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

Craig A. Jensen
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Abstract

It is not known whether or not the stable rational cohomology groups $\tilde H^*(Aut(F_\infty);\Q)$ 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 $H^5(\hat Q_m; \mathbb{Z})$, $H^6(\hat Q_m; \mathbb{Z})$, and $H^5(Q_m; \mathbb{Z})$ never stabilize as $m \to \infty$, where the moduli spaces $\hat Q_m$ and $Q_m$ are the quotients of the spines $\hat X_m$ and $X_m$ of ``outer space'' and ``auter space'', respectively, introduced by Culler and Vogtmann and by Hatcher and Vogtmann.

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Craig A. Jensen. 2001-12-18. Stable rational cohomology of automorphism groups of free groups and the integral cohomology of moduli spaces of graphs. https://arxiv.org/abs/math/0112188

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