Noncommutative geometry is a useful tool in the study of topology of Riemannian manifolds. Taking into account of the fundamental group in the formulation of a topological invariant, one can obtain a refined topological invariant involving the C*-algebra of the fundamental group. For example, The Novikov conjecture on homotopy invariance of higher signature has been developed extensively using noncommutative geometry.

In this research, we aim at introducing noncommutative geometry to Donaldson's theory of differential topology of smooth four manifolds. Dolnaldson's polynomial invariants are topological invariants for compact closed four manifolds and have important applications in smooth structures for four manifolds. We introduce the notion of twisted Donaldson invariants by implementing fundamental groups in the construction of Donaldson's invariants, together with examples and applications when the fundamental group is the group of integers.

This is joint work with T. Kato (Kyoto) and H. Sasahira (Kyushu).