On Casson-type instanton moduli spaces over negative definite four-manifolds

Abstract

Recently Andrei Teleman considered instanton moduli spaces over negative definite four-manifolds $X$ with $b_2(X)\ge 1$. If $b_2(X)$ is divisible by four and $b_1(X)=1$ a gauge-theoretic invariant can be defined; it is a count of flat connections modulo the gauge group. Our first result shows that if such a moduli space is non-empty and the manifold admits a connected sum decomposition $X \cong X_1 # X_2$ then both $b_2(X_1)$ and $b_2(X_2)$ are divisible by four; this rules out a previously natural appearing source of 4-manifolds with non-empty moduli space. We give in some detail a construction of negative definite 4-manifolds which we expect will eventually provide examples of manifolds with non-empty moduli space.

This version contains many improvements to the layout suggested by the referee; accepted for publication in Quarterly Journal of Mathematics

References in corpus (2)

• Harmonic sections in sphere bundles, normal neighborhoods of reduction loci, and instanton moduli spaces on definite 4-manifolds
• A vanishing result for a Casson-type instanton invariant

Cited by in corpus (2)

• A vanishing result for a Casson-type instanton invariant
• Seiberg-Witten equations, end-periodic Dirac operators, and a lift of Rohlin's invariant