Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs

2001/12/18 by Craig A. Jensen, Jensen, Craig A.
Mathematics · #05C25 #20F28 #20F32 #20J05 #55N91 #Algebraic Topology (math.AT) #FOS: Mathematics #Group Theory (math.GR) #math.AT #math.GR #msc:05C25 #msc:20F28 #msc:20F32 #msc:20J05 #msc:55N91

paper · pdf · doi:10.48550/arxiv.math/0112188

To appear in Publicacions Matematiques

arxiv created 2001/12/18 · arxiv updated 2009/11/30

Abstract

It is not known whether or not the stable rational cohomology groups H^*(Aut(F_∞);\Q) always vanish. We show that either the rational cohomology does not vanish in certain dimensions, or the integral cohomology of a moduli space of pointed graphs does not stabilize in certain other dimensions. Similar results are stated for groups of outer automorphisms. This yields that H⁵( Q_m; ℤ), H⁶( Q_m; ℤ), and H⁵(Q_m; ℤ) never stabilize as m → ∞, where the moduli spaces Q_m and Q_m are the quotients of the spines X_m and X_m of ``outer space'' and ``auter space'', respectively, introduced by Culler and Vogtmann and by Hatcher and Vogtmann.

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