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

Primary: 20F28, 20J06

Secondary: 20F14

References

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Publication

Received: 13 January 2005
Revised: 5 May 2005
Accepted: 21 June 2005
Published: 13 July 2005

Authors