An inequality for the \(h\)-invariant in instanton Floer theory.
type: Item
instance of: scholarly article
title: An inequality for the \(h\)-invariant in instanton Floer theory.
zbMATH Open document ID: 1045.57019
DOI: 10.1016/S0040-9383(03)00049-1
author: Kim A. Frøyshov
published in: Topology
publication date: 2004-03-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0111038
review text: In [Topology 41, 525--552 (2002; Zbl 0999.57032)], the author defines a non-trivial homomorphism \(h\) from the homology cobordism group of oriented integral homology \(3\)-spheres to the integers, using instanton Floer theory. In this paper some new properties of this invariant are obtained. Let \(X\) be a smooth, compact, oriented \(4\)-manifold with a homology sphere \(Y\) as boundary and \(b_2^+(X)=1\). Let \(\Sigma\subset X\) be a closed oriented embedded surface of genus \(g\) and self-intersection \(\Sigma\cdot\Sigma =1\). Let \(L\subset H^2(X;\mathbb{Z})/\text{torsion}\) be the sublattice annihilating \([\Sigma]\). Then the main result of the paper is the following formula: \[ h(Y)+\lceil g/2\rceil \geq e(L), \] where \(\lceil x \rceil\) denotes the smallest integer \(\geq x\) and \(e(L)\) is an invariant of the definite lattice \(L\) defined as follows: First \(w\in L\) is extremal if \(| w^2|\leq| z^2| \) for any \(z\in w + 2L\). Then \(e(L)\) is defined as the supremum of the quantities \(\lceil | w^2| -m \rceil/4\), where \(w\) is extremal, \(m\geq 0\), \(w^2\equiv m\pmod 2\) and \[ \eta(L,w,a,m)=\sum_{z\in w+2L} (-1)^{((z+w)/2)^2} (a\cdot z)^m \neq 0, \] for some \(a\in\Hom(L,\mathbb{Z})\). Note that \(e(L)\) has nice properties: \(e(L) \geq 0\), \(e(L)=0\) if \(L\) is diagonal, \(e(-k E_8\oplus L)=k\) if \(L\) is diagonal. The argument of proof consists on considering the blow-up \(X\# \overline{\mathbb C P}^2\) and the surfaces \(\Sigma^{\pm} = \Sigma \# (\pm E)\), of self-intersection zero, where \(E\) is the exceptional divisor. Then consider the moduli space of ASD connections for the family of metrics which stretch the metric along \(\Sigma^{\pm}\) for \(t\to \pm \infty\). Cut it down with a cycle of the form \(x^{n-2q} (x_1^2-4)^q\) until it is \(1\)-dimensional. Here \(x\) and \(x_1\) are cycles of the moduli space associated under the \(\mu\)-map to a fixed point in \(X\) well away from \(\Sigma^{\pm}\) and a point close to \(\Sigma^{\pm}\). Th…
reviewed by: Vicente Muñoz
Mathematics Subject Classification ID: 57R58
Mathematics Subject Classification ID: 57R57
zbMATH DE Number: 2056782
zbMATH Keywords: Floer homology
zbMATH Keywords: instantons
zbMATH Keywords: 4-manifolds
MaRDI profile type: Publication
Wikidata QID: Q124819823
arXiv ID: math/0111038
description: scientific article