The orientation of Yang-Mills moduli spaces and 4-manifold topology
type: Item
instance of: scholarly article
title: The orientation of Yang-Mills moduli spaces and 4-manifold topology
zbMATH Open document ID: 0683.57005
published in: Journal of Differential Geometry
publication date: 1987-01-01T00:00:00Z
review text: This important paper has two purposes. The first is to generalize the author's previous results concerning intersection forms of closed smooth 4-manifolds. The second is the construction of a canonical orientation of the Yang-Mills moduli space. Besides its application to the present paper, this is a fundamental information which is used in various other applications of gauge theory to 4-manifolds. The results concerning the intersection form are the following. Theorem 1: If X is a closed, oriented smooth 4-manifold whose intersection form Q: \(H^ 2(X;{\mathbb{Z}})/Torsion\to {\mathbb{Z}}\) is negative definite, then the form is equivalent over the integers to the standard form \((-1)\oplus (- 1)\oplus...\oplus (-1).\) Theorem 2: Let X be a closed, oriented smooth 4- manifold with the following three properties: (i) \(H_ 1(X;{\mathbb{Z}})\) has no 2-torsion. (ii) The intersection form Q on \(H^ 2(X)/Torsion\) has a positive part of rank 1 or 2. (iii) The intersection form is even. Then Q is equivalent over the integers to a hyperbolic form of rank 2 or 4. Theorem 1 generalizes the author's corresponding result for 1-connected manifolds [ibid. 18, 279-315 (1983; Zbl 0507.57010)], Theorem 2 was also proved previously by the author for 1-connected manifolds [ibid. 24, 275- 341 (1986; Zbl 0635.57007)]. The methods of the proofs are similar to those in the earlier mentioned papers and based on a detailed analysis of the moduli space of anti-self-dual (ASD) connection on certain SU(2) bundles over X. These moduli spaces are not compact but can be compactified by adding strata involving the moduli spaces coming from SU(2) bundles with smaller second Chern class. If X is not simply connected, the lowest stratum is more complicated creating new difficulties. These are resolved by means of a perturbation argument. This is the content of section 2. The construction of the canonical orientation on the moduli space of Yang-Mills instantons (ASD connections) is given in section 3…
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R80
Mathematics Subject Classification ID: 58J60
Mathematics Subject Classification ID: 58E15
Mathematics Subject Classification ID: 81T08
zbMATH DE Number: 4119253
zbMATH Keywords: moduli space of anti-self-dual connections on SU(2) bundles
zbMATH Keywords: intersection forms of closed smooth 4-manifolds
zbMATH Keywords: canonical orientation of the Yang- Mills moduli space
zbMATH Keywords: gauge theory
zbMATH Keywords: index theory for differential operators
zbMATH Keywords: complex Kähler surface
author: S. K. Donaldson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214441485
OpenAlex ID: W1783633298
Wikidata QID: Q115184368
DOI: 10.4310/JDG/1214441485
description: scientific article; zbMATH DE number 4119253