A survey of the foundations of four-manifold theory in the topological category

TL;DR

This survey provides a comprehensive overview of foundational theorems and literature on topological 4-manifolds, highlighting deep, open, and false results to guide researchers through complex theoretical terrain.

Contribution

It compiles and clarifies key theorems and proof strategies in topological 4-manifold theory, serving as a navigational guide for researchers.

Findings

Summarizes foundational theorems in topological 4-manifold theory.

Highlights deep and open problems in the field.

Provides references and proof strategies for key results.

Abstract

This survey aims to provide a guide to the literature on topological 4-manifolds. Foundational theorems on 4-manifolds are stated, especially in the topological category. Precise references are given, with indications of the strategies employed in the proofs. Where appropriate we give statements for manifolds of all dimensions. Many intuitively plausible theorems which are standard results in differential topology are either extraordinarily deep results in the topological category, are open, or are known to be false. Hence one must proceed with caution. This book seeks to help 4-manifold topologists navigate potential pitfalls, and to apply the many powerful results that do exist with confidence.