\(SO(3)\)-invariants for 4-manifolds with \(b_ 2^ +=1\). II
type: Item
instance of: scholarly article
title: \(SO(3)\)-invariants for 4-manifolds with \(b_ 2^ +=1\). II
zbMATH Open document ID: 0828.57013
author: John W. Morgan
author: D. Kotschick
published in: Journal of Differential Geometry
publication date: 1994-07-18T00:00:00Z
review text: This paper extends the definition of Donaldson polynomial invariants to the case of manifolds with \(b_1 = 0\) and \(b^+_2 = 1\) for \(SO (3)\)-bundles where \(w_2\) lifts to an integral class. The paper generalizes earlier work of the first author [Part I: Proc. Lond. Math. Soc., III. Ser. 63, No. 2, 426-448 (1991; Zbl 0699.53036)]. It completes the proof that the values of the invariants only depend on the chamber containing the self-dual harmonic 2-form for the metric used to define the ASD equation, and establishes more general properties of the difference of values as the self dual 2-form crosses a wall. It establishes the conjecture made there that the value of an invariant on every chamber is determined by its value on any one chamber, and the invariant is defined for all chambers. Still unresolved is finding an explicit wall crossing formula as given by \textit{S. K. Donaldson} [J. Differ. Geom. 26, 141-168 (1987; Zbl 0631.57010)] for \(SU (2)\)-bundles with \(c_2 = 1\). It conjectures that there are systematic formulae for these difference terms involving only the classes defining the wall and the self-intersection form of the manifold. The results here fill a gap in the earlier paper of the first author cited above, but by an altogether different approach. The new argument uses a generalized gluing construction for gluing concentrated ASD connections over \(S^4\) into not necessarily ASD connections on \(M\).
reviewed by: Terry Lawson
Mathematics Subject Classification ID: 57N13
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 590837
zbMATH Keywords: anti-self-dual
zbMATH Keywords: one-dimensional positive definite subspace of the second homology group
zbMATH Keywords: \(\text{spin}^ c\)-structure
zbMATH Keywords: Donaldson polynomial invariant
zbMATH Keywords: chamber
zbMATH Keywords: self-dual harmonic 2-form
zbMATH Keywords: ASD equation
zbMATH Keywords: wall crossing formula
zbMATH Keywords: self-intersection form
zbMATH Keywords: gluing construction
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214454879
OpenAlex ID: W1600741118
Wikidata QID: Q115179747
DOI: 10.4310/JDG/1214454879
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description: scientific article; zbMATH DE number 590837