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

Alexandra Pettet

Algebraic & Geometric Topology 5 (2005) 725–740

arXiv: math.GR/0501053


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

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

automorphisms of free groups, cohomology, Johnson homomorphism, descending central series
Mathematical Subject Classification 2000
Primary: 20F28, 20J06
Secondary: 20F14
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Received: 13 January 2005
Revised: 5 May 2005
Accepted: 21 June 2005
Published: 13 July 2005
Alexandra Pettet
Department of Mathematics
University of Chicago
Chicago IL 60637