The Johnson homomorphism and the second cohomology of IAn

Pettet, Alexandra

doi:10.2140/agt.2005.5.725

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The Johnson homomorphism and the second cohomology of $\mathrm{IA}_n$

Alexandra Pettet

Algebraic & Geometric Topology 5 (2005) 725–740

DOI: 10.2140/agt.2005.5.725

arXiv: math.GR/0501053

Abstract

Let $F_{n}$ be the free group on $n$ generators. Define ${IA}_{n}$ to be group of automorphisms of $F_{n}$ that act trivially on first homology. The Johnson homomorphism in this setting is a map from ${IA}_{n}$ to its abelianization. The first goal of this paper is to determine how much this map contributes to the second rational cohomology of ${IA}_{n}$ .

A descending central series of ${IA}_{n}$ is given by the subgroups $K_{n}^{(i)}$ which act trivially on $F_{n} ∕ F_{n}^{(i + 1)}$ , the free rank $n$ , degree $i$ nilpotent group. It is a conjecture of Andreadakis that $K_{n}^{(i)}$ is equal to the lower central series of ${IA}_{n}$ ; indeed $K_{n}^{(2)}$ is known to be the commutator subgroup of ${IA}_{n}$ . We prove that the quotient group $K_{n}^{(3)} ∕ {IA}_{n}^{(3)}$ is finite for all $n$ and trivial for $n = 3$ . We also compute the rank of $K_{n}^{(2)} ∕ K_{n}^{(3)}$ .

Keywords

automorphisms of free groups, cohomology, Johnson homomorphism, descending central series

Mathematical Subject Classification 2000

Primary: 20F28, 20J06

Secondary: 20F14

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Publication

Received: 13 January 2005

Revised: 5 May 2005

Accepted: 21 June 2005

Published: 13 July 2005

Authors

Alexandra Pettet
Department of Mathematics University of Chicago Chicago IL 60637 USA

Volume 5, issue 2 (2005)