The topology of four-dimensional manifolds

  title={The topology of four-dimensional manifolds},
  author={Michael H. Freedman},
  journal={Journal of Differential Geometry},
  • M. Freedman
  • Published 1982
  • Mathematics
  • Journal of Differential Geometry
0. Introduction Manifold topology enjoyed a golden age in the late 1950's and 1960's. Of the mysteries still remaining after that period of great success the most compelling seemed to lie in dimensions three and four. Although experience suggested that manifold theory at these dimensions has a distinct character, the dream remained since my graduate school days that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful… 
Low Dimensional Topology and Gauge Theory
The study of manifolds of dimensions at least 5 had remarkable success in the 1960’s, with the resolution of fundamental problems about existence and uniqueness of smooth or PL structures by a
Trisections of 4-manifolds
  • R. Kirby
  • Mathematics
    Proceedings of the National Academy of Sciences
  • 2018
The gauge theory invariants are very good at distinguishing smooth 4-manifolds that are homotopy equivalent but do not help at showing that they are diffeomorphic, so what is missing is the equivalent of the higher-dimensional s-cobordism theorem, a key to the successes in higher dimensions.
The Poincaré Conjecture 99 Years Later : A Progress Report
  • Mathematics
The topology of 2-dimensional manifolds or surfaces was well understood in the 19-th century. 1 In fact there is a simple list of all possible smooth compact orientable surfaces. Any such surface has
Surgery in cusp neighborhoods and the geography of irreducible 4-manifolds
Since its inception, the theory of smooth 4-manifolds has relied upon complex surface theory to provide its basic examples. This is especially true for simply connected 4-manifolds: a longstanding
A surgery sequence in dimension four; the relations with knot concordance
We present a systematic treatment of the classification problem for compact smooth 4-manifolds M. It is modeled on the surgery exact sequence, the central theorem in the classification of n-manifolds
The early part of the 1980s has experienced a vast increase in our understanding of smooth 4-manifolds. This has been accomplished principally through the work of S. Donaldson, namely Theorem 1.1
Smooth 4-manifolds and Symplectic Topology
One of the famous problems of topology is the classification problem for simply connected 4-manifolds. In the context of topological manifolds (up to homeomorphism), Freedman reduced the problem in
Fifty Years Ago: Topology of Manifolds in the 50's and 60's
The 1950’s and 1960’s were exciting times to study the topology of manifolds. This lecture will try to describe some of the more interesting developments. The flrst two sections describe work in
A survey of 4-manifolds through the eyes of surgery
Surgery theory is a method for constructing manifolds satisfying a given collection of homotopy conditions. It is usually combined with the s{cobordism theorem which constructs homeomorphisms or
Topology through the centuries: Low dimensional manifolds
This note will provide a lightning tour through the centuries, concentrating on the study of manifolds of dimension 2, 3, and 4. Further comments and more technical details about many of the sections


Surgery on Simply-Connected Manifolds
I. Poincare Duality.- 1. Slant Operations, Cup and Cap Products.- 2. Poincare Duality.- 3. Poincare Pairs and Triads Sums of Poincare Pairs and Maps.- 4. The Spivak Normal Fibre Space.- II. The Main
Surgery on compact manifolds
Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important
A proof of the generalized Schoenflies theorem
The following problem has been of interest for some time : Suppose h is a homeomorphic embedding of S~~X [0l] in S. Are the closures of the complementary domains of h(S~Xl/2) topological w-cells?
Upper Semicontinuous Decompositions of E 3
In this paper it is shown that monotone upper semicontinuous decompositions of E3 satisfying certain additional conditions have decomposition spaces which are topologically equivalent to E3. When
Transversality theories at dimension four
On Simply-Connected 4-Manifolds
This paper concerns (but does not succeed in performing) the diffeomorphism classification of closed, oriented, differential, simply-connected 4-manifolds. It arises out of the observation (due to
A fake S3 X R
  • Ann. of Math. 110
  • 1979