Circle action, lower bound of fixed points and characteristic numbers

TL;DR

This paper surveys how equivariant cohomology can unify approaches to understanding fixed points and characteristic numbers in circle actions on manifolds, providing a common framework for existing results.

Contribution

It introduces a unified method using equivariant cohomology to analyze fixed points and characteristic numbers in circle actions, simplifying and generalizing previous results.

Findings

Unified approach via equivariant cohomology for fixed point problems

Corollaries of known results derived from the method

Insights into the relationship between fixed points and characteristic numbers

Abstract

Given an $S^1$-manifold with isolated fixed points, some recent papers are concerned with the relationship between the least number of fixed points and the characteristic numbers of this manifold, and their proofs have some similar features. The main purpose of this short survey article is, by using the language of equivariant cohomology, to present a unified method to deal with such problems, of which the related known results are direct corollaries.