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Growth of Casson handles and transversality for ASD moduli spaces

Tsuyoshi Kato

Geometry & Topology 12 (2008) 1265–1311

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.

Yang-Mills theory, Casson handles, transversality
Mathematical Subject Classification 2000
Primary: 57M30, 57R57
Secondary: 14J80
Received: 12 May 2006
Revised: 22 April 2008
Accepted: 5 October 2007
Published: 3 June 2008
Proposed: Simon Donaldson
Seconded: Ron Stern, Rob Kirby
Tsuyoshi Kato
Department of Mathematics
Faculty of Science
Kyoto University
606-8502 Kyoto