Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant

Abstract

We introduce a gauge-theoretic integer lift of the Rohlin invariant of a smooth 4-manifold X with the homology of $S^1\times S^3$. The invariant has two terms; one is a count of solutions to the Seiberg-Witten equations on X, and the other is essentially the index of the Dirac operator on a non-compact manifold with end modeled on the infinite cyclic cover of X. Each term is metric (and perturbation) dependent, and we show that these dependencies cancel as the metric and perturbation vary in a 1-parameter family.

Revision after referee report; section on negative definite manifolds revised. To appear in JDG

References in corpus (4)

• Monopole Floer homology for rational homology 3-spheres
• Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds
• Dirac operators on manifolds with periodic ends
• On Casson-type instanton moduli spaces over negative definite four-manifolds

Cited by in corpus (1)

• A vanishing result for a Casson-type instanton invariant