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 .

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Let G be a compact Lie group. Consider the variety Hom(Z k , G) of representations of Z k into G. We can see this as a based space by taking as base point the trivial representation 1. The goal of this paper is to prove that π 1 (Hom(Z k , G)) is naturally isomorphic to π 1 (G) k .

By a theorem of Thurston, in the subgroup of the mapping class group generated by Dehn twists around two curves which fill, every element not conjugate to a power of one of the twists is pseudo-Anosov. We prove an analogue of this theorem for the outer automorphism group of a free group.