Circle actions on six dimensional oriented manifolds with isolated fixed points

Abstract

To classify a group action on a manifold, the data associated with the fixed point set is essential. In this paper, we classify the fixed point data of a circle action on a 6-dimensional compact connected oriented manifold with isolated fixed points, where the fixed point data consists of the collection of signs and weights at the fixed points. We show that this fixed point data can be reduced to the empty collection by performing a sequence of operations. Specifically, we prove that one can successively take equivariant connected sums at fixed points with $S^6$, $\mathbb{CP}^3$, or 6-dimensional analogues of the Hirzebruch surfaces (and their oppositely oriented counterparts), resulting in a fixed-point-free action on a compact connected oriented 6-manifold.