Hyperplanes in abelian groups and twisted signatures

Abstract

We investigate the following question: if A and A′ are products of finite cyclic groups, when does there exist an isomorphism f:A→A′ which preserves the union of coordinate hyperplanes (equivalently, so that f(x) has some coordinate zero if and only if x has some coordinate zero)?We show that if such an isomorphism exists, then A and A′ have the same cyclic factors; if all cyclic factors have order larger than 2, the map f is diagonal up to permutation, hence sends coordinate hyperplanes to coordinate hyperplanes.As a model application, we show using twisted signatures that there exists a family of compact 4-manifolds X(n) with H1X(n)=Z/n with the property that ∏X(ni)≅∏X(nj′) if and only if the factors may be identified (up to permutation), and that the induced map on first homology is represented by a diagonal matrix.