W. G. Dwyer

Homology, Massey products, and maps between groups

Math Reviews: 52:6710

Suppose that f is a group homomorphism from G to L. Let G(k) and L(k) denote the k'th lower central series quotients of G and L respectively. A theorem of Stallings states that if the map induced by f on group homology is an isomorphism in dimension one and an epimorphism in dimension two, then the map from G(k) to L(k) induced by f is an isomorphism for all k. The goal of this paper is to find some weaker homological condition on f which is enough to guarantee that the induced map from G(k) to L(k) is an isomorphism for all k less than or equal to some fixed N. The condition is expressed in terms of a certain subgroup of two-dimensional group homology which (roughly speaking) is dual to the set of two-dimensional group cohomology classes decomposable by Massey products of length less than N.

The paper also describes an explicit group-theoretic way to understand higher Massey products of one-dimensional cohomology classes.

There is some related work in the following papers.

• 88m:20111 Fenn, Roger; Sjerve, Denis Massey products and lower central series of free groups. Canad. J. Math. 39 (1987), no. 2, 322--337
• 81a:57021 Porter, Richard Milnor's $\bar µ $-invariants and Massey products. Trans. Amer. Math. Soc. 257 (1980), no. 1, 39--71