A spectral sequence determining the homology of \(\text{Out}(F_n)\) in terms of its mapping class subgroups. (Q2382015)

Let \(\Sigma\) be an orientable surface with boundary and \(P\Gamma(\Sigma)\) its mapping class group. The homology of this group is studied in details by \textit{J. L. Harer} [Ann. Math. (2) 121, 215-249 (1985; Zbl 0579.57005)], \textit{N. V. Ivanov} [Leningr. Math. J. 1, No. 3, 675-691 (1990); translation from Algebra Anal. 1, No. 3, 110-126 (1989; Zbl 0727.30036)], and \textit{J. L. Harer} [\texttt{http://www.math.duke.edu/preprints/93-09}]. On the other hand, if \(F_n\) is the free group of rank \(n\) and \(\text{Out}(F_n)\) its outer automorphism group, the virtual cohomological dimension of \(\text{Out}(F_n)\) is computed by \textit{M. Culler, K. Vogtmann} [Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)] and in [Algebr. Geom. Topol. 4, 1253-1272 (2004; Zbl 1093.20020)] by \textit{A. Hatcher} and \textit{K. Vogtmann} it is shown that the \(k\)-th integral homology of \(\text{Out}(F_n)\) is independent of \(n\) if \(n\geq 2k+5\). Since the mapping class groups of surfaces appear as subgroups of \(\text{Out}(F_n)\), in this paper the author attempts to clarify the relationship between the homology of mapping class groups and the homology of \(\text{Out}(F_n)\). This is obtained by constructing a covering of the spine of the Culler-Vogtmann outer space of \(\text{Out}(F_n)\). It is proved that the nerve of this covering is contractible and that the equivariant homology spectral sequence of the action of \(\text{Out}(F_n)\) on this nerve converges to \(H_*(\text{Out}(F_n))\); many of whose terms consist of the homology of mapping class groups.