An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants

Abstract

We prove an analogue of the Kotschick-Morgan conjecture in the context of SO(3) monopoles, obtaining a formula relating the Donaldson and Seiberg-Witten invariants of smooth four-manifolds using the SO(3)-monopole cobordism. The main technical difficulty in the SO(3)-monopole program relating the Seiberg-Witten and Donaldson invariants has been to compute intersection pairings on links of strata of reducible SO(3) monopoles, namely the moduli spaces of Seiberg-Witten monopoles lying in lower-level strata of the Uhlenbeck compactification of the moduli space of SO(3) monopoles [arXiv:dg-ga/9710032]. In this monograph, we prove --- modulo a gluing theorem which is an extension of our earlier work in [arXiv:math/9907107] --- that these intersection pairings can be expressed in terms of topological data and Seiberg-Witten invariants of the four-manifold. This conclusion is analogous to the Kotschick-Morgan conjecture concerning the wall-crossing formula for Donaldson invariants of a four-manifold with $b_2^{+}=1$; that wall-crossing formula and the resulting structure of Donaldson invariants for four-manifolds with $b_2^{+}=1$ were established, assuming the Kotschick-Morgan conjecture, by Goettsche [arXiv:alg-geom/9506018] and Goettsche and Zagier [arXiv:alg-geom/9612020]. In this monograph, we reduce the proof of the Kotschick-Morgan conjecture to an extension of previously established gluing theorems for anti-self-dual SO(3) connections (see [arXiv:math/9812060] and references therein). Since the first version of our monograph was circulated, applications of our results have appeared in the proof of Property P for knots by Kronheimer and Mrowka [arXiv:math/0311489] and work of Sivek on Donaldson invariants for symplectic four-manifolds [arXiv:1301.0377].

x + 229 pages

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Cited by in corpus (13)

• Chern-Simons Functional and the Homology Cobordism Group
• The SO(3) monopole cobordism and superconformal simple type
• Superconformal simple type and Witten's conjecture
• Characteristic classes via 4-dimensional gauge theory
• Enumerative invariants and wall-crossing formulae in abelian categories
• Donaldson invariants of symplectic manifolds
• On higher rank instantons & the monopole cobordism program
• A Smooth Compactification of the Moduli Space of Instantons and Its Application
• Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds
• Gluing in geometric analysis via maps of Banach manifolds with corners and applications to gauge theory
• The SO(3) Vortex Equations over Orbifold Riemann Surfaces
• $PU(N)$ monopoles, higher rank instantons, and the monopole invariants
• The universal Kobayashi-Hitchin correspondence on Hermitian manifolds