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

Stefan Friedl, Matthias Nagel, Patrick Orson, Mark Powell

TL;DR

This survey compiles foundational results on topological 4-manifolds, highlighting what in dimension four survives in the TOP category, where smooth techniques fail or are substantially deeper, and how algebraic invariants like intersection forms govern topological classification. It surveys collaring, isotopy extension, transversality, tubular neighbourhoods, and bundle theory in TOP, PL, and Diff settings, with a focus on four-manifolds and the dramatic contrasts between topological and smooth structures. Central themes include the Annulus and Stable Homeomorphism Theorems, smoothing theory via Kirby–Siebenmann, and the role of Wu/SW classes and twisted signatures in understanding spin, orientability, and the Viète of stable structures. Applications include classification of simply connected and non-simply connected 4-manifolds, stable diffeomorphism results, and insights into how large classes of groups (e.g., infinite cyclic, Baumslag–Solitar, finite groups) influence topological 4-manifold classification. Overall, the work provides a roadmap for navigating the interactions between topology, geometry, and algebra in dimension four, with precise proofs and references to guide researchers through potential pitfalls and available tools.

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.

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

TL;DR

This survey compiles foundational results on topological 4-manifolds, highlighting what in dimension four survives in the TOP category, where smooth techniques fail or are substantially deeper, and how algebraic invariants like intersection forms govern topological classification. It surveys collaring, isotopy extension, transversality, tubular neighbourhoods, and bundle theory in TOP, PL, and Diff settings, with a focus on four-manifolds and the dramatic contrasts between topological and smooth structures. Central themes include the Annulus and Stable Homeomorphism Theorems, smoothing theory via Kirby–Siebenmann, and the role of Wu/SW classes and twisted signatures in understanding spin, orientability, and the Viète of stable structures. Applications include classification of simply connected and non-simply connected 4-manifolds, stable diffeomorphism results, and insights into how large classes of groups (e.g., infinite cyclic, Baumslag–Solitar, finite groups) influence topological 4-manifold classification. Overall, the work provides a roadmap for navigating the interactions between topology, geometry, and algebra in dimension four, with precise proofs and references to guide researchers through potential pitfalls and available tools.

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.

Paper Structure

This paper contains 64 sections, 138 theorems, 158 equations, 14 figures.

Table of Contents

  1. Introduction
  2. High-dimensional topological manifolds
  3. A focus on dimension four
  4. What is in this survey?
  5. Manifolds
  6. Definition of manifolds
  7. Definition of submanifolds
  8. Immersions
  9. Collar neighbourhoods
  10. The Isotopy Extension Theorem
  11. CW structures, triangulations, and handle structures on manifolds
  12. The Annulus Theorem and the Stable Homeomorphism Theorem
  13. The Stable Homeomorphism Theorem
  14. The connected sum operation
  15. The Product Structure Theorem
  16. ...and 49 more sections

Key Result

Theorem 2.3

(Invariance of Domain Theorem) If $U\subseteq \mathbb{R}^n$ is an open subset and if $h\colon U\to \mathbb{R}^n$ is an injective map, then $h(U)$ is an open subset of $\mathbb{R}^n$.

Figures (14)

  • Figure 2.1: Definition of submanifolds.
  • Figure 2.2: Illustration of the proof of the Collar Neighbourhood Theorem \ref{['thm:collar-boundary']}.
  • Figure 3.1: Existence of structures on a compact manifold.
  • Figure 4.1: Illustration for the proof of Theorem \ref{['thm:connected-sum-well-defined']}.
  • Figure 5.1: Definition of $\partial_0$ and $\partial_1W$ of a submanifold.
  • ...and 9 more figures

Theorems & Definitions (349)

  • Remark 1.1
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Proposition 2.6
  • Definition 2.7
  • Remark 2.8
  • ...and 339 more