On an algebraic formula and applications to group action on manifolds

TL;DR

This paper presents an algebraic formula and applies it to estimate fixed points in cyclic group actions on manifolds, providing obstructions to certain group actions with isolated fixed points.

Contribution

It introduces a new algebraic approach to analyze group actions on manifolds and establishes obstructions for prime order cyclic group actions with isolated fixed points.

Findings

Derived lower bounds for fixed points under cyclic group actions

Identified obstructions to the existence of certain prime order actions

Applied algebraic methods to topological fixed point problems

Abstract

We consider a purely algebraic result. Then given a circle or cyclic group of prime order action on a manifold, we will use it to estimate the lower bound of the number of fixed points. We also give an obstruction to the existence of $\mathbb{Z}_p$ action on manifolds with isolated fixed points when $p$ is a prime.