The Johnson homomorphism and the second cohomology of IA_n

Abstract

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 .

The Johnson filtration of the McCool stabilizer subgroup of the automorphism group of a free group

“…Moreover, by independent works of Cohen-Pakianathan [4,5], Farb [6] and Kawazumi [10], it is known that gr 1 (A n ) is isomorphic to the abelianization of IA n . For k = 2 and 3, the rank of gr k (A n ) is determined by Pettet [19] and Satoh [21] respectively. For k ≥ 4, however, it seems that there are few results for the structure of gr k (A n ).…”

A reduction of the target of the Johnson homomorphisms of the automorphism group of a free group

“…The study of the Johnson homomorphisms was originally begun in 1980 by D. Johnson [10] who determined the abelianization of the Torelli subgroup of a mapping class group of a surface in [11]. Recently, the study of the Johnson filtration and the Johnson homomorphisms of Aut F n achieved good progress through the work of many authors, for example, [7], [12], [18], [19], [20], [24] and [26]. Through the images of the Johnson homomorphisms, we can study IA n using infinitely many pieces of a free abelian group of finite rank.…”

“…Now, for 1 ≤ k ≤ 3, the cokernel of τ k,Q is completely determined. (See [1], [24] and [26] for k = 1, 2 and 3, respectively.) Recently, Morita [19,20] showed that for each k ≥ 2, there appears the symmetric tensor product S k H Q of H Q := H Z ⊗ Q in the irreducible decomposition of Coker(τ k,Q ) using trace maps.…”

“…(See (3) below.) Furthermore, A n (3) has at most a finite index in A n (3) due to Pettet [24]. It is, however, also difficult to determine whether A n (k) coincides with A n (k) or not.…”

The second Johnson homomorphism and the second rational cohomology of the Johnson kernel

“…Our previous paper [24] treated the third cohomology of the Torelli group. Brendle-Farb [6] studied the second cohomology of the Torelli group and the Johnson kernel by using the Birman-Craggs-Johnson homomorphism, and Pettet [23] studied the second cohomology of the (outer-)automorphism group of a free group by using its first Johnson homomorphism.…”