Twisted Donaldson invariant

Abstract:

We define a twisted Donaldson’s invariant using the Dirac operator twisted by flat connections when the fundamental group of a four manifold is free abelian.  We also present its applications and verify that our formula is non trivial. This generalizes the Donaldson’s mu-map by twisting family of flat bundles along the Lusztig’s approach to Novikov conjecture.We also generalize the construction to non commutative case by use of Connes-Moscovici’s theory on Novikov conjecture and cyclic cohomology. This is a joint work with H.Sasahira and H.Wang.