Twisted Donaldson invariants

TL;DR

This paper introduces a new twisted Donaldson invariant influenced by the fundamental group of a four-manifold, extending it via non-commutative geometry to distinguish smooth structures on homeomorphic manifolds.

Contribution

It presents a novel variant of the Donaldson invariant twisted by the Picard group and generalizes it to arbitrary fundamental groups using non-commutative geometry.

Findings

Invariant distinguishes smooth structures on some homeomorphic four-manifolds

Extension from free abelian to general fundamental groups

Framework based on non-commutative geometry

Abstract

Fundamental group of a manifold gives a deep effect on its underlying smooth structure. In this paper we introduce a new variant of the Donaldson invariant in Yang-Mills gauge theory from twisting by the Picard group of a four manifold in the case when the fundamental group is free abelian. We then generalize it to the general case of fundamental groups by use of the framework of non commutative geometry. We also verify that our invariant distinguishes smooth structures between some homeomorphic four manifolds.