Surface slices and homology spheres

TL;DR

This paper develops a diagrammatic theory of surface cross sections to identify an infinite class of homology 3-spheres that can be embedded in homology 4-spheres but not in homotopy 4-spheres, revealing new embedding obstructions.

Contribution

It introduces a new diagrammatic approach to study surface slices, proving the existence of infinitely many homology 3-spheres with specific embedding properties.

Findings

Existence of infinitely many homology 3-spheres embeddable in homology 4-spheres

Obstruction to embedding in homotopy 4-spheres based on Taubes' work

Development of a diagrammatic theory for surface cross sections

Abstract

We develop the theory of the diagrammatics of surface cross sections to prove that there are an infinite number of homology 3-spheres smoothly embeddable in a homology 4-sphere but not in a homotopy 4-sphere. Our primary obstruction comes from work of Taubes.