Abstract
We describe the ring structure of the rational cohomology of the Torelli groups of the manifolds |$\#^g S^n \times S^n$| in a stable range, for |$2n \geq 6$|. Some of our results are also valid for |$2n=2$|, where they are closely related to unpublished results of Kawazumi and Morita.
1 Introduction
This paper can be considered as a (somewhat extensive) addendum to our earlier work with Kupers [
17]. We shall be concerned with the manifold
|$W_g:= \#^g S^n \times S^n$| generalising to higher dimensions the orientable surface of genus
|$g$|, its topological group
|$\textrm{Diff}^+(W_g)$| of orientation-preserving diffeomorphisms, and various subgroups of it. The first kind of subgroups are
|$\textrm{Diff}(W_g, D^{2n}) \leq \textrm{Diff}^+(W_g, *) \leq \textrm{Diff}^+(W_g)$|, the diffeomorphisms that fix a disk and a point, respectively. The second kind are their
Torelli subgroupsconsisting of those diffeomorphisms that in addition act trivially on
|$H_n(W_g;{\mathbb{Z}})$|. The intersection form on this middle cohomology group is nondegenerate and
|$(-1)^n$|-symmetric, giving a homomorphism
This map is not always surjective, but its image is a certain finite-index subgroup
|$G^{\prime}_g \leq G_g$|, even when restricted to
|$\textrm{Diff}(W_g, D^{2n})$|, so there is an outer
|$G^{\prime}_g$|-action on each of the Torelli subgroups. This makes the rational cohomology of each of the Torelli groups into
|$G^{\prime}_g$|-representations.
In [17], for |$2n \geq 6$|, we determined |$H^*(B\textrm{Tor}(W_g, D^{2n});{\mathbb{Q}})$| as a ring and as a |$G^{\prime}_g$|-representation in a range of degrees tending to infinity with |$g$|. Using the Serre spectral sequence associated to various simple fibrations relating the different Torelli groups, we were able to also determine |$H^*(B\textrm{Tor}^+(W_g, *);{\mathbb{Q}})$| and |$H^*(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$| as |$G^{\prime}_g$|-representations. This kind of argument was not able to determine the ring structure, however, as multiplicative information gets lost when passing to the associated graded of the Serre filtration. Here, we shall determine |$H^*(B\textrm{Tor}^+(W_g, *);{\mathbb{Q}})$| and |$H^*(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$| as |${\mathbb{Q}}$|-algebras too: this is achieved in Theorem 4.1. The statement given there is more powerful, but just as in [17, Section 5], one can extract from it the following presentation for |$H^*(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$|, which is easier to parse. (A presentation for |$H^*(B\textrm{Tor}^+(W_g, *);{\mathbb{Q}})$| can be extracted in a similar way.)
Let us write
|$H(g):= H^n(W_g;{\mathbb{Q}})$|, on which
|$G^{\prime}_g$| operates in the evident way. Let
|$\lambda : H(g) \otimes H(g) \to{\mathbb{Q}}$| denote the intersection form, and
|$\{a_i\}_{i=1}^{2g}$| be a basis of
|$H(g)$| with dual basis
|$\{a_i^\#\}_{i=1}^{2g}$| in the sense that
|$\lambda (a_i^\#, a_j) = \delta _{ij}$|, so that the form dual to the pairing
|$\lambda $| is
|$\omega = \sum _{i=1}^{2g} a_i \otimes a_i^\#$|. In Section
2.2, we will construct certain “modified twisted Miller–Morita–Mumford classes”, which when restricted to the Torelli group yield
|$G^{\prime}_g$|-equivariant maps
for each
|$c \in{\mathbb{Q}}[e, p_1, p_2, \ldots , p_{n-1}] = H^*(B\textrm{SO}(2n);{\mathbb{Q}})$| and each
|$r \geq 0$|. When
|$r=0$|, we write
|$\bar{\kappa }_c = \bar{\kappa }_c(1)$|; these agree with the usual Miller–Morita–Mumford classes
|$\kappa _c$|. Finally, we write
|$\chi := \chi (W_g) = 2 + (-1)^n 2g$| and will always suppose that this is not zero (i.e., that
|$(n,g) \neq (\textrm{odd}, 1)$|).
Theorem A.If |$2n \geq 6$| then, in a range of degrees tending to infinity with |$g$|, |$H^{*}(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$| is generated as a |${\mathbb{Q}}$|-algebra by the classes |$\bar{\kappa }_c(v_1 \otimes \cdots \otimes v_r)$| for |$c$| a monomial in |$e, p_1, \ldots , p_{n-1}$|, and |$r \geq 0$|, such that |$n(r-2) + |c|> 0$|. A complete set of relations in this range is given by
(i) |$\bar{\kappa }_c(v_{\sigma (1)} \otimes \cdots \otimes v_{\sigma (r)}) = \textrm{sign}(\sigma )^n \cdot \bar{\kappa }_c(v_1 \otimes \cdots \otimes v_r)$|,
(ii) |$\bar{\kappa }_e(v_1) = 0$|,
(iii) $$ \begin{aligned}[t] \sum _i \bar{\kappa }_x(v \otimes a_i) \cdot \bar{\kappa }_y(a_i^\# \otimes w) &= \bar{\kappa }_{x\cdot y}(v \otimes w) + \tfrac{1}{\chi ^2}\bar{\kappa }_{e^2} \cdot \bar{\kappa }_x(v) \cdot \bar{\kappa }_y(w)\\ &\nonumber \quad \quad - \tfrac{1}{\chi }\bigl (\bar{\kappa }_{e \cdot x}(v) \cdot \bar{\kappa }_y(w) + \bar{\kappa }_{x}(v) \cdot \bar{\kappa }_{e \cdot y}(w)\bigr ), \end{aligned} $$
(iv) $$ \begin{aligned}[t]\sum _i \bar{\kappa }_x(v \otimes a_i \otimes a_i^\#) = \tfrac{\chi -2}{\chi } \bar{\kappa }_{e\cdot x}(v) + \tfrac{1}{\chi ^2}\bar{\kappa }_{e^2} \cdot \bar{\kappa }_{x}(v)\end{aligned} $$
,
(v) |$\bar{\kappa }_{{\mathcal{L}}_i} =0$|, where |${\mathcal{L}}_i$| denotes the |$i$|th Hirzebruch |${\mathcal{L}}$|-class,
for
|$v \in H(g)^{\otimes r}$| and
|$w \in H(g)^{\otimes s}$|.
As in [17, Section 5.4], this is not the smallest possible presentation: it can be simplified by manipulating graphs; we leave the details to the interested reader.
For |$2n=4$| or |$2n=2$|, there is still a map from the |${\mathbb{Q}}$|-algebra given by this presentation to |$H^{*}(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$|. If |$2n=2$|, then (in a stable range) this map is an isomorphism onto the maximal algebraic subrepresentation in degrees |$\leq N$|, assuming that |$H^{*}(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$| is finite-dimensional in degrees |$< N$| for all large enough |$g$|. This is known to hold for |$N=2$| by work of Johnson [12].
1.1 Outline
The overall strategy is parallel to [
17]. There we defined certain twisted Miller–Morita–Mumford classes and used them to describe the twisted cohomology groups
|$H^*(B\textrm{Diff}^{+}(W_g, D^{2n}) ; \mathcal{H}^{\otimes s})$| in a stable range of degrees, where
|$\mathcal{H}$| is the local coefficient system corresponding to
|$H_n(W_g; {\mathbb{Q}})$| with the action by diffeomorphisms of
|$W_g$|. This calculation was valid for
|$2n=2$| as well. Using that for
|$2n \geq 6$|, the
|$G^{\prime}_g$|-representations
|$H^*(B\textrm{Tor}^+(W_g, D^{2n});{\mathbb{Q}})$| extend to representations of the ambient algebraic group (namely
|$\textrm{Sp}_{2g}$| or
|$\textrm{O}_{g,g}$|) by [
16] (in fact, we did something more complicated in [
17] because this algebraicity result was not known at the time, but please allow some narrative leeway), the argument was completed by establishing the degeneration of the Serre spectral sequence
using work of Borel, and then using a categorical form of Schur–Weyl duality to extract the structure of
|$H^*(B\textrm{Tor}^+(W_g, D^{2n});{\mathbb{Q}})$| from the
|$H^{*}(B\textrm{Diff}^{+}(W_g, D^{2n}) ; \mathcal{H}^{\otimes s})$| for all
|$s$|’s and various structure maps between them.
The twisted Miller–Morita–Mumford classes may be defined on |$B\textrm{Diff}^{+}(W_g, *)$| too, but not on |$B\textrm{Diff}^{+}(W_g)$|. Our first task will be to define so-called “modified twisted Miller–Morita–Mumford classes” in |$H^{*}(B\textrm{Diff}^{+}(W_g) ; \mathcal{H}^{\otimes s})$| and analyse their behaviour: it turns out that their behaviour is significantly more complicated than the unmodified version, though still understandable. We will then use them to describe the twisted cohomology groups |$H^*(B\textrm{Diff}^{+}(W_g) ; \mathcal{H}^{\otimes s})$| in a stable range of degrees. This description will be in terms of a certain vector space of graphs with vertices labelled by monomials in Euler and Pontrjagin classes, which play the role here of the vector spaces of labelled partitions from [17]. The passage from this calculation to |$H^*(B\textrm{Tor}^{+}(W_g);{\mathbb{Q}})$| is as above.
The case of dimension |$2n=2$| is somewhat special, in precisely the same way as it was in [17]: the calculation of |$H^*(B\textrm{Diff}^{+}(W_g) ; \mathcal{H}^{\otimes s})$| is valid in this case, but as the cohomology of |$B\textrm{Tor}^+(W_g)$| is not even known to be finite-dimensional in a stable range, we cannot make a conclusion about it. (Instead, one can make a conclusion about the continuous cohomology of the Torelli group, i.e., the Lie algebra cohomology of its Mal’cev Lie algebra: see [18], [3], [11].) In addition, in this case, our modified twisted Miller–Morita–Mumford classes are essentially the same as those that have been defined by Kawazumi and Morita [13, 14, 19], and the graphical calculus that we employ is similar to theirs. In Section 5, we fully explain this connection, and also relate it to work of Garoufalidis and Nakamura [9, 10] and Akazawa [1].
To avoid a great deal of repetition, we have refrained from spelling out a lot of the background that was given in [17], and from giving in detail arguments that are very similar to those given there. As such, this paper should not be considered as attempting to be self-contained: given that its interest will be to readers of [17] this should not present a problem.
2 Characteristic Classes
2.1 Recollection on twisted Miller–Morita–Mumford classes
If |$\pi ^{\prime}: E^{\prime} \to X^{\prime}$| is an oriented smooth |$W_g^{2n}$|-bundle equipped with a section |$s: X^{\prime} \to E^{\prime}$|, and |${\mathcal{H}}$| denotes the local coefficient system |$x \mapsto H_n((\pi ^{\prime})^{-1}(x);{\mathbb{Q}})$| on |$X^{\prime}$|, then it is explained in [17, Section 3.2] that there is a unique class |$\varepsilon = \varepsilon _s \in H^n(E^{\prime} ; {\mathcal{H}})$| characterised by
(i) for each |$x \in X^{\prime}$| the element |$\varepsilon \vert _{(\pi ^{\prime})^{-1}(x)} \in H^n((\pi ^{\prime})^{-1}(x);{\mathbb{Q}}) \otimes H_n((\pi ^{\prime})^{-1}(x);{\mathbb{Q}})$| is coevaluation, and
(ii) |$s^* \varepsilon =0$|.
The proof is as follows. The Serre spectral sequence yields an exact sequence
and the section
|$s$| shows that the right-hand map
|$(\pi ^{\prime})^*$| is injective, so that the map
|$d_{n+1}$| is zero, and splits the left-hand map
|$(\pi ^{\prime})^*$|. The class
|$\textrm{coev} \in H^0(X^{\prime} ; {\mathcal{H}}^\vee \otimes{\mathcal{H}})$| then gives rise to a unique
|$\varepsilon $| satisfying the given properties.
We then defined the twisted Miller–Morita–Mumford class
2.2 Modified twisted Miller–Morita–Mumford classes
If
|$\pi : E \to X$| is an oriented smooth
|$W_g^{2n}$|-bundle but is not equipped with a section then, as long as
|$\chi := \chi (W_g) = 2 + (-1)^n 2g \neq 0$| (i.e.,
|$(n,g) \neq (\textrm{odd}, 1)$|, cf. Remark
2.1), the cohomological role of the section can instead be played by the
transfer mapwhere
|$e:= e(T_\pi E) \in H^{2n}(E;{\mathbb{Q}})$| denotes the Euler class of the vertical tangent bundle. The projection formula
shows that this map splits
|$\pi ^*$|. Thus, in this situation, there is a unique class
|$\bar{\varepsilon } \in H^n(E ; {\mathcal{H}})$| characterised by
(i) for each |$x \in X$|, the element |$\bar{\varepsilon }\vert _{\pi ^{-1}(x)} \in H^n(\pi ^{-1}(x);{\mathbb{Q}}) \otimes H_n(\pi ^{-1}(x);{\mathbb{Q}})$| is coevaluation, and
(ii) |$\tfrac{1}{\chi }\pi _!(e \cdot \bar{\varepsilon })=0$|.
Remark 2.1.If |$(n,g)=(\textrm{odd}, 1)$| then there is no class |$\bar{\varepsilon } \in H^n(E;\mathcal{H})$| satisfying (i) and natural under pullback. To see this, it suffices to give one example of a smooth oriented |$W_1$|-bundle for which |$\bar{\varepsilon }$| does not exist. Consider the Borel construction for the evident action of |$S^1 \times S^1$| on |$W_1 = S^n \times S^n$| given by considering |$S^n$| as the unit sphere in |${\mathbb{C}}^{(n+1)/2}$|. This gives a smooth oriented |$W_1$|-bundle over |$B(S^1 \times S^1)$| with total space |$E \simeq \mathbb{C}\mathbb{P}^{(n-1)/2} \times \mathbb{C}\mathbb{P}^{(n-1)/2}$|. Thus |$H^n(E;\mathcal{H})=0$|, as |$n$| is odd, but |$E$| has a cell structure with only even-dimensional cells.
By analogy with (
2.1), we may then define the modified twisted Miller–Morita–Mumford class
If |$\pi : E \to X$| does have a section |$s: X \to E$|, then the class |$\varepsilon \in H^n(E;{\mathcal{H}})$| is also defined, and we may compare it with |$\bar{\varepsilon }$| as follows:
Lemma 2.2.If |$\pi : E \to X$| has a section, then |$\bar{\varepsilon } = \varepsilon - \tfrac{1}{\chi } \pi ^*\kappa _{\varepsilon e}$|.
Proof.The classes |$\varepsilon , \bar{\varepsilon } \in H^n(E;{\mathcal{H}})$| are both defined, and agree when restricted to the fibres of the map |$\pi $|, so by considering the Serre spectral sequence for |$\pi $|, we must have |$\bar{\varepsilon }-\varepsilon = \pi ^*(x)$| for some class |$x \in H^n(X;{\mathcal{H}})$|. Applying |$\tfrac{1}{\chi }\pi _!(e \cdot -)$|, we see that |$x =\tfrac{1}{\chi } \pi _!(e \cdot (\bar{\varepsilon }-\varepsilon ))= 0 - \tfrac{1}{\chi } \pi _!(e \cdot \varepsilon ) = - \tfrac{1}{\chi } \kappa _{\varepsilon e}$|. (Here we have used, as we often will, the fact that |$e$| has even degree to commute it past |$\varepsilon $|.)
Remark 2.3(Splitting principle).
The pullback
where
|$\pi _i: E_i \to X$| are copies of the map
|$\pi $|, is equipped with a section given by the diagonal map
|$\Delta : E_1 \to E_1 \times _X E_2$|. As the maps
|$\pi _1^*$| and
|$\textrm{pr}_2^*$| are injective (they are split by their corresponding transfer maps), for the purpose of establishing identities between the characteristic classes we have discussed, it suffices to do so for bundles that do have a section.
There is another description of |$\bar{\varepsilon }$|, which is sometimes useful. Let |$\textrm{pr}_1: E \times _X E \to E$| be as in (2.3), which is an oriented |$W_g$|-bundle with section given by the diagonal map |$\Delta $|, and so has the class |$\kappa _{\varepsilon e}(\textrm{pr}_1, \Delta )$| defined.
Lemma 2.4.We have |$\bar{\varepsilon } = - \tfrac{1}{\chi } \kappa _{\varepsilon e}(\textrm{pr}_1, \Delta ) \in H^n(E;{\mathcal{H}})$|.
Proof.By Remark
2.3 we may suppose without loss of generality that
|$\pi : E \to X$| has a section
|$s: X \to E$|, defining a class
|$\varepsilon = \varepsilon _s \in H^n(E;{\mathcal{H}})$|. Consider the pullback square (
2.3); let
|$e_i = (\textrm{pr}_i)^*(e) \in H^{2n}(E_1 \times _X E_2;{\mathbb{Q}})$| be the Euler class of the vertical tangent bundle on the
|$i$|th factor. Considering
|$\textrm{pr}_1$| as a
|$W_g$|-bundle with section given by the diagonal map
|$\Delta $|, there is a class
|$\varepsilon _\Delta \in H^n(E_1 \times _X E_2 ; {\mathcal{H}})$| defined. As both
|$\varepsilon _\Delta $| and
|$\textrm{pr}_2^*(\varepsilon _s)$| restrict to coevaluation on the fibres of
|$\textrm{pr}_1$|, we have
|$\varepsilon _\Delta - \textrm{pr}_2^*(\varepsilon _s) = \textrm{pr}_1^*(x)$| for some class
|$x \in H^n(E_1;{\mathcal{H}})$|. Pulling this equation back along
|$\Delta $| shows that
|$x = -\varepsilon _s$|, so
|$\varepsilon _\Delta = \textrm{pr}_2^*(\varepsilon _s) - \textrm{pr}_1^*(\varepsilon _s)$|. Then we have
as required.
The intersection form of the fibres of |$\pi : E \to X$| provides a map of local coefficient systems |$\lambda : \mathcal{H} \otimes \mathcal{H} \to{\mathbb{Q}}$|; as we will often be concerned with applying it to two factors of a tensor power |$\mathcal{H}^{\otimes k}$| and will have to specify which factors we apply it to, we will denote |$\lambda $| by |$\lambda _{1,2}$| and more generally write |$\lambda _{i,j}: \mathcal{H}^{\otimes k} \to \mathcal{H}^{\otimes k-2}$| for the map that applies |$\lambda $| to the |$i$|th and |$j$|th factors. We call such operations contraction.
If
|$p: E_1 \times _X E_2 \to X$| denotes the fibre product of two copies of
|$\pi : E \to X$|, and if this has a section
|$s: X \to E$|, then in [
17, Lemma 3.9], we have established the formula
where
|$v = s_!(1) \in H^{2n}(E;{\mathbb{Q}})$| is the fibrewise Poincaré dual to the section
|$s$|, cf. [
17, Lemma 3.1]. The analogue of this formula for
|$\bar{\varepsilon }$| is as follows.
Proof.As in Remark 2.3, we may suppose without loss of generality that |$\pi : E \to X$| has a section |$s: X \to E$|, so that |$\varepsilon \in H^n(E;{\mathcal{H}})$| is defined.
By Lemma
2.2, we have
|$\bar{\varepsilon } = \varepsilon - \tfrac{1}{\chi } \pi ^*\kappa _{\varepsilon e} \in H^n(E;{\mathcal{H}})$|, and so
The first term is given by (
2.4), and using [
17, Proposition 3.10], the last term is given by
For the middle two terms, note that
so we need to calculate
|$\lambda _{1,2}(\varepsilon \cdot \pi ^*\kappa _{ \varepsilon e}) \in H^{2n}(E;{\mathbb{Q}})$|. The class
|$\varepsilon \cdot \kappa _{\varepsilon e}$| is the fibre integral along
|$\textrm{pr}_1: E_1 \times _{X} E_2 \to E_1$| of
|$\varepsilon \times (\varepsilon \cdot e) = (\varepsilon \times \varepsilon ) \cdot (1 \times e)$|, so
and hence
and similarly
Combining these gives
as required.
If in addition we have a lift
|$\ell : E \to B$| of the fibrewise Gauss map along some fibration
|$\theta : B \to B\textrm{SO}(2n)$|, then for any
|$c \in H^*(B;{\mathbb{Q}})$|, we can define modified twisted Miller–Morita–Mumford classes by the formula
Under the action of a permutation
|$\sigma \in \mathfrak{S}_a$| of the tensor factors, these classes transform as
|$\textrm{sign}(\sigma )^n$|, as
|$\bar{\varepsilon }$| has degree
|$n$|. Thus, for any finite set
|$S$|, there is a well-defined element
To keep track of signs, for an ordered set
|$S = \{s_1 < s_2 < \cdots < s_a\}$|, we will often write
|$\kappa _{\bar{\varepsilon }^{s_1, \ldots , s_a} c} \in H^{n (a-2) + |c|}(X ; {\mathcal{H}}^{\otimes S})$| for the corresponding element, understanding that if
|$\sigma $| is a reordering of
|$S$|, then
|$\kappa _{\bar{\varepsilon }^{\sigma (s_1), \ldots , \sigma (s_a)} c} = \textrm{sign}(\sigma )^n\kappa _{\bar{\varepsilon }^{s_1, \ldots , s_a} c}$|.
Using Lemma 2.5, we immediately see that these characteristic classes satisfy the following analogue of the contraction formula from [17, Proposition 3.10].
Proposition 2.6(Modified contraction formula).
In
|$H^*(X ; {\mathcal{H}}^{\otimes -})$|, we have the identities
and
Similarly, from Lemma 2.2, we immediately obtain the following:
Proposition 2.7.If the bundle
|$\pi : E \to X$| has a section, so that the class
|$\varepsilon $| and hence
|$\kappa _{\varepsilon ^S c}$| is defined, then
Let us give an example of using the modified contraction formula to evaluate an expression.
Example 2.8.Consider the class
|$\lambda _{1,5}\lambda _{2,6} \lambda _{3,4}(\kappa _{\bar{\varepsilon }^{1,2,3}} \cdot \kappa _{\bar{\varepsilon }^{4,5,6}})$|. Then
The first term is
The second term is
The third term is
and the fourth term is the same as the third by the evident symmetry.
2.3 Graphical interpretation
In [17, Section 5], it was found to be very convenient to adopt a graphical formalism, where |$\kappa _{\varepsilon ^a c}$| corresponds to a vertex with |$a$| half-edges incident to it and a formal label |$c$|, a product of |$\kappa _{\varepsilon ^a c}$|’s corresponds to a disjoint union of such vertices, and applying the contraction |$\lambda _{i,j}$| corresponds to pairing up the half-edges labelled |$i$| and |$j$|.
It will be convenient to adopt a similar formalism here. Let |$S$| be a finite set, and |$\mathcal{V}$| be a graded |${\mathbb{Q}}$|-algebra with a distinguished element |$e \in \mathcal{V}_{2n}$|. Slightly modifying the definition from [17, Proof of Theorem 5.1] (the difference is that we allow labelled vertices whose contribution to the degree is 0) , a marked oriented graph with legs |$S$| and labelled by |${\mathcal{V}}$| consists of the following data:
(i) a totally ordered finite set |$\vec{V}$| (of vertices), a totally ordered finite set |$\vec{H}$| (of half-edges), and a monotone function |$a \colon \vec{H} \to \vec{V}$| (encoding that a half-edge |$h$| is incident to the vertex |$a(h)$|),
(ii) an ordered matching |$m = \{(a_i, b_i)\}_{i \in I}$| of the set |$H \sqcup S$| (encoding the oriented edges of the graph),
(iii) a function |$c \colon V \to{\mathcal{V}}$| with homogeneous values, such that |$|c(v)| + n(|a^{-1}(v)|-2) \geq 0$|.
Convention.In later figures, to avoid clutter, we adopt the following ordering conventions: vertices are numbered starting from 1 from left to right, half-edges around each vertex are ordered clockwise starting from the marked half-edge, and edges are oriented from the smaller half-edge to the larger one. These conventions are already used in Figure 1, when the “marked half-edge” is taken to be the smallest half-edge at a given vertex.
Fig. 1.
An example of a marked oriented graph, with |$\vec{V} = (v_1, v_2, v_3)$|, |$c_i = c(v_i) \in{\mathcal{V}}$|, and |$\vec{H} = (h_1, h_2, \ldots , h_{10})$|.
Marked oriented graphs
|$\Gamma = (\vec{V}, \vec{H}, a, m, c)$| and
|$\Gamma ^{\prime} = (\vec{V}^{\prime}, \vec{H}^{\prime}, a^{\prime}, m^{\prime}, c^{\prime})$| with the same set of legs
|$S$| are
isomorphic if there are order-preserving bijections
|$\vec{V} \overset{\sim }\to \vec{V}^{\prime}$| and
|$\vec{H} \overset{\sim }\to \vec{H}^{\prime}$|, which intertwine
|$a$| and
|$a^{\prime}$|, intertwine
|$c$| and
|$c^{\prime}$|, and send
|$m$| to
|$m^{\prime}$|. An
oriented graph is an isomorphism class
|$[\Gamma ]$| of marked oriented graphs. We assign to a marked oriented graph
|$\Gamma = (\vec{V}, \vec{H}, a, m, c)$| the degree
Let
|$\pi : E \to X$| be an oriented
|$W_g$|-bundle with a lift
|$\ell : E \to B$| of the map classifying the vertical tangent bundle along
|$\theta : B \to B\textrm{SO}(2n)$|, and let
|$\mathcal{V}:= H^*(B;{\mathbb{Q}})$| and
|$e:= \theta ^* e \in \mathcal{V}_{2n}$|. Then, given a marked oriented graph
|$\Gamma = (\vec{V}, \vec{H}, a, m, c)$| with legs
|$S$|, we form a class
by the following recipe. Firstly, we may form
where we have used the ordering on
|$\vec{V}$| to order the product, and the ordering on
|$\vec{H}$| to trivialise the factor of
|$(\det{\mathbb{Q}}^H)^{\otimes n} = (\bigotimes _{v \in V} \det{\mathbb{Q}}^{a^{-1}(v)})^{\otimes n}$| that arises from (
2.5). Secondly, taking two copies
|$S_1$| and
|$S_2$| of the set
|$S$| and writing
|$s_i \in S_i$| for the element corresponding to
|$s \in S$|, we can form
As each
|$\kappa _{\bar{\varepsilon }^{s_1, s_2}}$| has degree 0, the product does not depend on how the factors are ordered. Taking the product of (
2.6) and (
2.7) and then applying
|$\lambda _{x,y}$| for each ordered pair
|$(x,y)$| in the matching
|$m$| on
|$H \sqcup S = H \sqcup S_1$| gives the required class
|$\bar{\kappa }(\Gamma ) \in H^{*}(X;{\mathcal{H}}^{\otimes S_2}) = H^{*}(X;{\mathcal{H}}^{\otimes S})$|.
Example 2.9.In this graphical interpretation, we recognise the class evaluated in Example 2.8 as that associated to the theta-graph 
with a certain ordering and orientation.
Clearly,
|$\bar{\kappa }(\Gamma )$| only depends on the underlying oriented graph
|$[\Gamma ]$|. We now describe how it transforms when the orderings on
|$V$|,
|$H$|, and the pairs
|$m$| are changed, without changing the underlying labelled graph. If
|$\Gamma ^{\prime}=(\vec{V}^{\prime}, \vec{H}^{\prime}, a^{\prime}, m^{\prime}, c^{\prime})$| is another marked oriented graph and there are bijections
|$f: H \to H^{\prime}$| and
|$g: V \to V^{\prime}$| intertwining
|$a$| and
|$a^{\prime}$| and
|$c$| and
|$c^{\prime}$| and such that under these bijections the matching
|$m^{\prime}$| differs from
|$m$| by reversing the order of
|$k$| pairs, then
for certain signs described on [
17, pages 55–6].
Graphs considered as representing |$\bar{\kappa }$|’s behave differently to those representing |$\kappa $|’s described in [17, Section 5]. To distinguish them, we will depict the graphs representing |$\kappa $|’s in red, as we did in that paper, and the graphs representing |$\bar{\kappa }$|’s in blue. The contraction formula of [17, Proposition 3.10] was interpreted in [17, Section 5] as giving relations among red graphs that yield equivalent |$\kappa $|-classes. In the generality of a smooth oriented |$W_g$|-bundle |$\pi : E \to X$| with section |$s: X \to E$|, these may be depicted as follows:
Fig. 2.
The contraction formula, displayed graphically.
Here, the negative terms only arise when they make sense, that is, when the vertex has valence 2 in the first case, when the vertex labelled |$c$| has valence 1 in the second case, and when the vertex labelled |$c^{\prime}$| has valence 1 in the third case. We have used the ordering convention described above, where the marked half-edge is indicated by a bar.
Similarly, the modified contraction formula of Proposition 2.6 can be interpreted as giving the following relations among blue graphs that yield equivalent |$\bar{\kappa }$|-classes:
Fig. 3.
The modified contraction formula, displayed graphically.
3 Twisted Cohomology of Diffeomorphism Groups
The main goal of this section is to describe the twisted cohomology groups
in a stable range of degrees, of the classifying space
|$B\textrm{Diff}^{+}(W_g)$| of the group of orientation-preserving diffeomorphisms of
|$W_g$| (which classifies oriented
|$W_g$|-bundles), and the classifying space
|$B\textrm{Diff}^{+}(W_g, *)$| of the group of orientation-preserving diffeomorphisms of
|$W_g$|, which fix a point
|$* \in W_g$| (which classifies oriented
|$W_g$|-bundles with section). In [
17, Theorem 3.15], the analogous calculation was given for the classifying space
|$B\textrm{Diff}(W_g, D^{2n})$| of the group of diffeomorphisms of
|$W_g$|, which fix a disk
|$D^{2n} \subset W_g$|.
In order to do this, we will also discuss the manifolds
|$W_g$| equipped with
|$\theta $|-structures for the tangential structure
|$\theta : B\textrm{SO}(2n)\langle n \rangle \to B\textrm{O}(2n)$|, that is, the
|$n$|-connected cover of
|$B\textrm{O}(2n)$|. In this case, we will consider the homotopy quotients
where
|$\textrm{Bun}(TW_g, \theta ^*\gamma _{2n})$| denotes the space of vector bundle maps
|$TW_g \to \theta ^*\gamma _{2n}$| from the tangent bundle of
|$W_g$| to the bundle classified by
|$\theta $|. The group
|$\textrm{Diff}(W_g)$| acts on the space of bundle maps by precomposing with the derivative.
There is a factorisation
|$\theta : B\textrm{SO}(2n)\langle n \rangle \overset{\theta ^{\textrm{or}}}\to B\textrm{SO}(2n) \overset{\sigma }\to B\textrm{O}(2n)$|, and by obstruction theory, one sees that the space
|$\textrm{Bun}(TW_g, \sigma ^*\gamma _{2n})$| has two contractible path components corresponding to the two orientations of
|$W_g$|. In particular, there are equivalences
and so
|$\theta ^{\textrm{or}}$| induces maps
It is shown in [
7, Section 5.2] that these are principal
|$\textrm{SO}[0,n-1]$|-fibrations. In particular, the spaces
|$B\textrm{Diff}^\theta (W_g)$| and
|$B\textrm{Diff}^\theta (W_g, *)$| are path-connected.
3.1 Spaces of graphs
Our description of the twisted cohomology groups of
|$B\textrm{Diff}^{+}(W_g)$|,
|$B\textrm{Diff}^{+}(W_g, *)$|,
|$B\textrm{Diff}(W_g, D^{2n}) $|,
|$B\textrm{Diff}^{\theta }(W_g)$|, and
|$B\textrm{Diff}^{\theta }(W_g, *)$| in a stable range will be—via the graphical interpretation given in Section
2.3—in terms of graded vector spaces of labelled graphs, modulo certain relations. (Readers of [
17] may have been expecting vector spaces of labelled partitions instead: here we have found spaces of graphs more convenient for formulating results, cf. Remark
3.2, though spaces of labelled partitions will still play a role in the proofs.) To describe these spaces of graphs, we will use the graded
|${\mathbb{Q}}$|-algebras
with distinguished elements
|$e$| of degree
|$2n$| given by the Euler class. In order to work in a way that is agnostic about the genus
|$g$| of the manifold
|$W_g$| under consideration, we will work over the ring
|${\mathbb{Q}}[\chi ^{\pm 1}]$| instead of
|${\mathbb{Q}}$|, where
|$\chi $| is an invertible formal parameter, which will—later—be set to the Euler characteristic
|$2 + (-1)^n 2g$| of
|$W_g$|.
Definition 3.1.(a) imposes the sign rule for changing orderings of vertices and half-edges and for reversing orientations of edges;
(b) imposes linearity in the labels, and sets a graph containing an |$a$|-valent vertex labelled by |$c$| with |$|c| + n(a-1) < 0$| to zero;
(c) sets the 0-valent vertex labelled by |$e \in{\mathcal{V}}_{2n}$| equal to |$\chi $|, and if |$2n \equiv 0 \mod 4$| sets the 0-valent vertex labelled by |$p_{n/2} \in{\mathcal{V}}_{2n}$| equal to 0;
(d) imposes the contraction relations (which are the relations from Figure 2 in the case that |$s^*$| kills all positive-degree classes)
where the negative terms only arise when they makes sense, that is, in the first case when the vertex has valence 2 and its label |$c$| is a scalar multiple of |$1 \in{\mathcal{V}}_0$|, in the second case when |$c$| is a scalar multiple of |$1 \in \mathcal{V}_0$| and has valence 1, and similarly in the third case.
(ii) Let
where
|$\sim $| imposes (a)–(c) as well as
(iii) Let
where
|$\sim $| imposes (a) and (b), as well as
- (c|$^{\prime\prime} $|)
sets the 0-valent vertex labelled by |$e \in{\mathcal{W}}_{2n}$| equal to |$\chi $|, sets the 0-valent vertex labelled by any degree |$2n$| monomial in Pontrjagin classes equal to 0, and for any |$1 \leq i \leq \lfloor n/4 \rfloor $| sets
and (d
|$^\prime $|).
(iv) Let
where
|$\sim $| imposes (a) and (b), as well as
- (c|$^{\prime\prime\prime} $|)
sets the 0-valent vertex labelled by |$e \in{\mathcal{V}}_{2n}$| equal to |$\chi $|, if |$2n \equiv 0 \mod 4$| sets the 0-valent vertex labelled by |$p_{n/2} \in{\mathcal{V}}_{2n}$| equal to 0, and sets the 1-valent vertex labelled by |$e \in{\mathcal{V}}_{2n}$| equal to 0,
- (d|$^{\prime\prime\prime} $|)
imposes the contraction relations of Figure 3.
(v) Let
where
|$\sim $| imposes (a), (b), as well as
- (c|$^{\prime\prime\prime\prime} $|)
sets the 0-valent vertex labelled by |$e \in{\mathcal{W}}_{2n}$| equal to |$\chi $|, sets the 0-valent vertex labelled by any degree |$2n$| monomial in Pontrjagin classes equal to 0, sets the 1-valent vertex labelled by |$e \in{\mathcal{W}}_{2n}$| equal to 0, and for any |$1 \leq i \leq \lfloor n/4 \rfloor $| sets
and (d
|$^{\prime \prime \prime }$|).
Remark 3.2(Graphs and partitions).
In all cases, one can apply the (modified) contraction formula to pass from a graph to a sum of graphs with strictly fewer edges, and so by iterating to a sum of graphs with no edges. These are disjoint unions of labelled corollas, and so correspond to partitions of |$S$| with labels in |$\mathcal{V}$| or |$\mathcal{W}$|, plus additional external labels in cases (ii) and (iii). There are two issues with this. The first is that in cases (iv) and (v), it is not clear that the resulting sum of disjoint unions of labelled corollas is unique, as one has to choose an order in which to eliminate edges. The second is that even if it is, then the functoriality on the Brauer category that we describe below would involve gluing in edges and then eliminating them, leading to a complicated formula. This is why we have found it convenient to work with spaces of graphs.
We wish to consider each of the above as defining functors on the (signed) Brauer category as in [17, Section 2.3], but to take into account the parameter |$\chi $|, we must slightly generalise to a |${\mathbb{Q}}[\chi ]$|-linear version of the (signed) Brauer category.
Definition 3.3.For finite sets |$S$| and |$T$| let |$\textsf{preBr}^\chi (S,T)$| be the free |${\mathbb{Q}}[\chi ]$|-module on tuples |$(f,m_S, m_T)$| of a bijection |$f$| from a subset |$S^\circ \subset S$| to a subset |$T^\circ \subset T$|, an ordered matching |$m_S$| of |$S \setminus S^\circ $|, and an ordered matching |$m_T$| of |$T \setminus T^\circ $|.
Let |$\textsf{Br}^\chi (S,T)$| be the quotient of |$\textsf{preBr}^\chi (S,T)$| by the span of |$(f,m_S, m_T) - (f,m^{\prime}_S, m^{\prime}_T)$| whenever |$m_S$| agrees with |$m^{\prime}_S$| after reversing some pairs, and |$m_T$| agrees with |$m^{\prime}_T$| after reversing some pairs.
Let |$\textsf{sBr}^\chi (S,T)$| be the quotient of |$\textsf{preBr}^\chi (S,T)$| by the span of |$(f,m_S, m_T) - (-1)^{k+l}(f,m^{\prime}_S, m^{\prime}_T)$| whenever |$m_S$| agrees with |$m^{\prime}_S$| after reversing |$k$| pairs, and |$m_T$| agrees with |$m^{\prime}_T$| after reversing |$l$| pairs.
Let |$\textsf{(s)Br}^\chi $| be the |${\mathbb{Q}}[\chi ]$|-linear category whose objects are finite sets, and whose morphisms are the |${\mathbb{Q}}[\chi ]$|-modules |$\textsf{(s)Br}^\chi (S,T)$| defined above. In the case of |$\textsf{Br}^\chi $|, we think of |$[f,m_S, m_T]$| as representing 1-dimensional cobordisms with no closed components: then the composition law is given by composing cobordisms and then replacing each closed 1-manifold by a factor of |$\chi -2$|. In the case of |$\textsf{sBr}^\chi $|, we think of |$(f,m_S, m_T)$| as representing oriented 1-dimensional cobordisms with no closed components: then the composition law is given by composing cobordisms and then replacing each compatibly oriented closed 1-manifold by a factor of |$-(\chi -2)$|.
Let |$\textsf{d(s)Br}^\chi $| denote the subcategories having all objects and morphisms spanned by |$[f,m_S, m_T]$| with |$T^\circ =\emptyset $|.
For a central charge|$d \in{\mathbb{Q}}$|, let |$\textsf{(d)(s)Br}_d$| denote the |${\mathbb{Q}}$|-linear category obtained by specialising the |${\mathbb{Q}}[\chi ]$|-linear category |$\textsf{(d)(s)Br}^\chi $| to |$\chi =2 + d$| for |$\textsf{(d)Br}$| or |$\chi =2 - d$| for |$\textsf{(d)sBr}$|. (This notation then agrees with [17, Definition 2.14, 2.19].)
Fig. 4.
A graphical representation of a basis element |$(f,m_S, m_T)$| in |$\textsf{preBr}^\chi (S,T)$| from a |$6$|-element set |$S$| to a |$4$|-element set |$T$|, with |$m_T = \emptyset $|. (The order of crossings is irrelevant.) Reversing the arrow gives the same element in |$\textsf{Br}^\chi (S,T)$| and minus the element in |$\textsf{sBr}^\chi (S,T)$|.
We consider the spaces of graphs above as defining
|${\mathbb{Q}}[\chi ]$|-linear functors
in the evident way, by gluing of oriented graphs (after orientations have been arranged to be compatible). We endow them with a lax symmetric monoidality by disjoint union of graphs. We write
|$\mathcal{G}\textrm{raph}_1(-)_{g}: \textsf{(s)Br}_{2g} \to \textsf{Gr}({\mathbb{Q}}\textrm{-}\textsf{mod})$| and so on for their specialisations at
|$\chi =2+(-1)^n2g$| (defined for
|$(n,g) \neq (\textrm{odd}, 1)$|).
3.2 The isomorphism theorem
Theorem 3.4 below extends [17, Theorem 3.15] to |$B\textrm{Diff}^{\theta }(W_g, *)$|, |$B\textrm{Diff}^{+}(W_g, *)$|, |$B\textrm{Diff}^{\theta }(W_g)$|, and |$B\textrm{Diff}^{+}(W_g)$|.
To formulate it, we first observe that when
|$\pi : E \to X$| is a smooth oriented
|$W_g$|-bundle and
|$\mathcal{H}$| is the local coefficient system over
|$X$| given by the fibrewise
|$n$|th homology of this bundle, the fibrewise intersection form
|$\lambda : \mathcal{H} \otimes \mathcal{H} \to{\mathbb{Q}}$| and its dual
|$\omega : {\mathbb{Q}} \to \mathcal{H} \otimes \mathcal{H}$| are
|$(-1)^n$|-symmetric and satisfy
|$\lambda \circ \omega = (-1)^n2g \cdot \textrm{Id}$|, so provide a
|${\mathbb{Q}}$|-linear functor
|$S \mapsto \mathcal{H}^{\otimes S}$| from
|$\textsf{(s)Br}_{2g}$| to the category of local coefficient systems of
|${\mathbb{Q}}$|-modules over
|$X$|. (Strictly speaking our definitions require
|$\chi = 2 + (-1)^n 2g$| to be invertible, so we omit the case
|$(n,g) = (\textrm{odd}, 1)$|.) Composing this with taking cohomology gives a functor
The relations in the various spaces of graphs defined in Section 3.1 were chosen precisely to match the contraction formula of [17, Proposition 3.10] (in the case of |$\mathcal{G}\textrm{raph}_1$|) and the modified contraction formula of Proposition 2.6 (in the other cases), so that assigning to a graph its associated |$\kappa $|- or |$\bar{\kappa }$|-class provides natural transformations
(i) |$\kappa : \mathcal{G}\textrm{raph}_1(-)_g \to H^*(B\textrm{Diff}(W_g, D^{2n}); {\mathcal{H}}^{\otimes -})$|,
(ii) |$\kappa : \mathcal{G}\textrm{raph}_*^\theta (-)_g \to H^*(B\textrm{Diff}^{\theta }(W_g, *); {\mathcal{H}}^{\otimes -})$|,
(iii) |$\kappa : \mathcal{G}\textrm{raph}_*(-)_g \to H^*(B\textrm{Diff}^{+}(W_g, *); {\mathcal{H}}^{\otimes -})$|,
(iv) |$\bar{\kappa }: \mathcal{G}\textrm{raph}^\theta (-)_g \to H^*(B\textrm{Diff}^{\theta }(W_g); {\mathcal{H}}^{\otimes -})$|,
(v) |$\bar{\kappa }: \mathcal{G}\textrm{raph}(-)_g \to H^*(B\textrm{Diff}^{+}(W_g); {\mathcal{H}}^{\otimes -})$|,
of functors |$\textsf{(s)Br}_{2g} \to \textsf{Gr}({\mathbb{Q}}\textrm{-}\textsf{mod})$|.
Theorem 3.4.For |$2n =2$| or |$2n \geq 6$|, the maps (i)–(v) are isomorphisms in a range of cohomological degrees tending to infinity with |$g$|.
We will first give the proof in cases (i), (ii), (iii), and in case (v) assuming case (iv); the much more involved case (iv) will be treated afterwards.
Proof of Theorem 3.4 (i), (ii), (iii), (v).For case (i), observe that |$\mathcal{G}\textrm{raph}_1(-)_g$| is naturally isomorphic to the functor |$\mathcal{G}(-,{\mathcal{V}})$| from [17, Proof of Theorem 5.1], which is shown there to be isomorphic to the functor |$\mathcal{P}(-, {\mathcal{V}})_{\geq 0} \otimes \det ^{\otimes n}$|. This case then follows from [17, Theorem 3.15].
For case (ii), we first construct a homotopy fibre sequence
The left-hand term may be written as the homotopy quotient of
|$\textrm{Diff}(W_g, *)$| acting on the Stiefel manifold
|$\textrm{Fr}(T_*W_g)$| given by the space of frames in the tangent space to
|$W_g$| at the point
|$* \in W_g$|, as this action is transitive and its stabiliser is the subgroup that fixes a point and its tangent space, which is homotopy equivalent to fixing a disk. The middle term was defined as the homotopy quotient of
|$\textrm{Diff}(W_g, *)$| acting on
|$\textrm{Bun}(TW_g, \theta ^*\gamma _{2n})$|. Evaluation at
|$* \in W_g$| defines a
|$\textrm{Diff}(W_g, *)$|-invariant map
This is easily seen to be a Serre fibration. If we choose a point
|$x \in B\textrm{SO}(2n)\langle n \rangle $| and a framing
|$\xi : (\theta ^*\gamma _{2n})_x \overset{\sim }\to{\mathbb{R}}^{2n}$|, then there is a map
|$\xi _*: ev^{-1}(x) \to \textrm{Fr}(T_* W_g)$| given by sending a bundle map
|$\hat{\ell }: TW_g \to \theta ^*\gamma _{2n}$| whose underlying map sends * to
|$x$| to the framing
|$\xi \circ \hat{\ell }_x: T_* W_g \to (\theta ^*\gamma _{2n})_x \to{\mathbb{R}}^{2n}$|. One verifies by obstruction theory that
|$\xi _*: ev^{-1}(x) \to \textrm{Fr}(T_* W_g)$| is a weak equivalence. Taking homotopy orbits for
|$\textrm{Diff}(W_g, *)$| then gives the required homotopy fibre sequence (
3.1).
As
|$H^*(B\textrm{Diff}(W_g, D^{2n}); {\mathcal{H}}^{\otimes S})$| is spanned by products of twisted Miller–Morita–Mumford classes
|$\kappa _{\varepsilon ^a c}$| with
|$c \in \mathcal{V}$| in a stable range by (i), and these classes may be defined on
|$B\textrm{Diff}^{\theta }(W_g, *)$|, the Serre spectral sequence
for the homotopy fibre sequence (
3.1) collapses in a stable range. The result then follows by observing that the analogue of the Serre filtration of
|$\mathcal{G}\textrm{raph}_*^\theta (-)_g$|, induced by the descending filtration by degrees of
|$H^*(B\textrm{SO}(2n)\langle n \rangle ;{\mathbb{Q}}) = {\mathcal{V}}$|, has
because modulo
|$\mathcal{V}_{>0}$| the formula of (d
|$^\prime $|) specialises to that of (d). The induced map
therefore has the form
|$\mathcal{V} \otimes \{\textrm{the map }\kappa \text{ in case (i)}\}$| so is an isomorphism in a stable range by case (i). Case (ii) follows.
Case (iii) is just like the above, using the homotopy fibre sequence
instead, which is established in the analogous way, and
|${\mathcal{W}}$| in place of
|${\mathcal{V}}$|.
Case (v) can be deduced from case (iv) by applying the same method to the homotopy fibre sequence
established in [
7, Section 5.2]. The filtration step is a little different, so we give some details. It follows from (iv) that
|$H^*(B\textrm{Diff}^\theta (W_g); {\mathcal{H}}^{\otimes S})$| is spanned by products of twisted Miller–Morita–Mumford classes
|$\kappa _{\bar{\varepsilon }^a c}$| with
|$c \in \mathcal{V}$| in a stable range, and these may be defined on
|$B\textrm{Diff}^{+}(W_g)$| (in fact they may be defined even for
|$c \in \mathcal{W}$|) so the corresponding Serre spectral sequence
degenerates in a stable range. In this case, the analogue of the Serre filtration on
|$\mathcal{G}\textrm{raph}(-)_g$| is induced by giving the graph
|$\Upsilon _i:= (\{0\}, \emptyset , \emptyset \to \{0\}, \emptyset , c(0) = ep_i)$|—which is a single vertex labelled by
|$ep_i$| and no edges—filtration
|$4i$| for
|$1 \leq i \leq \lfloor n/4 \rfloor $|, giving all other connected graphs filtration 0, and extending multiplicatively. The associated graded of this filtration has the form
because the relation in (c
|$^{\prime \prime \prime \prime }$|) shows that any graph with a vertex labelled
|$cp_i$| for
|$1 \leq i \leq \lfloor n/4 \rfloor $| is equivalent to a graph of strictly larger filtration, unless the vertex is 0-valent and the label is
|$ep_i$|. As
|$\bar{\kappa }(\Upsilon _i) = \kappa _{ep_i} = \chi \cdot \xi ^*(p_i)$| by [
7, Remark 5.5], it follows that the induced map
has the form
|$\{\textrm{an isomorphism}\} \otimes \{\textrm{the map}\ {\bar{\kappa }}\ \text{in case (iv)}\}$| so is an isomorphism in a stable range by case (iv).
3.3 Proof of Theorem 3.4 (iv)
The proof of Theorem 3.4 (iv) is of a less formal nature. It will be parallel to that of [17, Theorem 3.15], but algebraically more complicated. An important tool will be the following lemma, inspired by [20, page 566].
Lemma 3.5.Let |$G$| be a topological group and |$p: P \to X$| be a principal |$G$|-bundle with action |$a: G \times P \to P$|, which satisfies the Leray–Hirsch property in cohomology over a field |${\mathbb{F}}$|. Then
is an equaliser diagram.
Proof.Let us leave |${\mathbb{F}}$| implicit. By the Leray–Hirsch property |$H^*(P)$| is a free |$H^*(X)$|-module and hence is faithfully flat. Thus, it suffices to prove that the diagram is an equaliser diagram after applying |$- \otimes _{H^*(X)} H^*(P)$|. By the Leray–Hirsch property, we also have |$H^*(P) \otimes _{H^*(X)} H^*(P) \overset{\sim }\to H^*(P \times _X P)$|. Thus, it suffices to show that
is an equaliser diagram, which is the same question for the principal
|$G$|-bundle
|$\textrm{pr}_2: P \times _X P \to P$|. But this principal
|$G$|-bundle has a section given by the diagonal map, which trivialises it: this trivialisation identifies the diagram with
which is indeed an equaliser diagram as it has a contraction induced by
|$a^*$|.
We adapt the proof of [17, Theorem 3.15], supposing for concreteness that |$n$| is odd. Consider the tangential structure |$\theta \times Y: B\textrm{SO}(2n) \langle n \rangle \times Y \to B\textrm{SO}(2n)$| with |$Y=K(W^\vee , n+1)$| and |$W$| a generic rational vector space. Then we have |$H^*(Y ; {\mathbb{Q}}) \cong \textrm{Sym}^*(W[n+1])$|, the symmetric algebra on the vector space |$W$| placed in (even) degree |$n+1$|. If |$n$| is even, then like at the end of the proof of [17, Theorem 3.15], we would take |$Y=K(W^\vee , n+2)$| instead, so |$H^*(Y;{\mathbb{Q}})$| would still be a symmetric algebra. Apart from this, there is no essential difference, and we will not comment further on the differences in the case |$n$| even.
There are associated universal
|$W_g$|-bundles
and an evaluation map
|$\ell : E^{\theta \times Y} \to Y$|. Neglecting the “maps to
|$Y$|” part of the tangential structure gives a homotopy fibre sequence
We can take
|$Y$| to be a topological abelian group, which then acts fibrewise on the map
|$\theta \times Y$| and hence acts compatibly on
|$E^{\theta \times Y}$| and
|$B\textrm{Diff}^{\theta \times Y}(W_g)$|. Using this, we can form the homotopy fibre sequence
The space
|$\textrm{map}(W_g, Y) /\!\!/ Y$| is a
|$K(H^n(W_g;{\mathbb{Q}}) \otimes W^\vee ,1)$|, so there is an identification of graded local coefficient systems
This is natural in the vector space
|$W$|, and scaling by
|$u \in{\mathbb{Q}}^\times $| acts on
|$\Lambda ^k({\mathcal{H}} \otimes W[1])$| by
|$u^k$|. It follows that it acts this way on the
|$k$|th row of the Serre spectral sequence
As the differentials in this spectral sequence must be equivariant for this
|${\mathbb{Q}}^\times $|-action, it follows that they must all be trivial. Furthermore, this action gives a weight decomposition of both sides, which identifies
the weight-
|$k$| subspace.
To access the latter groups, we use that there is a map
which by the main theorems of [
2,
8,
21] (for
|$2n=2$|) and [
4–6] for (
|$2n \geq 6$|) is an isomorphism on cohomology in a stable range of degrees. Here
|$\textrm{MT}\theta $| is the Thom spectrum of
|$-\theta ^*\gamma _{2n}$|, so writing
|$u_{-2n} \in H^{-2n}(\textrm{MT}\theta ;{\mathbb{Q}})$| for its Thom class, by the Thom isomorphism, we have
The rational cohomology of
|$\Omega ^\infty _0(\textrm{MT}\theta \wedge Y_+)$| is then given by
which can be considered as the free (graded-)commutative algebra on the even-degree classes
|$\kappa _{c, w_1 \cdots w_r}$| with
|$c \in{\mathbb{Q}}[p_{\lceil \tfrac{n+1}{4}\rceil }, \ldots , p_{n-1}, e]$| and
|$w_i \in W$|, modulo linearity in
|$c$| and in the
|$w_i$|, and modulo commutativity of the
|$w_i$|. The pullbacks of these classes along
|$\alpha $| we again denote
|$\kappa _{c, w_1 \cdots w_r}$|, and they may be described intrinsically as the fibre integrals
|$\pi ^Y_!(c(T_{\pi ^Y} E^{\theta \times Y}) \cdot \ell ^*(w_1 \cdots w_r))$|.
Lemma 3.6.There are unique classes
|$\bar{\kappa }_{c, w_1 \cdots w_r} \in H^*(B\textrm{Diff}^{\theta \times Y}(W_g) /\!\!/ Y;{\mathbb{Q}})$|, which pull back to
and in a stable range of degrees
|$H^*(B\textrm{Diff}^{\theta \times Y}(W_g) /\!\!/ Y;{\mathbb{Q}})$| is the free graded-commutative algebra on the classes
|$\bar{\kappa }_{c, w_1 \cdots w_r}$|, modulo linearity in
|$c$| and in the
|$w_i$|, commutativity of the
|$w_i$|, and modulo
|$\bar{\kappa }_{e, w_1}=0$|.
Proof.We wish to apply Lemma
3.5 to the principal
|$Y$|-bundle
First observe that the fibre inclusion
|$j: Y \to B\textrm{Diff}^{\theta \times Y}(W_g)$| classifies the
|$W_g$|-bundle
|$\textrm{pr}_1: Y \times W_g \to Y$| equipped with the product
|$\theta $|-structure and the map
|$\ell = \textrm{pr}_1: Y \times W_g \to Y$|. Thus, for any
|$w \in W$|, we have
and so (
3.2) satisfies the Leray–Hirsch property.
Lemma
3.5 then describes
|$H^*(B\textrm{Diff}^{\theta \times Y}(W_g) /\!\!/ Y;{\mathbb{Q}})$| as the equaliser of
In a stable range,
|$H^*(B\textrm{Diff}^{\theta \times Y}(W_g);{\mathbb{Q}})$| is described in terms of the classes
|$\kappa _{c, w_1 \cdots w_r}$|, so to make use of this equaliser description we must determine how these classes pull back along the action map
This map classifies the
|$W_g$|-bundle
|$Y \times \pi ^Y: Y \times E^{\theta \times Y} \to Y \times B\textrm{Diff}^{\theta \times Y}(W_g)$| equipped with the structure map
|$Y \times E^{\theta \times Y} \overset{Y \times \ell }\to Y \times Y \overset{\cdot }\to Y$|. As the
|$w_i \in W = H^{n+1}(Y;{\mathbb{Q}})$| are primitive with respect to the coproduct induced by the multiplication on
|$Y$|, we have
Our goal now is to show that the classes defined by
are equalised by the maps (
3.3), so by Lemma
3.5 descend to unique classes of the same name in
|$H^*(B\textrm{Diff}^{\theta \times Y}(W_g) /\!\!/ Y;{\mathbb{Q}})$|. To see this, we calculate using the formula above that
For each
|$A \subseteq \{1,2,\ldots , r\}$|, the coefficient of
|$w_A$| is
and
|$\sum _{U \subseteq A}(-1)^{|U|}$| is the binomial expansion of
|$(1-1)^{|A|}$| so vanishes if
|$A \neq \emptyset $| and is
|$1$| if
|$A=\emptyset $|, which shows that
|$a^*(\bar{\kappa }_{c, w_1 \cdots w_r}) = 1 \times \bar{\kappa }_{c, w_1 \cdots w_r}$| as required.
Finally, that these classes (except |$\bar{\kappa }_{e, w_1}=0$|) freely generate the |${\mathbb{Q}}$|-algebra |$H^*(B\textrm{Diff}^{\theta \times Y}(W_g) /\!\!/ Y;{\mathbb{Q}})$| in a stable range follows from the fact that the |$\kappa _{c, w_1 \cdots w_r}$| freely generate |$H^*(B\textrm{Diff}^{\theta \times Y}(W_g);{\mathbb{Q}})$| in a stable range, together with the observation that |$\bar{\kappa }_{c, w_1 \cdots w_r} \equiv{\kappa }_{c, w_1 \cdots w_r}$| modulo the ideal generated by classes |$\kappa _{e, w}$| and the Leray–Hirsch property again.
Let us provide a “fibre-integral” interpretation of the classes we have just constructed. Consider the map of principal
|$Y$|-bundles
The composition
|$\ell \circ i: Y \to Y$| is the identity, so
|$i^* \ell ^*(w) = w \in H^{n+1}(Y;{\mathbb{Q}})$|. We showed in the proof above that
|$j^* \kappa _{e, w} = \chi w \in H^{n+1}(Y;{\mathbb{Q}})$|, so in particular both these principal
|$Y$|-bundles satisfy the Leray–Hirsch property. Together these give that
As
|$Y$| is
|$n$|-connected, it follows from the Serre spectral sequence that there exists a unique class
|$\bar{\ell }^*(w) \in H^{n+1}(E^{\theta \times Y} /\!\!/ Y;{\mathbb{Q}})$|, which pulls back to
|$\ell ^*(w) - \tfrac{1}{\chi } (\pi ^Y)^*\kappa _{e, w}$|.
Proof.As the lower of the above principal
|$Y$|-bundles satisfies the Leray–Hirsch property, this identity may be verified after pulling back to
|$B\textrm{Diff}^{\theta \times Y}(W_g)$|. In
|$H^*(E^{\theta \times Y};{\mathbb{Q}})$|, we have
|$\bar{\ell }^*(w) =\ell ^*(w) - \tfrac{1}{\chi } (\pi ^Y)^*\kappa _{e, w}$|, so expanding out gives
as required.
The classes
|$\bar{\kappa }_{c, w_1 \cdots w_r}$| provide an isomorphism
in a stable range, natural in
|$W$|, which with the discussion above gives an identification of graded vector spaces
natural in
|$W$|.
We now proceed as in the proof of [
17, Theorem 3.15], which we briefly summarise. As the identification just given is natural in
|$W$|, we can consider it as an identification of (graded) Schur functors of
|$W$|, and therefore identify their coefficients (i.e., Schur–Weyl duality). This is implemented by letting
|$\textrm{GL}:= \operatorname *{colim}_{n \to \infty } \textrm{GL}_n({\mathbb{Q}})$| and considering the two sides as defining (graded) polynomial representations of
|$\textrm{GL}$| by setting
|$W:= \operatorname *{colim}_{n \to \infty } {\mathbb{Q}}^n$|. Writing
|$\hat{W}_*:= \lim _{n \to \infty } \textrm{Hom}({\mathbb{Q}}^n, {\mathbb{Q}})$|, Schur–Weyl duality may be expressed as the equivalence of categories
where
|$({\mathbb{Q}}\textrm{-}\textsf{mod}^{\textsf{FB}})^f$| denotes the functors from the category
|$\textsf{FB}$| of finite sets and bijections to the category of finite-dimensional
|${\mathbb{Q}}$|-modules; it gives a similar equivalence between categories of graded objects. Using that
|$n$| is odd, the left-hand side above can be expressed as
up to a small reindexing of degrees. It is elementary to see that the right-hand side may be expressed as
|$W^{\otimes -} \otimes ^{\textsf{FB}} \mathcal{P}^{\textrm{bis}}(-,{\mathcal{V}})_{\geq 0}$| up to the analogous reindexing of degrees, where
|$ \mathcal{P}(-,{\mathcal{V}})_{\geq 0} \to \mathcal{P}^{\textrm{bis}}(-,{\mathcal{V}})_{\geq 0}$| is the quotient by those partitions containing a part of size 1 labelled by
|$e \in{\mathcal{V}}_{2n}$|. Thus, carrying the factor of
|${\det }^{\otimes n}$| to the other side, the above identification yields a natural transformation
of lax symmetric monoidal functors
|$\textsf{FB} \to \textsf{Gr}({\mathbb{Q}}\textrm{-}\textsf{mod})$|, which is an isomorphism in a stable range.
On the other hand, assigning to a labelled part the corolla with that label gives a natural transformation
of lax symmetric monoidal functors
|$\textsf{FB} \to \textsf{Gr}({\mathbb{Q}}\textrm{-}\textsf{mod})$|, and we claim that using this (
3.4) factors through the map
|$\bar{\kappa }: \mathcal{G}\textrm{raph}^\theta (-)_g \to H^*(B\textrm{Diff}^{\theta }(W_g); {\mathcal{H}}^{\otimes -})$|. Assuming this claim for now, observe that using the contraction relations in Definition
3.1 (iv) (d
|$^{\prime \prime \prime }$|) to contract all edges shows that (
3.5) is surjective, which with the fact that (
3.4) is an isomorphism in a stable range will show that the map
|$\bar{\kappa }$| is an isomorphism in a stable range too (as well as the map (
3.5)).
It remains to show the factorisation, that is, that the map (
3.4) sends a part of size
|$a$| labelled by
|$c \in{\mathcal{V}}$| to the class
|$\kappa _{\bar{\varepsilon }^a c}$|. We again proceed as in the relevant step of the proof of [
17, Theorem 3.15]. There is a fibration sequence
and so, taking homotopy orbits for the fibrewise
|$Y$|-action, a fibration sequence
Again by functoriality in
|$W$| the associated Serre spectral sequence collapses to identify the weight decomposition as
Given the description in Lemma
3.7, we must show that the map
is given by
|$w \mapsto \bar{\varepsilon } \otimes w$|, which is the analogue of [
17, Claim 3.16]. As it is natural in the vector space
|$W$|, it must certainly be given by
|$\bar{\ell }(w) = x \otimes w$| for some
|$x \in H^n(E^\theta ; {\mathcal{H}})$|, and we must show that
|$x = \bar{\varepsilon }$|. That the restriction of
|$x$| to the fibre
|$W_g$| of
|$\pi : E^\theta \to B\textrm{Diff}^{\theta }(W_g)$| is given by coevaluation may be done precisely as in [
17, Claim 3.16]. By the characterisation of
|$\bar{\varepsilon }$|, it remains to check that
By the Leray–Hirsch property, this may be checked after pulling back to
|$B\textrm{Diff}^{\theta \times Y}(W_g)$|, but as
|$\bar{\ell }^*(w) = \ell ^*(w) - \tfrac{1}{\chi } (\pi ^Y)^*\kappa _{e, w} \in H^{n+1}(E^{\theta \times Y};{\mathbb{Q}})$| by definition, the vanishing is immediate.
3.4 Comparisons
There are natural maps
which each induce maps on
|$H^*(-; \mathcal{H}^{\otimes S})$|. There are corresponding maps of spaces of graphs
given as follows. The maps
|$c^*$| and
|$d^*$| are induced by the projections
|$\mathcal{W} \to \mathcal{V}$|. The maps
|$a^*$| and
|$e^*$| are induced by applying the augmentations
|$\mathcal{V} \to{\mathbb{Q}}$| and
|$\mathcal{W} \to{\mathbb{Q}}$| to the second tensor factor. The maps
|$b^*$| and
|$f^*$| are more subtle, as they involve converting between blue graphs and red graphs, via the formula of Proposition
2.7. Graphically, it is given by
with certain orderings.
The maps
|$b$| and
|$f$| are also oriented
|$W_g$|-bundles, so they also induce fibre-integration maps
|$b_!$| and
|$f_!$| on cohomology. These are
|$b^*$|- and
|$f^*$|-linear, respectively, so are determined by the maps (of degree
|$-2n$|)
which each send a monomial
|$c$| in
|$p_i$|’s and
|$e$| to the graph given by a single vertex labelled by
|$c$|.
4 Cohomology of Torelli Groups
The isomorphisms provided by Theorem
3.4 can be converted into information about the spaces
just as [
16, Theorem 4.1] is deduced from [
16, Theorem 3.15]. Let us give the definition of these spaces and formulate the result: the following is largely a reminder of some points from [
16], and we do not spell out all details again.
The group
|$\textrm{Diff}^{+}(W_g)$| acts on
|$H_n(W_g;{\mathbb{Z}})$| preserving the nondegenerate
|$(-1)^n$|-symmetric intersection form
|$\lambda : H_n(W_g;{\mathbb{Z}}) \otimes H_n(W_g;{\mathbb{Z}}) \to{\mathbb{Z}}$|. This provides a homomorphism
This map is not always surjective, but its image is a certain finite-index subgroup
|$G^{\prime}_g \leq G_g$|, an arithmetic group associated to the algebraic group
|$\textbf{Sp}_{2g}$| or
|$\textbf{O}_{g,g}$|. This subgroup has been determined by Kreck [
15]: it is the whole of
|$G_g$| if
|$n$| is even or
|$n=1,3,7$|, and otherwise is the subgroup
|$\textrm{Sp}_{2g}^q({\mathbb{Z}}) \leq \textrm{Sp}_{2g}({\mathbb{Z}})$| of those matrices that preserve the standard quadratic refinement (of Arf invariant 0).
We define
|$\textrm{Tor}^{+}(W_g)$| to be the kernel of this homomorphism, and
|$\textrm{Tor}^{+}(W_g, *)$| and
|$\textrm{Tor}(W_g, D^{2n})$| to be the kernel of its restriction to the subgroups
|$\textrm{Diff}^{+}(W_g, *)$| and
|$\textrm{Diff}(W_g, D^{2n})$|, respectively (these restrictions still have image
|$G^{\prime}_g$|). Furthermore, we define
where
|$\textrm{Bun}^+(TW_g, \theta ^*\gamma _{2n}) \subset \textrm{Bun}(TW_g, \theta ^*\gamma _{2n})$| consists of the orientation-preserving bundle maps (for some choice of orientation of
|$\theta ^*\gamma _{2n}$| that we make once and for all). By the discussion at the beginning of Section
3, the spaces
|$\textrm{Bun}^+(TW_g, \theta ^*\gamma _{2n})$| are path-connected, so each of the
|$B\textrm{Tor}$|’s we have defined are principal
|$G^{\prime}_g$|-bundles over the corresponding
|$B\textrm{Diff}$|’s. In particular, their rational cohomologies are both
|${\mathbb{Q}}$|-algebras and
|$G^{\prime}_g$|-representations, and we will describe them as such in a stable range. Before doing so, we recall that the work of Borel identifies
in a stable range of degrees, where
|$\sigma _{4i-2n}$| may be chosen so that it pulls back to the Miller–Morita–Mumford class
|$\kappa _{{\mathcal{L}}_i} \in H^{4i-2n}(B\textrm{Diff}^+(W_g;{\mathbb{Q}})$| associated to the
|$i$|th Hirzebruch
|$\mathcal{L}$|-class. In particular, the
|$\kappa _{{\mathcal{L}}_i}$| vanish in the cohomology of
|$B\textrm{Tor}^+(W_g)$|.
Let us write
|$H(g):= H^n(W_g ; {\mathbb{Q}})$|, which is the standard representation of
|$G^{\prime}_g$|. Pulled back from
|$B\textrm{Diff}^+(W_g)$| to
|$B\textrm{Tor}^+(W_g)$| the coefficient system
|${\mathcal{H}}$| is canonically trivialised, but has an action of
|$G^{\prime}_g$|: it can be identified with the dual
|$H(g)^\vee $|. The edge homomorphism of the Serre spectral sequence
allows us to consider the modified twisted Miller–Morita–Mumford classes
|$\bar{\kappa }_{\varepsilon ^S c}$| as providing
|$G^{\prime}_g$|-equivariant homomorphisms
The identities from the modified contraction formula correspond to identities among these maps: this will give relations analogous to [17, Section 5.2], which we will spell out after the proof of Theorem 4.1 below. First, we explain how these relations can be organised in a categorical way, as follows.
Considering (
4.1) as a natural transformation of functors on
|$\textsf{(s)Br}_{2g}$|, we may precompose it with the map
(which is an isomorphism in a stable range for
|$n \neq 2$| by Theorem
3.4). This gives
|$G^{\prime}_g$|-equivariant maps
|$H(g)^{\otimes S} \otimes \mathcal{G}\textrm{raph}_g(S) \to H^{*}(B\textrm{Tor}^+(W_g);{\mathbb{Q}})$|, which assemble to a map
out of the coend, where
|$K: \textsf{(s)Br}_{2g} \to \textsf{Rep}(G^{\prime}_g)$| sends
|$S$| to
|$H(g)^{\otimes S}$|. The domain obtains a graded-commutative
|${\mathbb{Q}}$|-algebra structure coming from the lax symmetric monoidality of
|$\mathcal{G}\textrm{raph}_g(-)$| and strong symmetric monoidality of
|$K(-)$|. Theorem
4.1 below will say that this is surjective in a stable range, with kernel the ideal generated by the
|$\kappa _{{\mathcal{L}}_i}$|, but before stating it we explain a simplification.
Let us write |$i: \textsf{d(s)Br} \to \textsf{(s)Br}_{2g}$| for the inclusion of the downward (signed) Brauer category. This subcategory is independent of |$g$|, as no circles can be created by composing morphisms in the downward Brauer category. Write |$\mathcal{G}\textrm{raph}_1(-)^{\prime} \subset i^*\mathcal{G}\textrm{raph}_1(-)_g$| for the subfunctor where we forbid bivalent vertices labelled by |$1 \in{\mathcal{V}}$| both of whose half-edges are legs; similarly, this functor is independent of |$g$|. Like just after [17, Proposition 3.11], |$\mathcal{G}\textrm{raph}_1(-)_g$| is then the left Kan extension |$i_* \mathcal{G}\textrm{raph}_1(-)^{\prime}$| of |$\mathcal{G}\textrm{raph}_1(-)^{\prime}$| along |$i$|. We similarly define |$\mathcal{G}\textrm{raph}_*^\theta (-)^{\prime}$|, |$\mathcal{G}\textrm{raph}_*(-)^{\prime}$|, |$\mathcal{G}\textrm{raph}^\theta (-)^{\prime}$|, and |$\mathcal{G}\textrm{raph}(-)^{\prime}$|, whose left Kan extensions again recover the original functors. The following is the analogue of [17, Theorem 4.1].
Theorem 4.1.There are
|$G^{\prime}_g$|-equivariant ring homomorphisms
which for
|$2n \geq 6$| are isomorphisms in a stable range of degrees.
If |$2n=2$| then, in a stable range of degrees and assuming that the target is finite-dimensional in degrees |$* < N$| for all large enough |$g$|, these maps are isomorphisms onto the maximal algebraic subrepresentations in degrees |$* \leq N$|, and monomorphisms in degrees |$* \leq N+1$|.
Proof.By the main theorem of [
16], as long as
|$2n \geq 6$|, the
|$G^{\prime}_g$|-representations
|$H^i(B\textrm{Tor}(W_g, D^{2n}); {\mathbb{Q}})$| are algebraic. Using the inheritance properties for algebraic representations from [
16, Theorem 2.2], the Serre spectral sequences for the homotopy fibre sequences
show that the cohomology groups of
|$B\textrm{Tor}^{+}(W_g, *)$| and
|$B\textrm{Tor}^{\theta }(W_g, *)$| are also algebraic
|$G_g^{\prime}$|-representations, and the same for the homotopy fibre sequences
show that the cohomology groups of
|$B\textrm{Tor}^{+}(W_g)$| and
|$B\textrm{Tor}^{\theta }(W_g)$| are algebraic
|$G_g^{\prime}$|-representations too.
Using this algebraicity property, case (i) is precisely [17, Theorem 4.1], using that by [17, Proof of Theorem 5.1] |$\mathcal{G}\textrm{raph}_1(-)_g$| is isomorphic to the functor |$\mathcal{P}(-, {\mathcal{V}})_{\geq 0} \otimes \det ^{\otimes n}$|. The other cases follow in the same way, using [17,Proposition 2.16], from Theorem 3.4, with one elaboration that we describe below. The addendum in the case |$2n=2$| is precisely as in [17, Theorem 4.1].
The elaboration comes when verifying the first hypothesis of [
17, Lemma 4.3], which in case (v) for example requires us to know that
|$H^*(B\textrm{Diff}^+(W_g) ; \mathcal{H}^{\otimes S})$| is a free
|$H^*(G^{\prime}_g;{\mathbb{Q}})$|-module in a stable range. But by transfer
|$H^*(B\textrm{Diff}^+(W_g) ; \mathcal{H}^{\otimes S})$| is a summand of
|$H^*(B\textrm{Diff}^+(W_g, *) ; \mathcal{H}^{\otimes S})$| (as
|$H^*(G^{\prime}_g;{\mathbb{Q}})$|-modules), and similarly with
|$\theta $|-structures, so cases (ii) and (iii) imply cases (iv) and (v). On the other hand, in case (iii) for example, we have discussed in the proof of Theorem
3.4 the degeneration of the Serre spectral sequence in a stable range, giving
The Serre filtration is one of
|$H^*(G^{\prime}_g;{\mathbb{Q}})$|-modules, so as the associated graded is a free
|$H^*(G^{\prime}_g;{\mathbb{Q}})$|-module in a stable range (because
|$H^*(B\textrm{Diff}(W_g, D^{2n}) ; \mathcal{H}^{\otimes S})$| is the case treated in [
17, Theorem 4.1]), it follows that
|$H^*(B\textrm{Diff}^+(W_g, *) ; \mathcal{H}^{\otimes S})$| is too. The same argument applies in case (ii).
This quite categorical description can be used to get a more down-to-earth presentation for these cohomology rings: in case (v), this is the presentation we have recorded in Theorem A. This is deduced just as in [17, Section 5], though a certain amount of the work can be avoided as we have already expressed things in terms of graphs rather than partitions. We outline the argument.
Let
|$R_{\textrm{pres}}$| denote the
|${\mathbb{Q}}$|-algebra whose presentation is given in Theorem
A but
omittting the relations (v) |$\bar{\kappa }_{{\mathcal{L}}_i}=0$|. By the universal property of coends, for any finite set
|$S$|, there is a map
|$\bar{\kappa }: H(g)^{\otimes S} \otimes \mathcal{G}\textrm{raph}(S)^{\prime} \to i^*(K^\vee ) \otimes ^{\textsf{d(s)Br}} \mathcal{G}\textrm{raph}(-)^{\prime}$|, and so in particular there are maps
corresponding to the graph given by the corolla with legs
|$\{1,2,\ldots ,r\}$| and label
|$c \in{\mathcal{W}}$|. The identifications made in a coend mean that if
|$\omega = \sum _{i=1}^{2g} a_i \otimes a_i^\# \in H(g)^{\otimes 2}$| is the form dual to the intersection pairing then
|$\sum _i \bar{\kappa }_c(v_1 \otimes \cdots \otimes v_{r-2} \otimes a_i \otimes a_i^\#)$| is identified with
Using the modified contraction formula (relation (d
|$^{\prime \prime \prime }$|)), and permuting the ordering of some half-edges, we have the identity
in
|$\mathcal{G}\textrm{raph}(\{1,2,\ldots ,r-2\})^{\prime}$|, showing that
|$\sum _i \bar{\kappa }_c(v_1 \otimes \cdots \otimes v_{r-2} \otimes a_i \otimes a_i^\#) = \tfrac{\chi -2}{\chi } \bar{\kappa }_{ce}(v_1 \otimes \cdots \otimes v_{r-2}) + \tfrac{1}{\chi ^2} \bar{\kappa }_{e^2} \bar{\kappa }_c(v_1 \otimes \cdots \otimes v_{r-2})$|, that is, that relation (iv) in the presentation of
|$R_{\textrm{pres}}$| holds in the algebra
|$i^*(K^\vee ) \otimes ^{\textsf{d(s)Br}} \mathcal{G}\textrm{raph}(-)^{\prime}$|. Similarly, one shows that relations (i)–(iii) in the presentation of
|$R_{\textrm{pres}}$| hold, so that there is defined an algebra map
This is in addition a map of algebraic
|$G^{\prime}_g$|-representations, so to show it is an isomorphism in a stable range it suffices to show that it induces an isomorphism on applying
|$[- \otimes H(g)^{\otimes S}]^{G^{\prime}_g}$| in a stable range, for all finite sets
|$S$|. This is done, as in the proof of [
17, Theorem 5.1], by constructing a factorisation
of the canonical map, which is an isomorphism in a stable range. The map
|$\alpha $| is surjective by construction, so is an isomorphism in a stable range, and hence
|$\phi _*$| is too. Thus,
|$\phi $| is an isomorphism in a stable range: quotienting out both sides by the ideal generated by the
|$\bar{\kappa }_{{\mathcal{L}}_i}$| gives Theorem
A. This finishes the outline of the argument.
5 The Case |$2n=2$|
Although Theorem 4.1 is only known to hold in a limited range of degrees in the case |$2n=2$| (|$N=2$| is currently the best known constant for |$g \geq 3$|, using the work of Johnson [12]), Theorem 3.4 does hold in a range of cohomological degrees tending to infinity with |$g$|. In this case, our discussion is closely related to the work of Kawazumi and Morita [13, 14, 19], and in this section, we take the opportunity to revisit that work from our perspective. Throughout this section, we assume that |$g \geq 2$|, so that |$\chi (W_g) = 2-2g \neq 0$|.
In terms of Kawazumi and Morita’s notation, we have
Under our assumption
|$g \geq 2$|, the groups
|$\textrm{Diff}^+(W_g)$|,
|$\textrm{Diff}^+(W_g, *)$|, and
|$\textrm{Diff}^+(W_g, D^2)$| all have contractible path-components, so the group cohomology of
|$\mathcal{M}_g$| is the cohomology of
|$B\textrm{Diff}^+(W_g)$|, and so on. Theorem
3.4 gives a natural transformation
of functors
|$\textsf{sBr}_{2g} \to \textsf{Gr}({\mathbb{Q}}\textrm{-}\textsf{mod})$|, which is an isomorphism in a stable range of degrees. Note that in this case
|$H^*(B\textrm{SO}(2);{\mathbb{Q}})={\mathbb{Q}}[e]$| so
|$\mathcal{V} = \mathcal{W} = {\mathbb{Q}}[e]$|, and there is no difference between the tangential structure
|$\theta $| and an orientation. In particular, if we denote by
|$\Gamma _i \in \mathcal{G}\textrm{raph}(\emptyset )$| the graph with a single vertex, no edges, and labelled by
|$e^{i+1}$|, then
|$\bar{\kappa }(\Gamma _i) = \kappa _i \in H^{2i}(\mathcal{M}_g;{\mathbb{Q}})$| is the usual Miller–Morita–Mumford class (our
|$\kappa _i$| is denoted
|$e_i$| in the work of Kawazumi and Morita).
Our goal in this section is to relate our |$\mathcal{G}\textrm{raph}(-)$| to the work of Kawazumi and Morita mentioned above as well as work of Akazawa [1] and Garoufalidis and Nakamura [9, 10]. A brief overview is as follows. The modified contraction formula can be used to simplify elements of |$\mathcal{G}\textrm{raph}(S)$| in two different way. Firstly, in Section 5.1, the modified contraction formula is used to express a graph as a linear combination of graphs with fewer edges, and hence by induction to express it as a linear combination of graphs with no edges. In particular, this shows that |$\mathcal{G}\textrm{raph}(\emptyset )_g \cong{\mathbb{Q}}[\Gamma _1, \Gamma _2, \ldots ]$| for the graphs |$\Gamma _i$| described above. Secondly, in Section 5.2, the modified contraction formula is used “in reverse” to express a graph as a linear combination of graphs with smaller powers of |$e$| in a label, or with smaller valence, and hence by induction to express it as a linear combination of trivalent graphs with all labels 1. Reduced to this setting of trivalent graphs, in Section 5.3, it is explained how the decorations by ordering and orientation data can be neglected, due to the existence of a normal form for it. This allows for all elements of |$\mathcal{G}\textrm{raph}(S)$| to be expressed in terms of undecorated trivalent graphs, and Section 5.4 describes the complete set of relations among undecorated trivalent graphs that need to be imposed in order to present |$\mathcal{G}\textrm{raph}(S)$|: we rediscover a modified form of the “|$I = H$|” relation among trivalent graphs, which has arisen in the work of Akazawa [1] and Garoufalidis and Nakamura [10]. Finally, in Section 5.5, we explain how this discussion completes a calculation in symplectic invariant theory, which has been outstanding since [10].
5.1 Reduction to corollas
The possible labels for the vertices of graphs in |$\mathcal{G}\textrm{raph}(S)$| are powers of the Euler class |$e$|. Given any graph, we may iteratedly apply the modified contraction formula to write it as a linear combination of graphs with fewer edges, and hence any graph is equivalent to a linear combination of graphs with no edges: these are disjoint unions of corollas. Of these, by definition of |$\mathcal{G}\textrm{raph}$|: the 0-valent corolla labelled by |$e$| is equal to the scalar |$\chi $|, the 1-valent corolla labelled by |$1 \in{\mathcal{V}}$| is trivial, and the 1-valent corolla labelled by |$e \in{\mathcal{V}}$| is trivial. Define a labelled partition of a finite set |$S$| to be a partition |$\{S_\alpha \}_{\alpha \in I}$| of |$S$| into (possibly empty) subsets and a label |$e^{n_\alpha }$| for each part, such that
(i) If |$|S_\alpha |=0$| then |$n_\alpha \geq 2$|,
(ii) If |$|S_\alpha |=1$| then |$n_\alpha \geq 1$|.
We give a part
|$(S_\alpha , n_\alpha )$| degree
|$2n_\alpha + |S_\alpha |-2$|, and a labelled partition the degree given by the sums of the degrees of its parts. Similarly to the proof of Theorem
3.4 (iv) (particularly around equation (
3.5)), let
|$\mathcal{P}^{\textrm{bis}}(S, {\mathcal{V}})_{\geq 0}$| denote the free
|${\mathbb{Q}}[\chi ^{\pm 1}]$|-module with basis the set of labelled partitions of
|$S$|. Assigning to a labelled part
|$(S_\alpha , e^{n_\alpha })$| the corolla with legs
|$S_\alpha $| and label
|$e^{n_\alpha }$| defines a map
natural in
|$S$| with respect to bijections.
Lemma 5.1.The map (5.1) is an isomorphism.
Proof.It is surjective, as explained above, by repeatedly applying the modified contraction formula to express a graph in terms of graphs without edges.
If it were not injective, then it would have some nontrivial |${\mathbb{Q}}[\chi ^{\pm 1}]$|-linear combination of labelled partitions in its kernel, of a given degree |$d$|, and this would remain a nontrivial |${\mathbb{Q}}$|-linear combination of labelled partitions when specialised to |$\chi = 2-2g$| for all |$g \gg 0$| (as a Laurent polynomial in |$\chi $| has finitely-many roots). But in the proof of Theorem 3.4 (iv), in the discussion after equation (3.5), it is explained that when specialised to |$\chi = 2-2g$| this map is an isomorphism in a range of degrees tending to infinity with |$g$|; for large enough |$g$| the degree |$d$| will be in this stable range, a contradiction.
In particular, using the graphs
|$\Gamma _i$| described above, there is an isomorphism
5.2 Reduction to trivalent graphs without labels
In this section, we will prove the following.
Theorem 5.2.Using the modified contraction formula, any marked oriented graph is equivalent to a |${\mathbb{Q}}[\chi ^{\pm 1}, (\chi -2)^{-1}, (\chi -3)^{-1}, (\chi -4)^{-1}]$|-linear combination of trivalent graphs with all vertices labelled by |$1 \in \mathcal{V}_0$|.
Let |$\mathcal{G}\textrm{raph}^{\textrm{tri}}(S) \leq \mathcal{G}\textrm{raph}(S)$| denote the sub-|${\mathbb{Q}}[\chi ^{\pm 1}]$|-module spanned by those marked oriented graphs that are trivalent and all of whose labels are |$1\in \mathcal{V}$|.
Corollary 5.3.The monomorphism |$i: \mathcal{G}\textrm{raph}^{\textrm{tri}}(-) \to \mathcal{G}\textrm{raph}(-)$| becomes an isomorphism upon inverting |$\chi -2$|, |$\chi -3$|, and |$\chi -4$|. In particular, |$\mathcal{G}\textrm{raph}^{\textrm{tri}}(-)_g = \mathcal{G}\textrm{raph}(-)_g$|.
Remark 5.4(2-valent vertices labelled by 1).
Using the relation
we can always remove 2-valent vertices labelled by 1. It is sometimes convenient when writing formulas for 3-valent graphs to also allow 2-valent vertices labelled by 1: we allow ourselves to do so, noting that the above can always be used to eliminate the 2-valent vertices.
Proof of Theorem 5.2.As a matter of notation, we will formally manipulate modified twisted Miller–Morita–Mumford classes, but this is equivalent to manipulating marked oriented graphs. Rearranging the first contraction formula gives
Rearranging the second contraction formula gives
and using (
5.3) to eliminate the Euler classes from the last two terms gives
It suffices to show that each corolla |$\kappa _{\bar{\varepsilon }^a e^b}$| may be represented by a linear combination of trivalent graphs. By Example 2.8, the class |$\kappa _{e^2}$| may be represented by a trivalent graph (after inverting |$\chi -3$|) so by iteratedly applying (5.3), it suffices to show that each |$\kappa _{\bar{\varepsilon }^n}$| can too. By Remark 5.4, we may as well show that classes can be represented by 2- and 3-valent graphs. To get started, we have |$\kappa _{\bar{\varepsilon }}=0$| as it has negative degree.
Consider the class
|$\lambda _{2,5} \lambda _{3,4}(\kappa _{\bar{\varepsilon }^{1,2,3}} \cdot \kappa _{\bar{\varepsilon }^{4,5,6}})$|. Using the form of the relations above, which avoid creating Euler classes, this is
Renumbering legs and rearranging, this shows that
|$\lambda _{1,2}(\kappa _{\bar{\varepsilon }^4})$| may be represented by 2- and 3-valent graphs.
Applied with
|$(a,b)=(2,2)$|, the second relation gives
which with the above shows that
|$\kappa _{\bar{\varepsilon }^{4}}$| may be represented by 2- and 3-valent graphs.
Similarly to the above, consider
|$\lambda _{2,5} \lambda _{3,4}(\kappa _{\bar{\varepsilon }^{1,2,3}} \cdot \kappa _{\bar{\varepsilon }^{4,5,6,7}})$|, which is
Renumbering legs and rearranging, this shows that
|$\lambda _{1,2}(\kappa _{\bar{\varepsilon }^5})$| may be represented by 2-, 3-, and 4-valent graphs; with the above it follows that it can also be represented by 2- and 3-valent graphs.
Applied with
|$(a,b)=(2,3)$|, the second relation gives
so it follows that
|$\kappa _{\bar{\varepsilon }^{5}}$| may be represented by 2- and 3-valent graphs.
If |$n \geq 6$|, then we can write |$n=a+b$| with |$a, b \geq 3$|, so |$a+2, b+2 < n$| and so the second relation expresses |$\kappa _{\bar{\varepsilon }^{n}}$| in terms of |$\kappa _{\bar{\varepsilon }^{m}}$|’s with |$m < n$|. Thus, all |$\kappa _{\bar{\varepsilon }^{n}}$|’s may be represented by 2- and 3-valent graphs as required.
It is worth observing that we have the relation
using that
|$\kappa _{\bar{\varepsilon } e}=0$| (by definition) and that
|$\kappa _{\bar{\varepsilon }}=0$| (as it has negative degree). This means that any graph having a trivalent vertex with a loop is trivial in
|$\mathcal{G}\textrm{raph}(-)$|.
5.3 A remark on orderings
A curious normalisation is possible when considering trivalent graphs, allowing one to neglect the orderings of vertices, of half-edges, and the orientations of edges. In [13, 14, 19], this is implemented ab initio and (marked) oriented graphs play no role. Let us explain this normalisation, extended to trivalent graphs with legs.
A trivalent graph |$\tilde{\Gamma }$| with legs |$S$| consists of a set |$V$| of vertices, a set |$H$| of half-edges, a 3-to-1 map |$a: H \to V$| recording to which vertex each half-edge is incident, and an unordered matching |$\mu $| on |$H \sqcup S$| recording which half-edges span an edge, and which half-edges are connected to which legs.
Given a trivalent graph
|$\tilde{\Gamma } = (V, H, a: H \to V, \mu )$| with legs
|$S$|, we may choose an ordering of
|$V$| and choose an ordering of
|$H$| such that
|$a: H \to V$| is weakly monotone (equivalently, choose an ordering of the half-edges incident at each vertex). We also choose an ordering of
|$S$|. There is an induced ordering of
|$H \sqcup S$| by putting
|$\vec{S}$| after
|$\vec{H}$|, and we form an ordered matching
|$m$| of
|$H \sqcup S$| by taking those pairs
|$(a,b)$| with
|$a < b$| and
|$\{a,b\} \in \mu $|. Using this, we form an oriented trivalent graph
|$\Gamma _{\textrm{choice}} = (\vec{V}, \vec{H}, a: H \to V, m)$|, depending on these choices. The normalisation is as follows. Let
|$x_1 < x_2 < x_3 < x_4 < \ldots < x_{2k} \in H \sqcup S$| be the total order on
|$H \sqcup S$|, and let
|$a_1 < b_1, \ldots , a_k < b_k$| be the ordered pairs, which span an edge, with
|$a_1 < a_2 < \ldots < a_k \in H \sqcup S$|. Then there is a bijection given by
and we define
|$\Gamma := \textrm{sign}(\rho ) \cdot \Gamma _{\textrm{choice}}$|.
Claim. As long as |$\tilde{\Gamma }$| has no vertices with loops, the element |$\Gamma $| does not depend on the choice of ordering of |$V$| or |$H$|, and depends on the ordering of |$S$| precisely as the sign representation.
In [
13,
14,
19], Kawazumi and Morita restrict to “trivalent graphs without loops”, but we find it more natural to allow loops but to set graphs with a loop to zero. Thus we define
so that the claim, together with the relation (
5.4), provides an epimorphism
of
|${\mathbb{Q}}[\chi ^{\pm 1}]$|-modules, natural with respect to bijections in
|$S$|. This can be extended to a natural transformation of functors on
|$\textsf{sBr}^\chi $| by letting an ordered matching
|$(a,b)$| of elements of
|$S$| act by adding an edge to the trivalent graph connecting
|$a$| and
|$b$|, and contracting the determinant by
|$a \wedge b$|. Doing so might create a circle with no vertices, which should be replaced by the scalar
|$\chi -2$|.
Proof of Claim.If |$(h_1, h_2, h_3)$| are the half-edges incident at a vertex |$v$| and we change their ordering to |$(h_{\sigma (1)}, h_{\sigma (2)}, h_{\sigma (3)})$| giving |$\Gamma _{\textrm{choice}}^{\prime}$|, then (under the assumption that |$\Gamma $| does not have loops) the relative ordering of half-edges forming an edge has not changed, so |$m^{\prime}=m$|. Thus |$\Gamma _{\textrm{choice}}^{\prime} = \textrm{sign}(\sigma ) \cdot \Gamma _{\textrm{choice}}$|. On the other hand, |$\rho ^{\prime}$| is obtained from |$\rho $| by postcomposing with |$\sigma $|, and precomposing with a permutation that permutes some |$(a_i < b_i)$|’s, which is an even permutation. Thus, |$\textrm{sign}(\rho ^{\prime}) = \textrm{sign}(\sigma ) \cdot \textrm{sign}(\rho )$|, so |$\Gamma ^{\prime} = \Gamma $|.
Suppose a vertex |$v^1$| has half edges |$(h_1^1, h_2^1, h_3^1)$| and |$v^2$| has half edges |$(h_1^2, h_2^2, h_3^2)$|, and |$v^1 < v^2 \in \vec{V}$| are adjacent in the ordering on |$V$|, and consider transposing the ordering of these vertices. For edges between a |$u<v^1$| and a |$v^i$| or between a |$v^i$| and a |$u> v^2$|, the relative ordering of their half-edges does not change. Edges between |$v^1$| and |$v^2$| have the relative ordering of their half-edges reversed. Thus, if there are |$N$| such edges, we have |$\Gamma _{\textrm{choice}}^{\prime} = (-1)^{1+N} \cdot \Gamma _{\textrm{choice}}$|. But the permutation |$\rho $| is changed by permuting |$(h_1^2, h_2^2, h_3^2)$| past |$(h_1^1, h_2^1, h_3^1)$|, which has sign |$-1$|, and |$N$| transpositions |$(a_i b_i)$|, which has sign |$(-1)^N$|. Thus again |$\Gamma ^{\prime} = \Gamma $|.
Finally, changing the order of |$S$| by a permutation |$\tau $| changes |$\rho $| by postcomposition with |$\tau $|, so acts as |$\textrm{sign}(\tau )$|.
Example 5.5.For the ordering of vertices and half-edges corresponding to the theta-graph in Example 2.8, the associated permutation is |$\rho =(1)(2 3 5)(4 6)$|, which is odd, so the undecorated theta-graph yields |$\tfrac{\chi -3}{\chi }\kappa _{e^2}$|. This is precisely minus the evaluation of |$\beta _{\Gamma _2}$| on [14, page 39] (unfortunately the theta-graph is denoted |$\Gamma _2$| in that paper). This minus comes from the use of a different sign convention, see the discussion at [17, top of page 33].
5.4 Relations among trivalent graphs
The modified contraction formula describes relations among graphs involving contracting an edge, but this necessarily involves graphs with vertices of different valencies. In Theorem 5.2, we have explained that, in the case of surfaces, all graphs may be expressed purely in terms of trivalent graphs: one may ask what relations among trivalent graphs |$\Gamma $| are imposed by the contraction formula.
For the unmodified contraction formula discussed in [17], the answer is that it imposes the “|$I=H$|” relation among trivalent graphs: this is because both the |$I$|- and |$H$|-graphs admit contractions to the |$X$|-graph. Furthermore, as all connected trivalent graphs with the same number of legs and of the same genus are equivalent under the “|$I=H$|” relation, and the contraction formula never changes the genus or number of legs, there are no further relations.
In the setting of the modified contraction formula discussed here, it is more complicated. It is best given in the setting of undecorated trivalent graphs.
Theorem 5.6.After inverting
|$\chi -2$|,
|$\chi -3$|, and
|$\chi -4$|, undecorated trivalent graphs that differ locally by
give the same elements in
|$\mathcal{G}\textrm{raph}^{\textrm{tri}}[(\chi -2)^{-1}, (\chi -3)^{-1}, (\chi -4)^{-1}]$|.
Proof.We establish this relation in |$\mathcal{G}\textrm{raph}^{\textrm{tri}}(\{a,b,c,d\}) \otimes \det{\mathbb{Q}}^{\{a,b,c,d\}}$|, and it then follows in general using functoriality on the signed Brauer category. We order the legs as |$a < b < c < d$|.
Consider first the
|$H$|-shaped graph shown in Figure
5 (i), with the depicted names of half edges, ordered as
|$3<1<5<6<2<4$|. Its corresponding permutation is
$$\bigl (\begin{smallmatrix} 3 & c & 1 & a & 5 & 6 & 2 & b & 4 & d \\ 3 & 1 & 5 & 6 & 2 & 4 & a & b &c & d \end{smallmatrix}\bigr )$$
, which is even. Thus, this ordering data represents the underlying undecorated
|$H$|-shaped trivalent graph. Ignoring for now the matchings to the legs (which are given by matching 1 with
|$a$|, 2 with
|$b$|, and so on), it corresponds to
|$\lambda _{5,6}(\kappa _{\bar{\varepsilon }^{3, 1, 5}} \cdot \kappa _{\bar{\varepsilon }^{6,2,4}})$|. Using the form of the relations that avoid creating Euler classes from the proof of Theorem
5.2, we have
Consider now the
|$I$|-shaped graph shown in Figure
5 (ii), with the depicted names of the half-edges, ordered as
|$4 < 3 < 5 < 6< 1 < 2$|. Its corresponding permutation is
$$\bigl (\begin{smallmatrix} 4 & d & 3 & c & 5 & 6 & 1 & a & 2 & b \\ 4 & 3 & 5 & 6 & 1 & 2 & a & b &c & d \end{smallmatrix}\bigr )$$
, which is odd. Thus, this ordering data represents
minus the underlying undecorated
|$I$|-shaped trivalent graph. Ignoring again the matchings to the legs, it corresponds to
The sum of these two expressions therefore represents the image under |$\Phi $| of the difference |$H-I$| of the underlying undecorated trivalent graphs. Furthermore, |$\kappa _{\bar{\varepsilon }^{4,3,1,2}} = - \kappa _{\bar{\varepsilon }^{3,1,2,4}}$| so these terms cancel.
From the proof of Theorem
5.2, we have the identity
expressing terms of the form
|$\lambda _{u,v}(\kappa _{\bar{\varepsilon }^{u,v,s,t}})$| in terms of (2- and) 3-valent vertices. Applying it to the sum of the two expressions above, and collecting terms, therefore gives
Using that
|$\kappa _{e^2} =\Phi (\tfrac{\chi }{\chi -3}\Theta )$| and carefully putting the graphs corresponding to the other terms into the normal form of Section
5.3 gives the identity in the statement of the theorem.
Our relation |$IH^{\textrm{mod}}$| is graphically identical to the relation called |$IH_0^{\textrm{bis}}$| by Akazawa [1, page 100] and in the corrigendum [10] to the paper of Garoufalidis and Nakamura [9]. In those papers, it is emphasised that |$IH_0^{\textrm{bis}}$| means this identity is imposed only when the 4 half-edges belong to distinct edges, but in fact this is redundant: if the 4 half-edges do not belong to distinct edges, then the identity already holds in |$\mathcal{G}\textrm{raph}^{\textrm{undec}}$|. So in fact imposing our relation |$IH^{\textrm{mod}}$| is identical to imposing their relation |$IH_0^{\textrm{bis}}$|.
Theorem 5.7.Upon inverting
|$\chi -2$|,
|$\chi -3$|, and
|$\chi -4$|, the maps
are isomorphisms.
Proof.Let |$R:={\mathbb{Q}}[\chi ^{\pm 1}, (\chi -2)^{-1}, (\chi -3)^{-1}, (\chi -4)^{-1}]$| and implicitly base change to this ring. We have already shown in Corollary 5.3 that the second map is an isomorphism, and |$\Phi $| is certainly an epimorphism, so it remains to show that the composition is a monomorphism.
For an undecorated trivalent graph
|$\Gamma $|, define a
double edge to be an unordered pair of vertices that share precisely two edges, and a
triple edge to be an unordered pair of vertices that share precisely three edges, that is, form a theta-graph. Define
filter
|$\mathcal{G}\textrm{raph}^{\textrm{undec}}$| by letting
|$F^k \mathcal{G}\textrm{raph}^{\textrm{undec}}$| be spanned by those
|$\Gamma $| with
|$\mu (\Gamma ) \geq k$|, and give
|$\mathcal{G}\textrm{raph}^{\textrm{undec}/(IH^{\textrm{mod}}})$| the induced filtration.
If |$\Gamma = \Gamma _H$| is a graph with |$\mu (\Gamma )=k$| and a distinguished “|$H$|” subgraph, and |$\Gamma _I$| is obtained by replacing this “|$H$|”-subgraph by “|$I$|”, then by applying the relation |$IH^{\textrm{mod}}$| to this subgraph we find that
(i) if the edge involved is not part of a double or triple edge then the relation gives |$\Gamma _H - \Gamma _I \in F^{k+1} \mathcal{G}\textrm{raph}^{\textrm{undec}}/(IH^{\textrm{mod}})$|,
(ii) if the edge involved is part of a double or triple edge then the relation is trivial (i.e., already holds in |$\mathcal{G}\textrm{raph}^{\textrm{undec}}$|).
Thus the associated graded of the induced filtration on
|$\mathcal{G}\textrm{raph}^{\textrm{undec}}/(IH^{\textrm{mod}})$| can be described as
|$\mathcal{G}\textrm{raph}^{\textrm{undec}}/(IH_0)$|, where as in [
9] the relation
|$IH_0$| means imposing the “
|$I=H$|” relation when the four half-edges belong to different edges. Now
|$IH_0$| is an equivalence relation on the set of isomorphism classes of trivalent graphs without loops, and similarly to [
9, Proof of Proposition 2.3 (c)] it is easy to see that all connected trivalent graph without loops of the same rank and with the same legs are equivalent to each other: in other words, the equivalences classes of such are given by partitions of
|$S$| (the parts are the legs of each connected component) labelled by a power of
|$e$| (recording the rank of the graph). It follows that the rank of
|$\mathcal{G}\textrm{raph}^{\textrm{undec}}/(IH^{\textrm{mod}})$| in each degree, as an
|$R$|-module, is at most that of
|$\mathcal{G}\textrm{raph}(\emptyset )$| as determined in Lemma
5.1, and so the composition in the statement of the theorem, which is an epimorphism, must be an isomorphism.
5.5 On the work of Garoufalidis and Nakamura
The discussion of the last few sections can be used to complete the work of Garoufalidis and Nakamura [
9,
10], concerning the calculation of the invariants
|$[\Lambda ^* V_{1^3} / (V_{2^2})]^{\textrm{Sp}}$| in a stable range. Here we write
|$V_\lambda $| for the irreducible
|$\textrm{Sp}$|-representation corresponding to the partition
|$\lambda $|, which was written as
|$[\lambda ]_{\mathfrak{s}\mathfrak{p}}$| in those papers, and
|$V_{2^2}$| denotes the unique copy of this irreducible in
|$\Lambda ^2 V_{1^3}$|. Combining Theorem 1.1 and Proposition 2.3 (c) of [
9] was supposed to calculate
|$[\Lambda ^* V_{1^3} / (V_{2^2})]^{\textrm{Sp}}$| in a stable range, but for the corrected version of Theorem 1.1 in [
10], which expresses these invariants as
|$\mathcal{G}\textrm{raph}^{\textrm{undec}}(\emptyset )_g/(IH_0^{bis})$|, the authors say “it turns out that a simple stable structure of [these invariants] as in Proposition 2.3 (c) will not be easy to detect”. However, Theorem
5.7 and equation (
5.2) gives that
in a stable range. Thus, in fact, Proposition 2.3 (c) of [
9] is correct as stated.
Remark 5.8.This can also be obtained from the work of Felder, Naef, and Willwacher [3]. Specifically, the graded-commutative algebra |$A_{(g)}$| defined just before Theorem 6 of that paper is |$\Lambda ^* V_{1^3} / (V_{2^2})$|, and Theorem 6 together with Proposition 36 (3) also gives the above.
Funding
This work was supported by the ERC under the European Union’s Horizon 2020 research and innovation programme [756444 to O.R.W.]; and by a Philip Leverhulme Prize from the Leverhulme Trust [to O.R.W.].
Acknowledgments
I am grateful to Alexander Kupers for feedback on an earlier draft.
Communicated by Prof. Arthur Bartels
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