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

For a finite group $G$ not of prime power order, Oliver (1996) 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 to 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 $n \geq 1$.