On the cohomology of Torelli groups. II (Q6623127)

The family of the closed orientable surfaces \(W_g := \#^gS^1 \times S^1\) naturally generalizes to the family of manifolds \(W_g := \#^gS^n \times S^n\) for \(n\) a positive integer. The present work studies the cohomology of two kinds of subgroups of the topological group \(\mathrm{Diff}^+(W_g)\) of orientation-preserving diffeomorphisms, and various subgroups of it. The first kind of subgroups are \(\mathrm{Diff}(W_g, D^{2n}) \leq \mathrm{Diff}^+(W_g, * ) \leq \mathrm{Diff}^+(Wg)\), the diffeomorphisms that fix a disk and a point, respectively. The second kind are their \textit{Torelli subgroups} \N\[\N\mathrm{Tor}(W_g, D^{2n}), \mathrm{Tor}^+(W_g, *), \mathrm{Tor}^+(W_g.)\N\]\NAs explicitly mentioned by the author, ``this paper can be considered as a (somewhat extensive) addendum'' to the work \textit{A. Kupers} and \textit{O. Randal-Williams} [Forum Math. Pi 8, Paper No. e7, 83 p. (2020; Zbl 1445.55011)]. So the reader should have some familiarity with that paper. The intersection form on the middle cohomology group \(H_n(W_g;\mathbb{Z})\) provides a homomorphims \N\[\N\alpha_g:\mathrm{Diff}^+(W_g) \to G_g\N\]\Nhaving image denoted by \(G'_g\). Let us write \(H(g) := H^n(W_g;\mathbb{Q})\), on which \(G'_g\) operates in the evident way. It is constructed certain ``modified twisted Miller-Morita-Mumford classes'', which when restricted to the Torelli group yield \(G'_g\)-equivariant maps \N\[\N\bar {\kappa}_c:H(g)^{\otimes r} \to H^{n(r-2)+|c|}(B\mathrm{Tor}^{+}(W_g);\mathbb{Q})\N\]\Nfor each \(c \in \mathbb{Q}[e,p_1,p_2,\cdots,p_{n-1}] = H^*(BSO(2n);\mathbb{Q})\) and each \(r \geq 0\). Finally, we write \(\chi := \chi (W_g) = 2 + (-1)^n2g\) and will always suppose that this is not zero (i.e., that \((n, g)\ne (odd, 1)\)). The main result, which provides information about the ring structure of the cohomology of the classifying space of \(B\mathrm{Tor}^{+}(W_g)\), is as follows.\N\NTheorem A. If \(2n \geq 6\) then, in a range of degrees tending to infinity with \(g\), \(H^{*}(B\mathrm{Tor}^{+}(W_g);\mathbb{Q})\) 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 \(e,p_1,\cdots,p_{n-1}\), and \(r \geq 0\), such that \(n(r-2) + |c| > 0\). A complete set of relations in this range is given by\N\N(i) \(\bar {\kappa}_c(v_{\sigma(1)}\otimes \cdots\otimes v_{\sigma(r)}) =\mathrm{sign}(\sigma)^n \cdot \bar {\kappa}_c(v_1\otimes \cdots\otimes v_r)\),\N\N(ii) \(\bar {\kappa}_e(v_1)=0\),\N\N(iii) \( \sum_{i} \bar {\kappa}_{X}(v \otimes a_i) \cdot \bar {\kappa}_Y(a_i^{*} \otimes w) = \bar {\kappa}_{X\cdot Y}(v \otimes w) + \frac{1}{\chi^2} \bar {\kappa}_{e^2}\cdot \bar {\kappa}_{X}(v)\cdot \bar {\kappa}_{Y}(w) - \frac{1}{\chi}( \bar {\kappa}_{e\cdot X}(v)\cdot \bar {\kappa}_{Y}(w) + \bar {\kappa}_{X}(v)\cdot \bar {\kappa}_{e\cdot Y}(w))\),\N\N(iv) \(\sum_i \bar {\kappa}^{X}(v\otimes a_i\otimes a_i^{\#})= \frac{\chi-2}{\chi}\bar {\kappa}_{e\cdot X}(v) +\frac{1}{\chi^2}\bar {\kappa}_{e^2}\cdot \bar {\kappa}_{X}(v),\)\N\N(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 \(w \in H(g)^{\otimes s}.\)\N\NLet \(\theta : BSO(2n)\langle n \rangle \to BO(2n)\) be the \(n\)-connected cover of \(BO(2n)\). First the author provides a long and detailed description is of the twisted cohomology groups \(H^{*}(B\mathrm{Diff}^+(W_g); \mathcal{H}^{\otimes S})\) and \(H^{*}(B\mathrm{Diff}^+(W_g, *); \mathcal{H}^{\otimes S})\), as well of the twisted cohomology groups of \N\[\NB\mathrm{Tor}(W_g, D^{2n}), B\mathrm{Tor}^{\theta} (W_g, *), B\mathrm{Tor}^+(W_g, *), B\mathrm{Tor}^{\theta} (W_g), B\mathrm{Tor}^+(W_g).\N\]\NThis description makes use of the so called ``spaces of graphs''. \N\NThe details are too numerous to be state here.\N\NFinally many comments are made relating the present work with several of the literature, where the cases \(2n=2\) and \(2n=4\) are discussed.