3-manifolds that bound no definite 4-manifold

TL;DR

This paper constructs a rational homology 3-sphere that cannot bound any definite 4-manifold, revealing new constraints on 3-manifold and 4-manifold interactions and their cobordism properties.

Contribution

It introduces a specific example of a 3-manifold with unique bounding properties, expanding understanding of 4-manifold topology and cobordism limitations.

Findings

Constructed a rational homology 3-sphere with no definite 4-manifold bounding

Showed such a 3-manifold cannot be cobordant to certain classes of 3-manifolds

Analyzed characteristic covectors in bimodular lattices to prove results

Abstract

We produce a rational homology 3-sphere that does not smoothly bound either a positive or negative definite 4-manifold. Such a 3-manifold necessarily cannot be rational homology cobordant to a Seifert fibered space or any 3-manifold obtained by Dehn surgery on a knot. The proof requires an analysis of short characteristic covectors in bimodular lattices.