Algebraic & Geometric Topology

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

Alexandra Pettet

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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).

Article information

Algebr. Geom. Topol., Volume 5, Number 2 (2005), 725-740.

Received: 13 January 2005
Revised: 5 May 2005
Accepted: 21 June 2005
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F28: Automorphism groups of groups [See also 20E36] 20J06: Cohomology of groups
Secondary: 20F14: Derived series, central series, and generalizations

automorphisms of free groups cohomology Johnson homomorphism descending central series


Pettet, Alexandra. The Johnson homomorphism and the second cohomology of $\mathrm{IA}_n$. Algebr. Geom. Topol. 5 (2005), no. 2, 725--740. doi:10.2140/agt.2005.5.725.

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