Counterexamples in 4-manifold topology

TL;DR

This paper explores the complex landscape of 4-manifold topology by presenting counterexamples that clarify the relationships between various equivalence relations, highlighting where intuitive implications fail.

Contribution

It systematically organizes key counterexamples and clarifies the relationships among different equivalence relations in 4-manifold topology.

Findings

h-cobordant manifolds are stably homeomorphic

Counterexamples show failure of other implications

Flowchart and table organize complex relationships

Abstract

We illustrate the rich landscape of 4-manifold topology through the lens of counterexamples. We consider several of the most commonly studied equivalence relations on 4-manifolds and how they are related to one another. We explain implications e.g. that $h$-cobordant manifolds are stably homeomorphic, and we provide examples illustrating the failure of other potential implications. The information is conveniently organised in a flowchart and a table.