Holomorphic disks and topological invariants for closed three-manifolds
type: Item
instance of: scholarly article
title: Holomorphic disks and topological invariants for closed three-manifolds
zbMATH Open document ID: 1073.57009
author: Zoltán Szabó
author: Peter Ozsváth
published in: Annals of Mathematics. Second Series
publication date: 2005-03-11T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0101206
review text: The paper under review is to define topological invariants for closed oriented 3-manifolds \(Y\) with a Spin\(^c\) structure \(s\) via the Lagrangian Floer homology construction. For a Heegaard surface \(\Sigma\) of \(Y\), the \(g\)-fold symmetric product \(\text{Sym}^g(\Sigma)\) is a complex manifold, where \(g\) is the genus of the surface \(\Sigma\). The \(g\)-tuples of attaching circles \(\alpha =\{\alpha_1, \dots, \alpha_g\}\) and \(\beta =\{\beta_1, \dots, \beta_g\}\) for two handlebodies give a pair of smoothly embedded \(g\)-tori \(T_{\alpha}\) and \(T_{\beta}\) in \(\text{Sym}^g(\Sigma)\). The chain groups \(\hat{CF}(\alpha, \beta, s)\) are abelian groups freely generated by the intersection points \(T_{\alpha} \cap T_{\beta}\) with relative index defined by the Maslov index. The boundary map measures the number of \(J\)-holomorphic disks in \(\text{Sym}^g(\Sigma)\) with relative index 1 along two intersection points. Such a definition provides a chain complex for rational homology 3-spheres; by Theorem 4.1 of the paper its homology is the Floer homology \(\hat{HF}(\alpha, \beta, s)\). Taking an integral lift on the Maslov index, one can also formulate other chain groups \(CF^{\infty}(\alpha, \beta, s)\) generated from \([x, i]\) for \(x\in T_{\alpha} \cap T_{\beta}\) and \(i\) an integer. The similar boundary map makes \(CF^{\infty}(\alpha, \beta, s)\) a chain complex and hence there is a Floer homology \(HF^{\infty}(\alpha, \beta, s)\). The subgroup of \(CF^{\infty}(\alpha, \beta, s)\) with negative indices forms another chain subgroup \(CF^{-}(\alpha, \beta, s)\); denote its quotient group \(CF^{\infty}/CF^{-}(\alpha, \beta, s)\) by \(CF^{+}(\alpha, \beta, s)\). Hence there are four Floer homology groups. The similar chain groups with fixed orientation are defined for positive first Betti number of 3-manifolds. The paper is devoted to showing that the Floer homology groups are topological invariants, i.e. they are invariant under Heegaard moves (isotopie…
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 32Q65
Mathematics Subject Classification ID: 57R15
zbMATH DE Number: 2144173
zbMATH Keywords: Heegaard diagram
zbMATH Keywords: holomorphic disk
zbMATH Keywords: holomorphic triangle
zbMATH Keywords: Floer homology
zbMATH Keywords: spin\(^c\) structure
zbMATH Keywords: Heegaard moves
Wikidata QID: Q29026387
MaRDI profile type: Publication
OpenAlex ID: W2053508424
arXiv ID: math/0101206
DOI: 10.4007/ANNALS.2004.159.1027
description: scientific article