Geometry & Topology

Growth of Casson handles and transversality for ASD moduli spaces

Tsuyoshi Kato

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Abstract

In this paper we study the growth of Casson handles which appear inside smooth four-manifolds. A simply-connected and smooth four-manifold admits decompositions of its intersection form. Casson handles appear around one side of the end of them, when the type is even. They are parameterized by signed infinite trees and their growth measures some of the complexity of the smooth structure near the end. We show that with respect to some decompositions of the forms on the K3 surface, the corresponding Casson handles cannot be of bounded type in our sense. In particular they cannot be periodic. The same holds for all logarithmic transforms which are homotopically equivalent to the K3 surface. We construct Yang–Mills gauge theory over Casson handles of bounded type, and verify that transversality works over them.

Article information

Source
Geom. Topol., Volume 12, Number 3 (2008), 1265-1311.

Dates
Received: 12 May 2006
Revised: 22 April 2008
Accepted: 5 October 2007
First available in Project Euclid: 20 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513800097

Digital Object Identifier
doi:10.2140/gt.2008.12.1265

Mathematical Reviews number (MathSciNet)
MR2421128

Zentralblatt MATH identifier
1148.57022

Subjects
Primary: 57M30: Wild knots and surfaces, etc., wild embeddings 57R57: Applications of global analysis to structures on manifolds, Donaldson and Seiberg-Witten invariants [See also 58-XX]
Secondary: 14J80: Topology of surfaces (Donaldson polynomials, Seiberg-Witten invariants)

Keywords
Yang-Mills theory Casson handles transversality

Citation

Kato, Tsuyoshi. Growth of Casson handles and transversality for ASD moduli spaces. Geom. Topol. 12 (2008), no. 3, 1265--1311. doi:10.2140/gt.2008.12.1265. https://projecteuclid.org/euclid.gt/1513800097


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