Monopoles over 4–manifolds containing long necks I

Frøyshov, Kim A¹

¹ Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Geometry & topology, Tome 9 (2005) no. 1, pp. 1-93.

Résumé

We study moduli spaces of Seiberg–Witten monopoles over ${spin}^c$ Riemannian 4–manifolds with long necks and/or tubular ends. This first part discusses compactness, exponential decay, and transversality. As applications we prove two vanishing theorems for Seiberg–Witten invariants.

Détail

DOI : 10.2140/gt.2005.9.1

Keywords: Floer homology, Seiberg–Witten, Bauer–Furuta, compactness, monopoles

Bibliographie

[1] R A Adams, Sobolev spaces, Pure and Applied Mathematics 65, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London (1975)

[2] M F Atiyah, The index theorem for manifolds with boundary, from: "Seminar on the Atiyah–Singer index theorem" (editor R S Palais), Annals of Mathematics Studies 57, Princeton University Press (1965)

[3] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43

[4] C Bär, Zero sets of solutions to semilinear elliptic systems of first order, Invent. Math. 138 (1999) 183

[5] S Bauer, A stable cohomotopy refinement of Seiberg–Witten invariants II, Invent. Math. 155 (2004) 21

[6] S Bauer, Refined Seiberg–Witten invariants, from: "Different faces of geometry", Int. Math. Ser. (N. Y.) 3, Kluwer/Plenum, New York (2004) 1

[7] S Bauer, M Furuta, A stable cohomotopy refinement of Seiberg–Witten invariants I, Invent. Math. 155 (2004) 1

[8] J Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press (1960)

[9] S K Donaldson, The Seiberg–Witten equations and 4–manifold topology, Bull. Amer. Math. Soc. $($N.S.$)$ 33 (1996) 45

[10] S K Donaldson, Floer homology groups in Yang–Mills theory, Cambridge Tracts in Mathematics 147, Cambridge University Press (2002)

[11] S K Donaldson, P B Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press (1990)

[12] R Fintushel, R J Stern, Immersed spheres in 4–manifolds and the immersed Thom conjecture, Turkish J. Math. 19 (1995) 145

[13] A Floer, An instanton-invariant for 3–manifolds, Comm. Math. Phys. 118 (1988) 215

[14] K A Frøyshov, Monopole Floer homology for rational homology 3–spheres, in preparation

[15] K A Frøyshov, The Seiberg–Witten equations and four-manifolds with boundary, Math. Res. Lett. 3 (1996) 373

[16] R S Hamilton, Harmonic maps of manifolds with boundary, Lecture Notes in Mathematics 471, Springer (1975)

[17] L Hörmander, The analysis of linear partial differential operators III, Grundlehren der Mathematischen Wissenschaften, Springer (1985)

[18] M Ishida, C Lebrun, Curvature, connected sums, and Seiberg–Witten theory, Comm. Anal. Geom. 11 (2003) 809

[19] J L Kazdan, Unique continuation in geometry, Comm. Pure Appl. Math. 41 (1988) 667

[20] P B Kronheimer, T S Mrowka, Floer homology for Seiberg–Witten monopoles, book in preparation

[21] P B Kronheimer, T S Mrowka, The genus of embedded surfaces in the projective plane, Math. Res. Lett. 1 (1994) 797

[22] P Kronheimer, T Mrowka, P Ozsváth, Z Szabó, Monopoles and lens space surgeries, Ann. of Math. $(2)$ 165 (2007) 457

[23] H B Lawson Jr, M L Michelsohn, Spin geometry, Princeton Mathematical Series 38, Princeton University Press (1989)

[24] R B Lockhart, R C Mcowen, Elliptic differential operators on noncompact manifolds, Ann. Scuola Norm. Sup. Pisa Cl. Sci. $(4)$ 12 (1985) 409

[25] M Marcolli, Seiberg–Witten–Floer homology and Heegaard splittings, Internat. J. Math. 7 (1996) 671

[26] M Marcolli, B L Wang, Equivariant Seiberg–Witten Floer homology, Comm. Anal. Geom. 9 (2001) 451

[27] D Mcduff, D Salamon, $J$–holomorphic curves and quantum cohomology, University Lecture Series 6, American Mathematical Society (1994)

[28] J W Morgan, Z Szabó, C H Taubes, A product formula for the Seiberg–Witten invariants and the generalized Thom conjecture, J. Differential Geom. 44 (1996) 706

[29] L I Nicolaescu, Notes on Seiberg–Witten theory, Graduate Studies in Mathematics 28, American Mathematical Society (2000)

[30] D Salamon, Removable singularities and a vanishing theorem for Seiberg–Witten invariants, Turkish J. Math. 20 (1996) 61

[31] M E Taylor, Partial differential equations, Texts in Applied Mathematics 23, Springer (1996)

[32] K K Uhlenbeck, Connections with $L^{p}$ bounds on curvature, Comm. Math. Phys. 83 (1982) 31