Seiberg-Witten invariants of non-simple type and Einstein metrics

TL;DR

This paper constructs specific four-dimensional manifolds with non-trivial Seiberg-Witten moduli spaces that are not derived from almost complex structures, and demonstrates the existence of many such manifolds lacking Einstein metrics.

Contribution

It introduces new examples of 4-manifolds with non-simple Seiberg-Witten invariants and shows they do not admit Einstein metrics, expanding understanding of 4-manifold geometry.

Findings

Existence of 4-manifolds with non-trivial Seiberg-Witten moduli spaces not induced by almost complex structures

Construction of infinitely many non-homeomorphic 4-manifolds with prescribed invariants

Demonstration that these manifolds do not support Einstein metrics

Abstract

We construct examples of four dimensional manifolds with Spin$^c$-structures, whose moduli spaces of solutions to the Seiberg-Witten equations, represent a non-trivial bordism class of positive dimension, i.e. the Spin$^c$-structures are not induced by almost complex structures. As an application, we show the existence of infinitely many non-homeomorphic compact oriented 4-manifolds with free fundamental group and predetermined Euler characteristic and signature that do not carry Einstein metrics.