Volume 5, issue 2 (2005)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 14
Issue 5, 2511–3139
Issue 4, 1881–2509
Issue 3, 1249–1879
Issue 2, 627–1247
Issue 1, 1–625

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Editorial Interests
Author Index
Editorial procedure
Submission Guidelines
Submission Page
Author copyright form
Subscriptions
Contacts
G&T Publications
GTP Author Index
ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
The Johnson homomorphism and the second cohomology of $\mathrm{IA}_n$

Alexandra Pettet

Algebraic & Geometric Topology 5 (2005) 725–740

arXiv: math.GR/0501053

Abstract

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).

Keywords
automorphisms of free groups, cohomology, Johnson homomorphism, descending central series
Mathematical Subject Classification 2000
Primary: 20F28, 20J06
Secondary: 20F14
References
Forward citations
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