Dimension of the Torelli group for Out(F_n)

Abstract

Let T_n be the kernel of the natural map from Out(F_n) to GL(n,Z). We use combinatorial Morse theory to prove that T_n has an Eilenberg-MacLane space which is (2n-4)-dimensional and that H_{2n-4}(T_n,Z) is not finitely generated (n at least 3). In particular, this recovers the result of Krstic-McCool that T_3 is not finitely presented. We also give a new proof of the fact, due to Magnus, that T_n is finitely generated.

arXiv comment27 pages, 9 figures

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• Effective finite generation for [IA_n,IA_n] and the Johnson kernel
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• On the low dimensional cohomology groups of the IA-automorphism group of a free group of rank three
• Higher incoherence of the automorphism groups of a free group
• Secondary characteristic classes for subgroups of automorphism groups of free groups
• Partial Torelli groups and homological stability