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

TL;DR

This paper completes the proof of Witten's conjecture relating Donaldson and Seiberg-Witten invariants for a broad class of four-manifolds, focusing on low-degree terms and using PU(2) monopole techniques.

Contribution

It provides a partial verification of Witten's conjecture by computing low-degree Donaldson invariants via Seiberg-Witten invariants using PU(2) monopole cobordism.

Findings

Confirmed Witten's conjecture for degrees ≤ c-2 in many four-manifolds.

Computed Donaldson invariants in terms of Seiberg-Witten invariants.

Utilized Chern class calculations and orientation comparisons.

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.