Mod 2 Seiberg-Witten invariants of homology tori

TL;DR

This paper computes the mod 2 Seiberg-Witten invariants for spin 4-manifolds with homology like a 4-torus, linking the invariant to the cohomology ring structure, especially the cup product.

Contribution

It provides a method to determine the mod 2 Seiberg-Witten invariant based on the cohomology ring structure of homology tori, including examples.

Findings

Invariant depends on the 4-fold cup product on H^1(X)

Explicit calculation for examples of homology tori

Establishes relation between cohomology ring and Seiberg-Witten invariant

Abstract

We show that the mod 2 Seiberg-Witten invariant can be determined for a spin manifold X which has the same homology groups as the 4-torus. The value depends on the structure of the cohomology ring of X, and in particular on the 4-fold cup product on H^1(X). We also consider some examples of homology tori.