Growth of Casson handles and transversality for ASD moduli spaces
type: Item
instance of: scholarly article
title: Growth of Casson handles and transversality for ASD moduli spaces
zbMATH Open document ID: 1148.57022
author: Tsuyoshi Kato
published in: Geometry \& Topology
publication date: 2008-06-13T00:00:00Z
review text: After proving that various exotic \(4\)-manifolds exist it is natural to measure the complexity of these \(4\)-manifolds. Topologically, any simply-connected \(4\)-manifold with intersection form \(-2k\,E_8\oplus \ell H\) admits an open set homeomorphic to a punctured connected sum of \(\ell\) copies of \(S^2\times S^2\). By the work of Donaldson this homeomorphism cannot be a diffeomorphism and so we see that this open set is obtained by attaching \(2\ell\) Casson handles to the \(4\)-ball. Clearly all of these Casson handles cannot be standard, so one may ask how complicated are the resulting Casson handles. In [Geom. Topol. 8, 779--830 (2004; Zbl 1064.57022)], the present author introduced the notion of a Casson handle of bounded type. These Casson handles arise from sub trees of homogeneous signed trees. Homogeneous signed trees are defined inductively by calling the natural numbers with adjacent edges having a common sign type \((T_1,\pm)\) trees and setting type \((T_{n+1},\pm)\) trees to be obtained by attaching type \((T_n,\pm)\) trees (all with the same sign) to the vertices of a type \((T_1,\pm)\) tree. In this paper the author describes why Seiberg-Witten theory is insufficient to address the question of the existence bounded type Casson handle decompositions. He then addresses the transversality issues involved in constructing a Donaldson invariant for these open \(4\)-manifolds. (He addressed the index problem in [Contemporary Mathematics 347, 113--129 (2004; Zbl 1072.14052)]). Using his results he is able to prove for example that some of these open sets in single log transforms of a K3 surface cannot be constructed entirely from bounded type Casson handles.
reviewed by: David Auckly
Mathematics Subject Classification ID: 57M30
Mathematics Subject Classification ID: 57R57
Mathematics Subject Classification ID: 14J80
Mathematics Subject Classification ID: 57N13
zbMATH DE Number: 5288421
zbMATH Keywords: Casson handles
zbMATH Keywords: Gauge theory
MaRDI profile type: Publication
full work available at URL: gt.2008.12.1265
OpenAlex ID: W2068803352
cites work: Elliptic surfaces and some simple exotic \(\mathbb{R}^ 4\)'s
cites work: Symmetric symplectic homotopy K 3 surfaces
cites work: Q3998602
cites work: An instanton-invariant for 3-manifolds
cites work: The topology of four-dimensional manifolds
cites work: Complex versus differentiable classification of algebraic surfaces
cites work: Q4289665
cites work: Q5187597
cites work: Irreducible 4-manifolds need not be complex
cites work: ASD moduli spaces over four-manifolds with tree-like ends
cites work: Q4656874
cites work: Instanton invariants and flat connections on the Kummer surface
cites work: Q4283542
cites work: The Seiberg-Witten invariants and symplectic forms
cites work: More constraints on symplectic forms from Seiberg-Witten invariants
cites work: Connections with \(L^ p \)bounds on curvature
cites work: Removable singularities in Yang-Mills fields
cites work: Q5507718
DOI: 10.2140/GT.2008.12.1265
Recommended article: Constructions of smooth 4-manifolds
Recommended article: Q4332782
Recommended article: The Donaldson invariant and embedded 2-spheres.
Recommended article: Q3414433
Recommended article: ASD moduli spaces over four-manifolds with tree-like ends
Recommended article: Q3629273
Recommended article: A note on the wick of the Casson handles
Recommended article: Almost complex 4-manifolds with vanishing first Chern class
Recommended article: Abelian monopoles and complex geometry
Recommended article: Q4213298
description: scientific article; zbMATH DE number 5288421