1983/01/01 by Simon Donaldson · 3 citations
Mathematics · #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Geometric Analysis and Curvature Flows
paper · pdf · doi:10.1090/s0273-0979-1983-15090-5
openalex publication_date 1983/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/07
Introduction, statement of result. To any compact oriented 4-manifold X there is associated a quadratic form Q, defined on the cohomology group H 2 (X;Z) by Q(a) = ( U ) [I]. Poincar duality requires that it be a "unimodular" form-given by a symmetric matrix of determinant 1 with respect to any base for the torsion free part of H 2 . It is known from arithmetic that there are many such forms that are positive definite and not equivalent (over the integers) to the standard form [4, Chapter 5]. The problem of finding which forms are realised by simply-connected 4-manifolds was raised, for example, in IfX is a smooth, compact, simply-connected oriented 4-manifold with the property that the associated form Q is positive definite, then Q is equivalent, over the integers, to the standard diagonal form.