Splitting criteria for a definite 4-manifold with infinite cyclic fundamental group

TL;DR

This paper provides new criteria for determining when a closed, definite 4-manifold with infinite cyclic fundamental group is TOP-split, linking topological properties with algebraic conditions on Hermitian forms.

Contribution

It introduces extended splitting criteria and connects topological splitting with algebraic representation of Hermitian forms over Laurent polynomial rings.

Findings

New criteria for TOP-splitting of 4-manifolds with infinite cyclic fundamental group.

An algebraic characterization of positive definite Hermitian forms over Laurent polynomial rings.

Construction of an infinite family of orthogonally indecomposable unimodular odd definite symmetric Z-forms.

Abstract

Two criteria for a closed connected definite 4-manifold with infinite cyclic fundamental group to be TOP-split are given. One criterion extends a sufficient condition made in a previous paper. The result is equivalent to a purely algebraic result on the question asking when a positive definite Hermitian form over the ring of integral one-variable Laurent polynomials is represented by an integer matrix. As an application, an infinite family of orthogonally indecomposable unimodular odd definite symmetric $Z$-forms is produced.