Configuration-mapping spaces and homology stability
Martin Palmer, Ulrike Tillmann
备注：
37 pages, 11 figures. Final accepted version, to appear in Res. Math. Sci. Section 10 has been removed and will be revised and appear separately. Section 9 has been merged into sections 7 and 8
For a given bundle $ξ\colon E \to M$ over a manifold, configuration-section spaces on $ξ$ parametrise finite subsets $z \subseteq M$ equipped with a section of $ξ$ defined on $M \smallsetminus z$, with prescribed "charge" in a neighbourhood of the points $z$. These spaces may be interpreted physically as spaces of fields that are permitted to be singular at finitely many points, with constrained behaviour near the singularities. As a special case, they include the Hurwitz spaces, which parametrise branched covering spaces of the $2$-disc with specified deck transformation group.
We prove that configuration-section spaces are homologically stable (with integral coefficients) whenever the underlying manifold $M$ is connected and has non-empty boundary and the charge is "small" in a certain sense. This has a partial intersection with the work on Hurwitz spaces of Ellenberg, Venkatesh and Westerland.
In this paper, we deal with stable homology computations with twisted coefficients for mapping class groups of surfaces and of 3-manifolds, automorphism groups of free groups with boundaries and automorphism groups of certain right-angled Artin groups. On the one hand, the computations are led using semidirect product structures arising naturally from these groups. On the other hand, we compute the stable homology with twisted coefficients by FI-modules. This notably uses a decomposition result of the stable homology with twisted coefficients for pre-braided monoidal categories proved in this …
A spectral sequence calculating the homology groups of some spaces of maps equivariant under compact group actions is described. For the main example, we calculate the rational homology groups of spaces of even and odd maps $S^m \to S^M$, $m<M$, or, which is the same, the stable homology groups of spaces of non-resultant homogeneous polynomial maps ${\mathbb R}^{m+1} \to {\mathbb R}^{M+1}$ of growing degrees. Also, we find the homology groups of spaces of ${\mathbb Z}_r$-equivariant maps of odd-dimensional spheres for any $r$. As a technical tool, we calculate the homology groups of configurat…
We let the mapping class group $Γ_{g,1}$ of a genus $g$ surface $Σ_{g,1}$ with one boundary component act on the homology $H_*(F_{n}(Σ_{g,1});\mathbb{Q})$ of the $n^{th}$ ordered configuration space of the surface. We prove that the action is non-trivial when restricted to the $(n-1)^{st}$ stage of the Johnson filtration, for all $n\ge 1$ and $g\ge 2$. We deduce an analogous result for closed surfaces.
We provide a short proof that the dimensions of the mod $p$ homology groups of the unordered configuration space $B_k(T)$ of $k$ points in a torus are the same as its Betti numbers for $p>2$ and $k\leq p$. Hence the integral homology has no $p$-power torsion. The same argument works for the punctured genus $g$ surface with $g>0$, thereby recovering a result of Brantner-Hahn-Knudsen via Lubin-Tate theory.
In this paper we compute the mapping class group of simply-connected closed smooth manifolds $M$ with integral homology $H_{*}(M) \cong \mathbb Z \oplus \mathbb Z \oplus \mathbb Z$ provided that $\dim M \ne 4$.
We study stability patterns in the high dimensional rational homology of unordered configuration spaces of manifolds. Our results follow from a general approach to stability phenomena in the homology of Lie algebras, which may be of independent interest.
