$\mathbb{Z}_2$-actions on positively curved manifolds

TL;DR

This paper extends previous results on $Z_2$-actions on positively curved manifolds, showing that similar topological classifications hold when the fixed point set's dimension is reduced to about 2/5 of the manifold's dimension.

Contribution

It improves the known bounds on the fixed point set dimension for $Z_2$-actions on positively curved manifolds, maintaining the same topological classification results.

Findings

The fixed point set dimension bound is lowered to approximately 2n/5.

The classification of the manifold remains the same under the new fixed point set dimension bound.

The results generalize previous work with a weaker fixed point set dimension condition.

Abstract

Kennard, Khalili Samani, and Searle showed that for a $\mathbb{Z}_2$-torus acting on a closed, positively curved Riemannian $n$-manifold, $M^{n}$, with a non-empty fixed point set for $n$ large enough and $r$ approximately half the dimension of $M$, then $M^n$ is homotopy equivalent to $S^n$, $\mathbb{R}\mathrm{P}^{n}$, $\mathbb{C}\mathrm{P}^{\frac{n}{2}}$, or a lens space. In this paper, we lower $r$ to approximately $2n/5$ and show that we still obtain the same result.