Fixed points of circle actions on spaces with rational cohomology of $S^n V S^{2n} V S^{3n}$ or $P^2(n) V S^{3n}$

TL;DR

This paper investigates the fixed point sets of circle actions on spaces with specific rational cohomology types, providing classifications and examples based on their cohomological properties within the Borel fibration framework.

Contribution

It classifies possible fixed point sets of circle actions on spaces with given rational cohomology, depending on their non-homologous to zero property in the Borel fibration.

Findings

Classified fixed point sets based on cohomology types.

Determined conditions for fixed point set structures.

Provided examples for each possible case.

Abstract

Let $X$ be a finitistic space with its rational cohomology isomorphic to that of the wedge sum $P^2(n)\vee S^{3n} $ or $S^{n} \vee S^{2n}\vee S^{3n}$. We study continuous $\mathbb{S}^1$ actions on $X$ and determine the possible fixed point sets up to rational cohomology depending on whether or not $X$ is totally non-homologous to zero in $X_{\mathbb{S}^1}$ in the Borel fibration $X\hookrightarrow X_{\mathbb{S}^1} \longrightarrow B_{\mathbb{S}^1}$. We also give examples realizing the possible cases.