Cobordism-equivalence for codimension-one submanifolds

TL;DR

This paper establishes a homology-based criterion for when two hypersurfaces are connected via embedded cobordisms, and applies handle decompositions to relate Seifert surfaces through surgeries.

Contribution

It provides a new homological characterization of hypersurface cobordisms and offers a conceptual proof relating Seifert surfaces through surgeries.

Findings

Hypersurfaces are related by cobordisms iff they share the same homology class.

Handle decompositions convert cobordisms into sequences of surgeries.

Seifert surfaces of the same link are connected by tube attachments and removals.

Abstract

We show that two hypersurfaces in a manifold are related by a sequence of embedded cobordisms if and only if they represent the same homology class. By applying handle decompositions we turn these cobordisms into a sequence of embedded surgeries. Specializing to Seifert surfaces we obtain a conceptual proof that two Seifert surfaces of a fixed link are related by tube attachments and tube removals.