1982/01/01 by Michael Freedman, Michael Hartley Freedman · 18 citations
Computer Science · Mathematics · #Combinatorics #Curvature #Geometric and Algebraic Topology #Geometry #Homotopy and Cohomology in Algebraic Topology #Manifold (fluid mechanics) #Mathematics #Pure mathematics #Ricci-flat manifold #Scalar curvature #Topological and Geometric Data Analysis #Topology (electrical circuits)
paper · pdf · doi:10.4310/jdg/1214437136
crossref issued 1982/01/01 · crossref published 1982/01/01 · crossref published-print 1982/01/01 · openalex publication_date 1982/01/01 · crossref created 2017/03/16 · crossref deposited 2021/05/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/15 · crossref indexed 2026/08/03
To my teachers and friends 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 1 that some key principle from the high dimensional theory would extend, at least to dimension four, and bring with it the beautiful adherence of topology to algebra familiar in dimensions greater than or equal to five. There is such a principle. It is a homotopy theoretic criterion for imbedding (relatively) a topological 2-handle in a smooth four-dimensional manifold with boundary. The main impact, as outlined in 1, is to the classification of 1-connected 4-manifolds and topological end recognition. However, certain applications to nonsimply connected problems such as knot concordance are also obtained.