since the ground breaking work of donaldson in the 1980s, topologists has achieved huge success in using gauge theory to study smooth 4-manifolds with nonzero second homology. the case of 4-manifolds with trivial second homology is relatively less known. in particular, when the 4-manifold have the same homology as s1 cross s3, there are several gauge theoretic invariants. the first one is the casson-seiberg-witten invariant lsw(x) defined by mrowka-ruberman-saveliev; the second one is the fruta-ohta invariant lfo(x). it is conjecture that these two invariants are equal to each other (this is an analogue of witten's conjecture relating donaldson and seiberg-witten invariants.) in this talk, i will recall the definition of these two invariants, give some applications of them (including a new obstruction for metric with positive scalar curvature), and sketch a proof of this conjecture for finite-order mapping tori. this is based on a joint work with danny ruberman and nikolai saveliev.