Hyperplanes in abelian groups and twisted signatures

TL;DR

The paper characterizes when isomorphisms between products of finite cyclic groups preserve coordinate hyperplanes, and applies this to distinguish certain 4-manifolds using twisted signatures, revealing structural invariants.

Contribution

It establishes conditions under which isomorphisms preserve coordinate hyperplanes in products of cyclic groups and applies this to classify certain 4-manifolds via twisted signatures.

Findings

Isomorphisms preserve coordinate hyperplanes if cyclic factors have order > 2.

Such isomorphisms are diagonal up to permutation.

Application to classifying 4-manifolds with specific homology groups.

Abstract

We investigate the following question: if $A$ and $A'$ are products of finite cyclic groups, when does there exist an isomorphism $f: A \to A'$ 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. Thus one can recover the coordinate hyperplanes from knowledge of their union. This result is well-adapted for application to invariants with a certain multiplicativity property. As a model application, we show using twisted signatures that there exists a family of compact 4-manifolds $X(n)$ with $H_1 X(n) = \mathbb Z/n$ with the property that $\prod X(n_i) \cong \prod X(n'_j)$ if and only if the factors may be identified (up to permutation), and that the induced map on first homology is (up to permutation) represented by a diagonal matrix.