## Geometry & Topology

### ASD moduli spaces over four-manifolds with tree-like ends

Tsuyoshi Kato

#### Abstract

In this paper we construct Riemannian metrics and weight functions over Casson handles. We show that the corresponding Atiyah–Hitchin–Singer complexes are Fredholm for some class of Casson handles of bounded type. Using these, the Yang–Mills moduli spaces are constructed as finite dimensional smooth manifolds over Casson handles in the class.

#### Article information

Source
Geom. Topol., Volume 8, Number 2 (2004), 779-830.

Dates
Received: 16 October 2001
Revised: 29 March 2004
Accepted: 29 April 2004
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883417

Digital Object Identifier
doi:10.2140/gt.2004.8.779

Mathematical Reviews number (MathSciNet)
MR2087070

Zentralblatt MATH identifier
1064.57022

Keywords
Yang–Mills theory Casson handles

#### Citation

Kato, Tsuyoshi. ASD moduli spaces over four-manifolds with tree-like ends. Geom. Topol. 8 (2004), no. 2, 779--830. doi:10.2140/gt.2004.8.779. https://projecteuclid.org/euclid.gt/1513883417

#### References

• M Atiyah, N Hitchin, I Singer, Self-duality in four dimensional Riemannian geometry, Proc. R. Soc. London 362 (1978) 425 – 461
• M Atiyah, V Patodi, I Singer, Spectral asymmetry in Riemannian geometry, I, II and III, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 – 69; 78 (1975) 405–432; 79 (1976) 71–99
• M Atiyah, V Patodi, I Singer, Spectral asymmetry in Riemannian geometry, I, II and III, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 – 69; 78 (1975) 405–432; 79 (1976) 71–99
• M Atiyah, V Patodi, I Singer, Spectral asymmetry in Riemannian geometry, I, II and III, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43 – 69; 78 (1975) 405–432; 79 (1976) 71–99
• $\check{\text{Z}}$ Bi$\check{\text{z}}$aca, A re-imbedding algorithm for Casson handles, Trans. A.M.S. 345 (1994) 435–510
• $\check{\text{Z}}$ Bi$\check{\text{z}}$aca, An explicit family of exotic Casson handles, Proc. of the A.M.S. 123 (1995) 1297–1302
• $\check{\text{Z}}$ Bi$\check{\text{z}}$aca, R Gompf, Elliptic surfaces and some simple exotic $\mathbb{R}^4$'s, Journal of Differential Geometry 43 (1996) 458–504
• A Casson, Three lectures on new infinite constructions in $4$–dimensional manifolds,
• A Connes, Non-commutative geometry, Academic press (1994)
• S Demichelis, M Freedman, Uncountably many exotic $\mathbb{R}^4$'s in standard $4$–space, Journal of Differential Geometry 35 (1992) 219–254
• S Donaldson, Polynomial invariants for smooth $4$–manifolds, Topology 29 (1990) 257–315
• S Donaldson, An application of gauge theory to four-dimensional topology, Journal of Differential Geometry 18 (1983) 279–315
• S Donaldson, P Kronheimer, The geometry of four-manifolds, Oxford University Press (1990)
• Y Eliashberg, Topological characterization of Stein manifolds of dimension $>2$, Internat. J. Math. 98 (1990) 29–46
• A Floer, An instanton invariant for $3$ manifolds, Comm.Math.Phy. 118 (1988) 215–240
• S Freed, K Uhlenbeck Instantons and four–manifolds, Springer, 2nd edition (1991)
• M Freedman, The topology of four-dimensional manifolds, Journal of Differential Geometry 17 (1982) 357–454
• M Freedman, F Quinn, Topology of $4$–manifolds, Princeton Univ. Press (1990)
• M Freedman, L Taylor, A universal smoothing of four-space, Journal of Differential Geometry 24 (1986) 69–78
• R Gompf, Handlebody construction of Stein surfaces, Ann. Math. 148 (1998) 619–693
• R Gompf, A Stipsicz, $4$–manifolds and Kirby calculus, Volume 20 of Graduate Studies in Mathematics, AMS (1999)
• M Gromov, H Lawson, Positive scalar curvature on complete Riemannian manifolds, I.H.E.S Publ. Math. 58 (1983) 295–408
• T Kato, Spectral analysis on tree like spaces from gauge theoretic aspects, to appear in the Proceedings of Discrete Geometric Analysis, Contemporary Math., AMS
• R Kirby, The topology of $4$–manifolds, Volume 1374 of Lecture Notes in Mathematics, Springer–Verlag
• V Kondrat'ev Boundary value problems for elliptic equations in domains with conical or angular points, Trans. Moscow Math. Soc. 16 (1967)
• R Lockhart, R McOwen, Elliptic differential operators on non compact manifolds, Ann. Sci. Ec. Norm. Sup. Pisa 12 (1985) 409–446
• J Milnor On simply connected $4$–manifolds, Symp. Int. Top. Alg. Mexico (1958) 122–128
• F Quinn, Ends of maps in dimension $4$ and $5$ Journal of Differential Geometry 17 (1982) 503–521
• C Taubes, Gauge theory on asymptotically periodic 4 manifolds, Journal of Differential Geometry 25 (1987) 363–430
• C Taubes, Casson's invariant and gauge theory, Journal of Differential Geometry 31 (1990) 547–599
• L Taylor An invariant of smooth $4$–manifolds, Geometry and Topology 1 (1997) 71–89
• K Uhlenbeck Connections with $L^p$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31–42
• K Uhlenbeck Removable singularities in Yang–Mills fields, C.M.P. 83 (1982) 11–29
• F Warner Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, Springer–Verlag (1971).