Seiberg-Witten invariants of non-simple type and Einstein metrics
  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. 
Heberto del Rio Guerra