Homological stability for oriented configuration spaces (Q2846979)

For a manifold \(M\) and a path-connected space \(X\), we can define the unordered configuration space \(C_n(M,X) = Emb([n], M)\times_{\Sigma_n}X^n\), where \([n]\) is the discrete space \(\{1,\ldots, n\}\). This is the space of configurations of \(n\) distinct points (or ``particles'') in \(M\), each carrying a label (or ``parameter'') in \(X\). One considers a manifold \(M\) which is the interior of a connected manifold with boundary of dimension at least \(2\). By adding a point near the boundary to a configuration, we obtain a ``stabilization'' map \(s_n: C_n(M,X)\longrightarrow C_{n+1}(M,X)\) which is defined up to homotopy. It is a main theorem of \textit{O. Randal-Williams} [Q. J. Math. 64, No. 1, 303--326 (2013; Zbl 1264.55009)] that this map induces an isomorphism on homology up to degree \({n-1\over 2}\) and a surjection up to degree \({n\over 2}\). The proof proceeds by induction and uses the fact that the stabilization maps are split injective in homology.NEWLINENEWLINEThe paper at hand establishes a similar result for ``oriented'' configuration spaces \(C_n^+(M,X) = Emb([n], M)\times_{A_n}X^n\), where \(A_n\) is the alternating group. The main theorem states that for \(M\) and \(X\) as above, there are stabilization maps \(s_n^+: C^+_n(M,X)\longrightarrow C^+_{n+1}(M,X)\) which induce an isomorphism in homology up to degree \({n-5\over 3}\) and a surjection up to degree \({n-2\over 3}\). A similar stability bound is given if integral homology is replaced by any connective homology theory.NEWLINENEWLINETo prove his theorem the author follows the same steps as Randal-Williams, however new subtle arguments have to be provided to go around the fact that the maps \(s^+\) in the oriented case are not injective in homology (the author provides counter-examples) and that there are obstructions to the existence of certain self-homotopies of iterated stabilization maps between the homotopy cofibers of these maps. This second step is resolved by means of Lemma 6.2 (the factorization lemma) which is of independent interest and gives factorization properties for maps between the mapping cones of two maps \(A\rightarrow X\) and \(B\rightarrow Y\) if they sit in a homotopy commutative square of maps with \(f:A\rightarrow B\) and \(g: X\rightarrow Y\) such that the square can be filled in by two homotopy commutative triangles.NEWLINENEWLINEThe main theorem has corollaries for the homological stability of certain sequences of groups. Namely if \(G\) is any discrete group and \(S\) is the interior of a connected surface with boundary, then the maps between the wreath products \(G\wr A_n\rightarrow G\wr A_{n+1}\) and \(G\wr A\beta_n^S\rightarrow G\wr A\beta_{n+1}^S\) induce homology isomorphisms and epimorphisms through the same ranges as previously; \({n-5\over 3}\) and \({n-2\over 3}\) respectively. Here \(A\beta_n^S\) is the alternating braid group on \(n\) strands consisting of those braids whose induced permutation is even.NEWLINENEWLINEThe homological stability for oriented configuration spaces has been addressed earlier by Guest-Koslowski-Yamaguchi [\textit{M. A. Guest} et al., J. Math. Kyoto Univ. 36, No. 4, 809--814 (1996; Zbl 0891.55013)] who proved it for \(M\) a compact connected Riemann surface minus a non-empty finite set of points and \(X=pt\). The main theorem in this paper explains how to extend this result to arbitrary open manifolds \(M\) and labels in a path-connected \(X\).