamma _{2n}$ and consider only those $\theta $ -structures which extend it; we let $\mathrm {Bun}_{\partial }(TW_{g,1},\theta ^*\gamma _{2n})$ denote the space of such bundle maps. The topological group $\mathrm {Diff}_{\partial }(W_{g,1})$ of diffeomorphisms of $W_{g,1}$ which fix its boundary pointwise (in the $C^{\infty }$ -topology) acts on it through the derivative. We define

We will only use tangential structures which satisfy the assumptions of Section 8 of [Reference Kupers and Randal-WilliamsKRW20a]. By Lemma 8.5 (i) of [Reference Kupers and Randal-WilliamsKRW20a], there then exists up to homotopy a unique orientation-preserving boundary condition $\ell _{\partial } \colon TW_{g,1}|_{\partial W_{g,1}} \to \theta ^* \gamma _{2n}$ which extends to $W_{g,1}$ . For a $\theta $ -structure $\ell $ on $W_{g,1}$ , we denote by $B\mathrm {Diff}^{\theta }_{\partial }(W_{g,1})_{\ell }$ the path-component of $B\mathrm {Diff}^{\theta }_{\partial }(W_{g,1})$ which contains it.

The intersection product endows $H_n(W_{g,1};\mathbb {Z})$ with a nondegenerate $(-1)^n$ -symmetric form. Every diffeomorphism of $W_{g,1}$ induces an automorphism of this form, so we get a homomorphism

$$\begin{align*}\pi_1(B\mathrm{Diff}_{\partial}(W_{g,1})) \longrightarrow G_g = \begin{cases} \mathrm{Sp}_{2g}(\mathbb{Z}) & \text{if }n\text{ is odd,} \\ \mathrm{O}_{g,g}(\mathbb{Z}) & \text{if }n\text{ is even.}\end{cases}\end{align*}$$

We will let $\smash {G^{\theta ,[\ell ]}_g} \subset G_g$ denote the image of the composition

$$\begin{align*}\pi_1(B\mathrm{Diff}^{\theta}_{\partial}(W_{g,1})_{\ell}) \longrightarrow \pi_1(B\mathrm{Diff}_{\partial}(W_{g,1})) \longrightarrow G_g.\end{align*}$$

It is a finite index subgroup by [Reference Kupers and Randal-WilliamsKRW20a, Lemma 8.9]. There is an induced map $B\mathrm { Diff}^{\theta }_{\partial }(W_{g,1})_{\ell } \to BG^{\theta ,[\ell ]}_g$ , and we define

5.2 $Z_n$ and high-dimensional manifolds with framings

For $Z_n$ , we will use the above constructions with tangential structure $\theta _{\mathrm {fr}} \colon E\mathrm {O}(2n) \to B\mathrm {O}(2n)$ to obtain Torelli groups $B\mathrm {Tor}^{\mathrm {fr}}_{\partial }(W_{g,1})_{\ell }$ with framings. From this, we developed the commutative diagram [Reference Kupers and Randal-WilliamsKRW20b, (15)] with rows and columns fibration sequences

with overlines denoting a passage to a finite cover or finite index subgroup which has no effect on rational cohomology (we refer to loc. cit. for definitions and details). The proof of Theorem 5.1 uses the following facts about $X_1(g)$ , which combines Lemma 4.9, Theorem 4.11, Corollary 4.20 and Remark 6.5 of [Reference Kupers and Randal-WilliamsKRW20b].

5.3 Proof of Theorem 5.1

We first prove Theorem 5.1 for $Z_n$ , using the above theorem as input. As a consequence of Theorem 5.2 (i), there is a strongly convergent unstable rational Adams spectral sequence

$$\begin{align*}E^2_{s,t} = H^{\mathrm{Com}}_{s}(H^*(X_1(g);\mathbb{Q}))_t \Longrightarrow \mathrm{Hom}(\pi_{t-s}(\Omega X_1(g)),\mathbb{Q}),\end{align*}$$

with $d^r$ -differential of bidegree $(-r,-r+1)$ which, by naturality and Theorem 5.2 (ii), is a spectral sequence of $\overline {G}{}^{\mathrm {fr},[\ell ]}_g$ -representations.

By Theorem 5.2 (iii), in a stable range the rational cohomology of $X_1(g)$ is concentrated in degrees which are multiples of n. Thus, in a stable range the groups $E^2_{s,t}$ are supported along the lines $t = r \cdot n$ with $r \in \mathbb {N}$ and by the argument of Lemma 4.14 we have $E^2_{s,rn}=0$ for $r < s$ . Thus, $E^2_{s,t}$ with $t=r\cdot n$ is nontrivial only for $1 \leq s \leq r$ , contributing to degrees

$$\begin{align*}r\cdot n-r+1 \leq t-s+1 \leq r\cdot n.\end{align*}$$

Furthermore, as long as these ranges of total degrees are separated from each other there can be no differentials between the corresponding lines: In particular, as long as $n \geq r-2$ the spectral sequence must collapse in degrees $t-s+1 \leq r \cdot n$ . Thus, as long as $r \leq n+2$ , there is an isomorphism of $\overline {G}{}^{\mathrm {fr},[\ell ]}_g$ -representations in a stable range

$$\begin{align*}H^{\mathrm{Com}}_{s}(H^*(X_1(g);\mathbb{Q}))_{rn} \cong \mathrm{Hom}(\pi_{rn-s+1}(X_1(g)) , \mathbb{Q}).\end{align*}$$

By Theorem 5.2 (iv), in degrees $* \leq \tfrac {n(n-3)}{2}$ the nontrivial representations on the right side vanish except when $r=s$ . Hence the same is true for the left side.

By Theorem 5.2 (iii) there is a map

$$\begin{align*}K^{\vee} \otimes^{\mathsf{d(s)Br}} Z_n\longrightarrow H^*(X_1(g);\mathbb{Q})\end{align*}$$

of $\overline {G}{}^{\mathrm {fr},[\ell ]}_g$ -representations which is an isomorphism in a stable range. By Lemma 4.19

$$\begin{align*}K^{\vee} \otimes^{\mathsf{d(s)Br}} H^{\mathrm{Com}}_p(Z_n)_q \cong H^{\mathrm{Com}}_p(K^{\vee} \otimes^{\mathsf{d(s)Br}} Z_n)_q\end{align*}$$

as long as $\tfrac {q}{n} \leq \tfrac {1}{3} g$ , which is satisfied by taking g large enough. By the vanishing result established above, for all large enough g and for $q \leq \smash {\tfrac {n(n-3)}{2}}$ , this is a trivial representation whenever $q \neq n\cdot p$ . By the discussion in Section 2.4, the same is true as $\mathrm {O}_{\epsilon }(H(g))$ -representations.

By Lemma 4.17, if $H^{\mathrm {Com}}_p(Z_n)_q(S)$ were not zero for a nonempty finite set S, then for $g \geq |S|$ so would be $[H_{[S]} \otimes (K^{\vee } \otimes ^{\mathsf {d(s)Br}} H^{\mathrm {Com}}_p(Z_n)_q]^{\mathrm {O}_{\epsilon }(H(g))}$ . As $H_{[S]}$ for a nonempty set S is a direct sum of irreducibles that are not equal to the trivial representation, this implies that $K^{\vee } \otimes ^{\mathsf {d(s)Br}} H^{\mathrm {Com}}_p(Z_n)_q$ would contain a nontrivial subrepresentation. Using the above, we conclude that

$$\begin{align*}H^{\mathrm{Com}}_{p}(Z_n)_{q}(S) = 0 \text{ for } S \neq \varnothing \text{ and } q \neq n \cdot p \text{ as long as } q \leq \tfrac{n(n-3)}{2}.\end{align*}$$

We now use that for all n’s of the same parity the commutative algebra objects $Z_n$ are isomorphic up to a linear scaling of degrees. As the bound ‘ $q \leq \tfrac {n(n-3)}{2}$ ’ scales quadratically with n it can therefore be ignored, yielding Theorem 5.1 for $Z_n$ .

To deduce Theorem 5.1 for $E_n$ , we invoke Corollary 4.23.

5.4 $E_n$ and high-dimensional manifolds with Euler structure

Instead of using Corollary 4.23 to deduce Theorem 5.1 for $E_n$ from $Z_n$ , one may prove it directly using a custom tangential structure which mimics in high dimensions the moduli spaces of surfaces. We explain this in this extended remark, as it may be of independent interest. It was in fact our original approach to Theorem A.

The Pontrjagin classes $p_j \in H^{4j}(B\mathrm {SO}(2n);\mathbb {Z})$ for $1 \leq j < n$ induce a map

$$\begin{align*}B\mathrm{SO}(2n) \longrightarrow \prod_{1 \leq j<n} K(\mathbb{Z},4j).\end{align*}$$

We denote its homotopy fibre by $\widetilde {B}^e$ , which has n-connected cover .

The name is justified by the calculation $H^*(B^e;\mathbb {Q}) = \mathbb {Q}[e]$ where e denotes the Euler class, of degree $2n$ . By construction $B^e$ is n-connected, and since $B\mathrm {SO}(2n)$ has finitely generated homotopy groups, so does $B^e$ . Thus, this tangential structure satisfies the hypotheses for Section 8 of [Reference Kupers and Randal-WilliamsKRW20a]. From Section 5.1, we obtain $B\mathrm { Tor}^{\theta _e}_{\partial }(W_{g,1})$ which we shorten to $B\mathrm {Tor}^e_{\partial }(W_{g,1})$ . A straightforward adaptation of the techniques of [Reference Koike and TeradaKRW20c, Reference Kupers and Randal-WilliamsKRW20b] yields a homotopy-commutative diagram

with rows and columns fibration sequences, and an analogue of Theorem 5.2 sufficient to directly prove Theorem 5.1 for $E_n$ .

6.1 Red and black graphs

We will first describe explicit cofibrant replacements for the augmented unital commutative algebra objects $Z_n, E_n \in \mathsf {Alg}_{\mathsf {Com}^+}^{\mathrm {augm}}(\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch}))$ .

Definition 6.2. If $e = \{h_1, h_2\} \in E(\Gamma )$ is an edge, the graph $\Gamma ' = \Gamma /e$ obtained by contracting the edge e has vertices , half-edges , boundary map

$$\begin{align*}\partial_{\Gamma'}\colon H(\Gamma) \setminus \{h_1, h_2\} \overset{\mathrm{inc}}\longrightarrow H(\Gamma) \overset{\partial_{\Gamma}}\longrightarrow V(\Gamma) \xrightarrow{\mathrm{quot}} V(\Gamma)/(\partial_{\Gamma}(h_1) \sim \partial_{\Gamma}(h_2)),\end{align*}$$

and involution $\iota _{\Gamma '} = \iota _{\Gamma }\vert _{H(\Gamma ')}$ .

Note that if $\Gamma $ is admissible, so are $\Gamma /e$ and $\Gamma \backslash e$ . Recall that . Define 1-dimensional vector spaces

(11)

Isomorphisms of weighted red and black graphs induce isomorphisms of these vector spaces.

If $e \in E_B(\Gamma )$ , there are maps $-/e\colon \mathfrak {K}_B(\Gamma ) \to \mathfrak {K}_B(\Gamma /e)$ and $- {\setminus } e\colon \mathfrak {K}_B(\Gamma ) \to \mathfrak {K}_B(\Gamma {\setminus } e)$ given by contracting with e. As contracting or colouring an edge does not change the set of legs, there are also canonical identifications $\mathfrak {L}(\Gamma ) \overset {\sim }\to \mathfrak {L}(\Gamma /e)$ and $\mathfrak {L}(\Gamma ) \overset {\sim }\to \mathfrak {L}(\Gamma {\setminus } e)$ .

Definition 6.5. For an admissible weighted red and black graph $\Gamma $ , we set

and define a graded vector space

where the sum is over all admissible weighted red and black graphs with legs identified with S, and $\sim $ is the equivalence relation induced by isomorphism of such graphs.

We define a differential by summing over all ways of contracting a black edge or turning a black edge red, with certain signs. More formally, we define $d=d_{\text {con}}+d_{\text {col}}$ on this graded vector space by

$$ \begin{align*} d_{\text{con}}(\Gamma, \ell, \nu, \omega^{\otimes n}) &= \sum_{e \in E_B(\Gamma)} (\Gamma/e, \ell, \nu/e, \omega^{\otimes n})\\ d_{\text{col}}(\Gamma, \ell, \nu, \omega^{\otimes n}) &= -\sum_{e \in E_B(\Gamma)} (\Gamma {\setminus} e, \ell, \nu {\setminus} e, \omega^{\otimes n}). \end{align*} $$

We consider these complexes as defining a functor $RB^{E_n}\colon \mathsf {d(s)Br} \to \mathsf {Ch}$ , by letting the morphism $[(f, m_S)] \in \mathsf {d(s)Br}(S, T)$ induce the map $RB^{E_n}(S) \to RB^{E_n}(T)$ given by creating (oriented) red edges according to the (ordered) matching $m_S$ and then relabelling the remaining legs according to f. (This is well defined as a functor on $\mathsf {d(s)Br}$ because of the twisting by $\mathfrak {L}(\Gamma )^{\otimes n}$ .) Disjoint union makes $RB^{E_n}$ into a commutative algebra object.

Figure 5 A weighted red and black graph with two vertices, of weight $g_{ \alpha }$ and $g_{ \beta }$ respectively, and two legs, labelled by $ \underline {2}$ . We have also indicated the terms $d_{\mathrm {col}}$ and $d_{\mathrm {con}}$ of the differential, assuming n is even (if n were odd, then the graph would be zero by symmetry of the legs). In $d_{\mathrm {col}}$ , the reader maybe expected two terms obtained by turning red one of the two black edges connecting the two vertices, by these yield the same graph with opposite sign and hence cancel. The same happens in $d_{\mathrm {con}}$ , where two terms obtained by collapsing one of the two black edges connecting the two vertices cancel.

We let $RB^{Z_n}(S)$ be the quotient of $RB^{E_n}(S)$ by those weighted red and black graphs having some vertex of weight $>0$ . These assemble into a functor $RB^{Z_n}\colon \mathsf {d(s)Br} \to \mathsf {Ch}$ which is a quotient of $RB^{E_n}$ , having an induced augmented unital commutative algebra structure. Concretely $RB^{Z_n}(S)$ is given by weighted red and black graphs having all weights zero (equivalently, no weights), and the differential is given by the same formula with the understanding that $\Gamma /e$ is zero when e is a loop (as then $\Gamma /e$ has a vertex of weight $>0$ ).

Proposition 6.6. There are weak equivalences of unital commutative algebras

$$\begin{align*}RB^{Z_n} \overset{\sim}\longrightarrow Z_n\quad\quad\text{ and }\quad\quad RB^{E_n} \overset{\sim}\longrightarrow E_n,\end{align*}$$

and $RB^{Z_n}$ and $RB^{E_n}$ are cofibrant in $\mathsf {Alg}_{\mathsf {Com}^+}^{\mathrm {augm}}(\mathsf {Fun}(\mathsf {d(s)Br}, \mathsf {Ch}))$ .

Proof. Let $\phi \colon RB^{E_n} \to E_n$ be the map which

1. (i) kills weighted red and black graphs having any black edges, and

2. (ii) assigns to a graph having only red edges the induced partition of its set of legs given by the connected components of the graph, with the part corresponding to the connected subgraph $\Gamma _{\alpha }$ assigned weight $\left (\sum _{v \in V(\Gamma _{\alpha })} w(v)\right ) + 1 - \chi (\Gamma _{\alpha })$ , that is, the sum of the weights of its vertices and its genus.

This is a map of commutative algebra objects. It is also – when evaluated on any S – a map of chain complexes, as follows. If $(\Gamma , \ell , \omega ^{\otimes n}, \nu )$ is such that $\Gamma $ has at least two black edges, then every term of $d(\Gamma , \ell , \omega ^{\otimes n}, \nu )$ has a black edge and so vanishes under $\phi $ ; $(\Gamma , \ell , \omega ^{\otimes n}, \nu )$ does too. If $(\Gamma , \ell , \omega ^{\otimes n}, \nu )$ is such that $\Gamma $ has a single black edge e, then

$$\begin{align*}d(\Gamma, \ell, \nu, \omega^{\otimes n}) = (\Gamma/e, \ell, 1, \omega^{\otimes n}) - (\Gamma {\setminus} e, \ell, 1,\omega^{\otimes n}),\end{align*}$$

and $\Gamma /e$ and $\Gamma {\setminus } e$ induce the same partition on S with the same weights so this vanishes on applying $\phi $ ; $(\Gamma , \ell , \omega ^{\otimes n}, \nu )$ does too.

To check that $\phi $ is a weak equivalence, we consider the filtration given by letting $F^p RB^{E_n}(S)$ be spanned by those graphs with $\leq p$ edges. As $d_{\text {con}}$ reduces the number of edges and $d_{\text {col}}$ preserves the number of edges, we have

$$\begin{align*}\mathrm{Gr} (RB^{E_n}(S), d=d_{\text{con}}+d_{\text{col}}) \cong (RB^{E_n}(S), d_{\text{col}}).\end{align*}$$

This complex splits over isomorphism classes of weighted graphs $\Gamma $ as a sum of chain complexes $\mathrm {Colour}_*(\Gamma )_{\mathrm { Aut}(\Gamma )}$ , where $\mathrm {Colour}_*(\Gamma )$ has basis the set of subgraphs of $\Gamma $ , considered as those edges to be coloured red: Such a subgraph has degree given by $n\left (\sum _{v \in V(\Gamma )}(2w(v)+\mathrm {val}(v)-2)\right ) + \# E_B(\Gamma )$ , and the differential sums over all ways of adding a single edge to the red subgraph. In other words, up to a shift of degrees $\mathrm {Colour}_*(\Gamma )$ is the reduced simplicial chain complex of the simplex $\Delta ^{E(\Gamma )}$ on the set of edges of $\Gamma $ . It is therefore acyclic if $\Gamma $ contains any edges and has homology $\mathbb {Q}[n\sum _{v \in V(\Gamma )}(2w(v)+\mathrm {val}(v)-2)]$ otherwise. As it is supported on graphs with no edges, the differential $d_{\text {con}}$ has no effect and so the spectral sequence for this filtration collapses, to give

$$\begin{align*}H_*(RB^{E_n}(S), d) \cong \bigoplus_{\substack{\text{weighted graphs } \Gamma \\ \text{ with no edges}}} \mathbb{Q}[n\sum_{v \in V(\Gamma)}(2w(v)+\mathrm{val}(v)-2)],\end{align*}$$

and with this description we observe that $\phi _*\colon H_*(RB^{E_n}(S), d) \to E_n$ is an isomorphism as required. The argument for $Z_n$ is completely parallel.

Neglecting for a moment the differential, let $X \subset RB^{E_n}$ be the subobject spanned by those weighted red and black graphs whose black subgraphs are connected (we take this to also mean nonempty). The commutative unital algebra structure on $RB^{E_n}$ induces a map

(12) $$ \begin{align} F^{\mathrm{Com}^+}(X) = \bigoplus_{k \geq 0} (X^{\otimes k})_{\mathfrak{S}_k} \longrightarrow RB^{E_n} \end{align} $$

which we claim is an isomorphism. By definition of Day convolution, we have

$$\begin{align*}(X^{\otimes k})(S) = \operatorname*{\mathrm{colim}}_{\substack{f: S_1 \sqcup \cdots \sqcup S_k \to S \\ \in \mathsf{d(s)Br}}} X(S_1) \otimes \cdots \otimes X(S_k).\end{align*}$$

Spelling this out, it is the space of weighted red and black graphs with legs S whose black subgraph has precisely k components, and the components are ordered; $(X^{\otimes k})_{\mathfrak {S}_k}(S)$ is then the same without the ordering of components. We therefore see that (12) is indeed an isomorphism.

Reincorporating the differential, we have just shown that $RB^{E_n}$ is quasi-free, and it is also nonnegatively graded, so a standard induction over skeleta of X shows that it is cofibrant, cf. [Reference MillèsLV12, Proposition B.6.6]. The same goes for $RB^{Z_n}$ .

Let $RB^{E_n}_{\mathrm {conn}}$ be the quotient of $RB^{E_n}$ by the unit and those weighted red and black graphs whose black subgraph is not connected, and $RB^{Z_n}_{\mathrm {conn}}$ be the analogous quotient of $RB^{Z_n}$ . As the red and black graphs whose black subgraph is not connected are precisely the decomposables, we find the following.

Following Section 4.2.5, as $Z_n$ and $E_n$ have trivial differential the homology groups $\mathrm {AQ}_*(Z_n)$ and $\mathrm {AQ}_*(E_n)$ are equipped with an additional internal grading. This may be implemented in terms of the resolutions $RB^{Z_n}$ and $RB^{E_n}$ , and so in the models $RB^{Z_n}_{\mathrm {conn}}$ and $RB^{E_n}_{\mathrm {conn}}$ , by giving a (weighted) red and black graph $\Gamma $ internal degree

It is indeed preserved by the differential, and by the quasi-isomorphisms to $Z_n$ and $E_n$ . As the total degree of $\Gamma $ is $ n\left (\sum _{v \in V(\Gamma )}(2 w(v)+\mathrm {val}(v)-2)\right ) + \#E_B(\Gamma )$ , its Harrison degree is then $\# E_B(\Gamma ) + 1$ .