PU(2) monopoles. II: Top-level Seiberg-Witten moduli spaces and Witten's conjecture in low degrees

Abstract

In this article we complete the proof---for a broad class of four-manifolds---of Witten's conjecture that the Donaldson and Seiberg-Witten series coincide, at least through terms of degree less than or equal to c-2, where c is a linear combination of the Euler characteristic and signature of the four-manifold. This article is a revision of sections 4--7 of an earlier version, while a revision of sections 1--3 of that earlier version now appear in a separate companion article (math.DG/0007190). Here, we use our computations of Chern classes for the virtual normal bundles for the Seiberg-Witten strata from the companion article (math.DG/0007190), a comparison of all the orientations, and the PU(2) monopole cobordism to compute pairings with the links of level-zero Seiberg-Witten moduli subspaces of the moduli space of PU(2) monopoles. These calculations then allow us to compute low-degree Donaldson invariants in terms of Seiberg-Witten invariants and provide a partial verification of Witten's conjecture.

Journal fur die Reine und Angewandte Mathematik, to appear; 65 pages. Revision of sections 4-7 of version v1 (December 1997)

References in corpus (7)

• Superconformal invariance and the geography of four-manifolds
• PU(2) monopoles and relations between four-manifold invariants
• PU(2) Monopoles, I: Regularity, Uhlenbeck Compactness, and Transversality
• PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces
• Donaldson invariants and wall-crossing formulas. I: Continuity of gluing maps
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Cited by in corpus (12)

• An SO(3)-monopole cobordism formula relating Donaldson and Seiberg-Witten invariants
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• PU(2) monopoles and links of top-level Seiberg-Witten moduli spaces
• A theory of the invariants obtained from the moduli stacks of stable objects on a smooth polarized surface
• An SO(3)-version of 2-torsion instanton invariants
• Superconformal simple type and Witten's conjecture
• The SO(3) monopole cobordism and superconformal simple type
• What do Topologists want from Seiberg--Witten theory? (A review of four-dimensional topology for physicists)
• Donaldson = Seiberg-Witten from Mochizuki's formula and instanton counting
• A vanishing result for a Casson-type instanton invariant
• Degeneracy loci of families of Dirac operators
• Virtual Morse-Bott index, moduli spaces of pairs, and applications to topology of smooth four-manifolds