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Stable homology of automorphism groups of free groups. (English) Zbl 1268.20057

The permutation group \(\Sigma_n\) acts on a free group \(F_n\) by permuting the generators. This defines an injective homomorphisms \(\varphi_n\colon\Sigma_n\to\operatorname{Aut}\,F_n\).
The main result in this landmark paper is that the induced map in integral homology is an isomorphism up to degree \(k\) with \(n>2k+1\). It immediately follows that \(\operatorname{Aut}\,F_n\) has trivial rational homology in these degrees, settling a question raised by A. E. Hatcher [Comment. Math. Helv. 70, No. 1, 39-62 (1995; Zbl 0836.57003)].
Formulated in homotopy theoretic terms the theorem says \(\mathbb Z\times B\operatorname{Aut}\,F_\infty^+\simeq QS^0\) where \(\operatorname{Aut}\,F_\infty:=\lim_n\operatorname{Aut}\,_n\), \(X^+\) denotes \(X\) after Quillen’s plus construction, and \(QS^0:=\lim_n\text{map}_*(S^n,S^n)\).
The work can be understood as the analogue for graphs what had been done for surfaces. The above two statements are then the analogue of the Mumford conjecture and the Madsen-Weiss theorem. Conceptually the proof follows the approach via cobordism categories [as in S. Galatius, I. Madsen, M. Weiss and the reviewer, Acta Math. 202, No. 2, 195-239 (2009; Zbl 1221.57039)]. The main new tool are spaces \(\Phi(\mathbb R^n)\) of non-compact graphs in \(\mathbb R^n\) which define a spectrum that replaces the Thom spectrum \(MTSO(2)=\mathbb CP^\infty_{-1}\) used in the case of surfaces.
The proof proceeds by: (1) Identifying the space of compact graphs in \(\mathbb R^\infty\) as a classifying space for the outer automorphism group \(\text{Out}(F_n)\); this uses Culler-Vogtmann’s outer space [M. Culler and K. Vogtmann, Invent. Math. 84, 91-119 (1986; Zbl 0589.20022)]. (2) Showing that the ‘scanning map’ induces a homology equivalence from the space of compact graphs in \(\mathbb R^\infty\) of genus \(g\) to a component of the infinite loop space \(\Omega^\infty\Phi:=\lim_n\text{map}_*(S^n,\Phi(\mathbb R^n))\) in degrees increasing with \(g\). And (3) showing that the infinite loop space \(\Omega^\infty\Phi\) is homotopic to \(QS^0\).
The paper introduces new techniques that have subsequently played an important role in the further development of the subject area. In particular the role of the scanning map is emphasised and the proof of an unstable version of the main result of the author et al. [loc. cit., Zbl 1221.57039] is sketched.

MSC:

20J05 Homological methods in group theory
20F28 Automorphism groups of groups
20J06 Cohomology of groups
20E05 Free nonabelian groups
55N25 Homology with local coefficients, equivariant cohomology
55P47 Infinite loop spaces
55R40 Homology of classifying spaces and characteristic classes in algebraic topology
57R20 Characteristic classes and numbers in differential topology
55P42 Stable homotopy theory, spectra
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