An instanton-invariant for 3-manifolds
type: Item
instance of: scholarly article
title: An instanton-invariant for 3-manifolds
zbMATH Open document ID: 0684.53027
DOI: 10.1007/BF01218578
author: Andreas Floer
published in: Communications in Mathematical Physics
publication date: 1988-01-01T00:00:00Z
review text: In this seminal paper the author introduces new invariants for 3- manifolds defined by the instanton solutions of the Yang-Mills equations. Let M be an oriented homology 3-sphere and choose a Riemannian metric on M. Let X be the Riemannian product \(M\times {\mathbb{R}}\). Any finite energy solution of the SU(2) instanton equations over X (i.e. connection on an SU(2) bundle over X with anti-self-dual curvature tensor, and whose curvature lies in \(L^ 2)\) is asymptotically flat at the ends of X. Flat SU(2) connections over M correspond to representations of \(\pi_ 1(M)\) in SU(2). Suppose for the moment that there is only a finite set \(S\cup \{\theta \}\) of conjugacy classes of such representations, and also that apart from the trivial representation \(\theta\) these are ``acyclic'', in the sense that the cohomology of M in the coefficient system defined by the associated flat \({\mathfrak su}(2)\) bundle is trivial. Then the equivalence classes of instantons, modulo bundle automorphism and the action of translations on X, are parametrised by moduli spaces \({\mathcal M}_{\alpha \beta}\), for \(\alpha\),\(\beta\in S\cup \{\theta \}\). Generically these will be countable unions of smooth manifolds. There is a function d: \(S\to {\mathbb{Z}}/8\) such that the components of \({\mathcal M}_{\alpha,\beta}\) have dimension d(\(\alpha)\)-d(\(\beta)\)-1, mod 8. In this situation the author's invariants are groups formed from the cohomology of a complex whose chains are freely generated by the set S: \(C_*=Maps(S,{\mathbb{Z}})\), which is \({\mathbb{Z}}/8\)-graded by d. The differential \(\partial\) in the complex is defined so that the matrix element \(\partial_{\alpha,\beta}\) of a pair \(\alpha\) \(\beta\) with \(d(\alpha)=d(\beta)+1\) is given by the number of elements in the 0- dimensional piece of \({\mathcal M}_{a\beta}\), counted with appropriate signs. (To formulate this definition the author shows first that, in this situation, the 0-dimensional piece must be com…
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 81T08
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 4120934
zbMATH Keywords: invariants for 3-manifolds
zbMATH Keywords: instanton solutions of the Yang-Mills equations
zbMATH Keywords: homology 3-sphere
zbMATH Keywords: anti-self-dual curvature tensor
zbMATH Keywords: Flat SU(2) connections
zbMATH Keywords: representations of \(\pi _ 1(M)\) in SU(2)
zbMATH Keywords: moduli spaces
zbMATH Keywords: Chern-Simons function on the space of connections
zbMATH Keywords: Casson invariant
zbMATH Keywords: Euler characteristic of Floer homology
reviewed by: S. K. Donaldson
MaRDI profile type: Publication
Wikidata QID: Q57309150
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description: scientific article; zbMATH DE number 4120934