Fixed point sets of smooth $G$-manifolds pseudo-equivalent to a $G$-template

TL;DR

This paper extends Oliver's classification of fixed point sets of smooth finite group actions to a broader class of manifolds, establishing conditions based on Euler characteristics and Oliver numbers, and constructs fixed point free actions on real projective spaces.

Contribution

It generalizes Oliver's results to pseudo-equivalent manifolds and characterizes fixed point sets via Euler characteristics and Oliver numbers.

Findings

Fixed point sets characterized by Euler characteristic congruences.

Existence of fixed point free actions on real projective spaces for Oliver groups.

Extension of classification to pseudo-equivalent $G$-manifolds.

Abstract

For a finite group $G$ not of prime power order, Oliver (1996) has answered the question which manifolds occur as the fixed point sets of smooth actions of $G$ on disks (resp., Euclidean spaces). We extend Oliver's result to compact (resp., open) smooth $G$-manifolds $M$ pseudo-equivalent to $Y$, a finite $\mathbb{Z}$-acyclic $G$-CW complex such that the fixed point set $Y^G$ is non-empty, connected, and $\chi(Y^G) \equiv 1 \pmod{n_G}$, where $n_G$ is the Oliver number of $G$. We prove that the answer to the question above does not depend on the choice of $Y$. For a finite connected $G$-CW complex $Y$ such that $Y^G$ is non-empty and connected, called a $G$-template, we prove that a compact stably parallelizable manifold $F$ occurs as the fixed point set $M^G$ of a compact smooth $G$-manifold $M$ pseudo-equivalent to $Y$, if and only if $\chi(F) \equiv \chi(Y^G) \pmod{n_G}$. Moreover, there exists a compact smooth fixed point free $G$-manifold pseudo-equivalent to a $G$-template $Y$, if and only if $\chi(Y^G) \equiv 0 \pmod{n_G}$. In particular, similarly as for actions on disks, there exists a compact smooth fixed point free $G$-manifold pseudo-equivalent to the real projective space $\mathbb{R}{\rm P}^{2n}$ for an integer $n \geq 1$, if and only if $G$ is an Oliver group. Finally, we prove that each finite Oliver group $G$ has a smooth fixed point free action on $\mathbb{R}{\rm P}^{2n}$ itself for some integer $n \geq 1$.