Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary
type: Item
instance of: scholarly article
title: Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary
zbMATH Open document ID: 1025.57016
DOI: 10.1016/S0001-8708(02)00030-0
author: Zoltán Szabó
author: Peter Ozsváth
published in: Advances in Mathematics
publication date: 2003-04-02T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0110170
review text: In the papers [Holomorphic disks and topological invariants for closed three-manifolds, to appear in Ann. of Math.] and [Holomorphic disks and three-manifold invariants: properties and applications, to appear in Ann. of Math.], the authors introduced a beautiful new Floer homology theory with tremendous applications; this they extended to a 4-dimensional theory in [Holomorphic triangles and invariants for smooth four-manifolds, preprint (2001)]. The current article presents a number of impressive applications of this theory in dimensions 3 and 4. The authors begin by defining a numerical invariant for a rational homology sphere together with a Spin\(^c\) structure, which they call a correction term. This invariant is an analog of that of \textit{K. A. Frøyshov} defined in [Math. Res. Lett. 3, 373-390 (1996; Zbl 0872.57024)]. This invariant is additive with respect to connected sums and is a rational homology cobordism invariant. The authors show that for an integral homology sphere \(Y\), this correction term represents the difference between the Casson invariant of the three-manifold and the Euler characteristic of \(HF^+_{red}(Y)\). For an integral homology 3-sphere \(Y\), the authors define the \textit{complexity} of \(Y\) to be the rank of \(HF^+_{red}(Y)\). Previous results giving lower bounds for the complexity of a \(Y\) and for the manifold resulting from \(1/n\)-Dehn surgery on a knot in \(Y\) are improved. An integral homology 3-sphere is said to be invisible if both the complexity and the correction term vanish; it is shown that for any knot \(K\) in an invisible homology sphere \(Y\), if the Alexander polynomial of the knot is nontrivial then any nontrivial surgery on the knot yields a 3-manifold which is not invisible. Some restrictions on knots in \(S^3\) with lens space surgeries are obtained. A table of allowed Alexander polynomials for knots in \(S^3\) with lens space surgeries is provided. Calculations are performed for the complexity, the correcti…
Mathematics Subject Classification ID: 57M27
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 57M25
Mathematics Subject Classification ID: 57N10
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 1891234
zbMATH Keywords: Floer homology
zbMATH Keywords: intersection forms
zbMATH Keywords: lens space surgeries
zbMATH Keywords: Casson invariant
reviewed by: Cynthia L. Curtis
MaRDI profile type: Publication
arXiv ID: math/0110170
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OpenAlex ID: W1975428572
description: scientific article