Geometry & Topology

Growth of Casson handles and transversality for ASD moduli spaces

Tsuyoshi Kato

Full-text: Open access


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

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

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

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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)

Yang-Mills theory Casson handles transversality


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.

Export citation


  • Ž Bižaca, R E Gompf, Elliptic surfaces and some simple exotic $\mathbb{R}^4$s, J. Differential Geom. 43 (1996) 458–504
  • W Chen, S Kwasik, Symmetric homotopy K3 surfaces
  • S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford University Press (1990)
  • A Floer, An instanton-invariant for 3–manifolds, Comm. Math. Phys. 118 (1988) 215–240
  • M H Freedman, The topology of four-dimensional manifolds, J. Differential Geom. 17 (1982) 357–453
  • R Friedman, J W Morgan, Complex versus differentiable classification of algebraic surfaces, Topology Appl. 32 (1989) 135–139
  • R Friedman, J W Morgan, Smooth four-manifolds and complex surfaces, Ergebnisse series 27, Springer, Berlin (1994)
  • D Gilbarg, N S Trudinger, Elliptic partial differential equations of second order, second edition, Grundlehren series 224, Springer, Berlin (1983)
  • R E Gompf, T S Mrowka, Irreducible 4–manifolds need not be complex, Ann. of Math. $(2)$ 138 (1993) 61–111
  • T Kato, ASD moduli spaces over four-manifolds with tree-like ends, Geom. Topol. 8 (2004) 779–830
  • T Kato, Spectral analysis on tree like spaces from gauge theoretic view points, from: “Discrete geometric analysis”, Contemp. Math. 347, Amer. Math. Soc., Providence, RI (2004) 113–129
  • P B Kronheimer, Instanton invariants and flat connections on the Kummer surface, Duke Math. J. 64 (1991) 229–241
  • T Matumoto, On diffeomorphisms of a $K3$ surface, from: “Algebraic and topological theories (Kinosaki, 1984)”, Kinokuniya, Tokyo (1986) 616–621
  • C H Taubes, The Seiberg–Witten invariants and symplectic forms, Math. Res. Lett. 1 (1994) 809–822
  • C H Taubes, More constraints on symplectic forms from Seiberg–Witten invariants, Math. Res. Lett. 2 (1995) 9–13
  • K K Uhlenbeck, Connections with $L^p$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31–42
  • K K Uhlenbeck, Removable singularities in Yang–Mills fields, Comm. Math. Phys. 83 (1982) 11–29
  • C T C Wall, On the orthogonal groups of unimodular quadratic forms II, J. Reine Angew. Math. 213 (1963/1964) 122–136