Growth of Casson handles and transversality for ASD moduli spaces

Kato, Tsuyoshi

doi:10.2140/gt.2008.12.1265

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Growth of Casson handles and transversality for ASD moduli spaces

Tsuyoshi Kato

Geometry & Topology 12 (2008) 1265–1311

DOI: 10.2140/gt.2008.12.1265

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.

Keywords

Yang-Mills theory, Casson handles, transversality

Mathematical Subject Classification 2000

Primary: 57M30, 57R57

Secondary: 14J80

References

Publication

Received: 12 May 2006

Revised: 22 April 2008

Accepted: 5 October 2007

Published: 3 June 2008

Proposed: Simon Donaldson

Seconded: Ron Stern, Rob Kirby

Authors

Tsuyoshi Kato
Department of Mathematics Faculty of Science Kyoto University 606-8502 Kyoto Japan

Volume 12, issue 3 (2008)