ON CASSON-TYPE INSTANTON MODULI SPACES OVER NEGATIVE DEFINITE 4-MANIFOLDS

Recently Andrei Teleman considered instanton moduli spaces over negative definite 4-manifolds X with b(2)(X) >= 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 congruent to X(1)#X(2), then both b(2)(X(1)) and b(2)(X(2)) are divisible by four; this rules out a previously naturally 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.