Homology of configuration spaces of surfaces modulo an odd prime (Q6582265)

Let \(C_{\bullet}(M)\) denote the disjoint union of the unordered configuration spaces of \(M\). This paper studies its homological properties when \(M=\Sigma_{g,1}\), the compact orientable surface of genus \(g\) and one boundary component. First, the \(R\)-homology of \(C_{\bullet}(\Sigma_{g,1})\) is described as a bigraded module over \(H_*(C_{\bullet}(D^2);R)\), which was known to be isomorphic to \(R[\varepsilon] \otimes_R \text{Ext}_{\Gamma_R(y)}^{*-\bullet}(R,R)\) as a bigraded ring, for any commutative ring \(R\). The description includes derived functors \(\text{Ext}_{\Gamma_R(y)}(-,-)\), with the module structure coming from the Yoneda product. In the case of \(R=\mathbb{F}_p\), for an odd prime \(p\), the Ext terms are determined, obtaining explicit computations.\N\NThe method to determine the homology of \(C_{\bullet}(\Sigma_{g,1})\) is worth highlighting. The homology of \(C_n(\Sigma_{g,1})\) is isomorphic to the shifted reduced cohomology of \(C_n(\Sigma_{g,1})^{\infty}\) by Poincaré-Lefschetz duality, even as \(\Gamma_{g,1}\)-modules. Here the space \(C_n(\Sigma_{g,1})^{\infty}\) is the result of collapsing to a point the \(n\)-tuples in \(\Sigma_{g,1}^n\) with repeated coordinates or with some coordinate in the boundary, and then taking the orbit space under the action of the symmetric group. The authors provide \(C_n(\Sigma_{g,1})^{\infty}\) with a cellular structure, and the corresponding cellular chain complex is isomorphic to a reduced bar complex involving the cellular chain complexes for similar spaces constructed from bouquets of circles.\N\NAnother aspect considered is the kernel of the action of the mapping class group \(\Gamma_{g,1}\) on homology. The subgroup \(\mathcal{K}_{g,1}\) of \(\Gamma_{g,1}\) generated by Dehn twists around separating simple closed curves, also known as the Johnson kernel, acts trivially on the homology of \(C_{\bullet}(\Sigma_{g,1})\) over any commutative ring \(R\), and when \(R=\mathbb{Z}\) or \(\mathbb{Q}\), it coincides with the kernel of the action. When \(R=\mathbb{F}_p\), for an odd prime \(p\), the kernel of the action is shown to be the subgroup of \(\Gamma_{g,1}\) generated by \(\mathcal{K}_{g,1}\) and the \(p\)th powers of all Dehn twists.\N\NThe rest of the article is dedicated to the proof of some consequences. For instance, one can recover previous homological stability results from these computations and relate, for an odd prime \(p\), the asymptotic growth of the dimension of \(H_{n-i}(C_n(\Sigma_{g,1});\mathbb{F}_p)\) with \(\text{log}_p(n)^i n^{2g-1}\). Another interesting corollary shows that the embedding of \(\Sigma_{g,1} \amalg \Sigma_{1,1}\) inside \(\Sigma_{g+1,1}\) as the complement of a pair of pants induces a split injective map\N\[\NH_*(C_n(\Sigma_{g,1});R) \otimes_R H_1(\Sigma_{1,1};R) \to H_{*+1}(C_{n+1}(\Sigma_{g+1,1});R)\N\]\Nfor any commutative ring \(R\).