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

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.

arXiv comment155 pages, 14 figures. Comments very welcome. Version 2 includes corrections to Theorems 2.7, 8.2, and 11.6. Version 3 has been extensively rewritten following a referee report, including new chapters 3 and 7

References in corpus (5)

• Subexponential groups in 4-manifold topology
• An elementary illustrated introduction to simplicial sets
• Stable classification of 4-manifolds with 3-manifold fundamental groups
• On calculating the slice genera of 11- and 12-crossing knots
• Topological 4-manifolds with right-angled Artin fundamental groups

Cited by in corpus (10)

• Spacetime foam: a review
• Embedded surfaces with infinite cyclic knot group
• Four-manifolds up to connected sum with complex projective planes
• Connected sum decompositions of high-dimensional manifolds
• Counterexamples in 4-manifold topology
• Balanced algebraic unknotting, linking forms, and surfaces in three- and four-space
• Homotopy ribbon concordance, Blanchfield pairings, and twisted Alexander polynomials
• Gluck twists on concordant or homotopic spheres
• A relative version of Rochlin's theorem
• Stable diffeomorphism classification of some unorientable 4-manifolds