THE JOHNSON HOMOMORPHISM AND THE SECOND COHOMOLOGY OF IAn
نویسنده
چکیده
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 K (i) n which act trivially on Fn/F (i+1) n , the free rank n , degree i nilpotent group. It is a conjecture of Andreadakis that K (i) n is equal to the lower central series of IAn ; indeed K (2) n is known to be the commutator subgroup of IAn . We prove that the quotient group K (3) n /IA (3) n is finite for all n and trivial for n = 3. We also compute the rank of K (2) n /K (3) n . AMS Classification 20F28, 20J06; 20F14
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