Circle actions with two fixed points

TL;DR

This paper classifies certain smooth circle actions on 2n-dimensional stably complex manifolds with exactly two fixed points, showing such actions only occur in dimensions 2 and 6.

Contribution

It establishes a classification result for circle actions with two fixed points on stably complex manifolds, using rigid Hirzebruch genera.

Findings

Only dimensions 2 and 6 admit such circle actions.

The actions are not bound equivariantly in these dimensions.

The proof employs rigid Hirzebruch genera techniques.

Abstract

We prove that if the circle group acts smooth and unitary on 2n-dimensional stably complex manifold with two isolated fixed points and it is not bound equivariantly, then n=1 or 3. Our proof relies on the rigid Hirzebruch genera.