The Johnson homomorphism and the second cohomology of IAn

Pettet, Alexandra¹

¹Department of Mathematics, University of Chicago, Chicago IL 60637, USA

Algebraic and Geometric Topology, Tome 5 (2005) no. 2, pp. 725-740

Résumé

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 Fn∕Fn(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).

Détail

DOI : 10.2140/agt.2005.5.725

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

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