Stable homology of automorphism groups of free groups

By Søren Galatius. 2006

124 citations recorded by the source corpus.

Homology of the group Aut(F_n) of automorphisms of a free group on n generators is known to be independent of n in a certain stable range. Using tools from homotopy theory, we prove that in this range it agrees with homology of symmetric groups. In particular, Aut(F_n) has no stable rational homology, settling a conjecture of Hatcher and Vogtmann.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a0c4ee313a6a3c94fdb0061704d096b4eb019a84

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

A note on the (∞,n)–category of cobordisms (2015, background): “subsection. 5.2.1. The topological space pPBordnqk 1,...,kn We endow the set pPBordV nqk 1,...,kn with the following topologycoming from modiﬁcations of the Whitney C8-topology on EmbpM,Vˆp0,1qnq. In [Gal11], spelled out in more detail in [GRW10], a topology is constructed3 on the set of closed (not”
Representation stability and finite linear groups (2014, methodology): “he free group Fn of rank n. Just like for GLn(OK), work of Hatcher [Hat] and Hatcher–Vogtmann [HatVo1] shows that Aut(Fn) satisﬁes homological stability. These stable values were computed by Galatius [Ga], who proved in particular that they rationally vanish. For ℓ ≥ 2, the level ℓ congruence subgro”
Homological stability for automorphism groups (2014, background): “In general, the stable homology is though non-trivial, as is for example the stable homology with constant coefficients [20].”
Homological stability for mapping class groups of surfaces (2010, unknown): “Given two disjoint sets of points ∆0,∆1 ⊂ ∂S, define B(S,∆0,∆1) ⊂ A(S,∆0∪∆1) to be the subcomplex of arcs with one boundary point in ∆0 and one in ∆1.”
Hairy graphs and the unstable homology of Mod(g, s), Out(Fn) and Aut(Fn) (2011, background): “ave been shown to be non-trivial in homology ([2, 3, 10, 18]), and they are all conjectured to be non-trivial. We remark that the stable homology of both Mod(g;s) and Out(F n) is well understood (see [7, 17]) but the unstable homology remains quite mysterious, and all of Morita’s classes lie in the ”
Representation theory and homological stability (2010, background): “ller classes ei, or combinations thereof, are nonzero in H∗(Ig,1;Q). Thus even an aﬃrmative answer to Morita’s Conjecture would not imply Conjecture 6.1 for the trivial representation. Since Galatius [Ga] has proven that Hi(Aut(Fn);Q) = 0 for n≫i, it is natural to make the following conjecture. Conj”