Let $K$ be a finite simplicial complex with an admissible action of a finite group $G$.

(*Terminology:* By an action of a group $G$ on $K$ I mean an action by simplicial automorphisms. The action is called admissible if for all $g\in G$ and all simplices $\sigma\in K$ the equality $g\sigma=\sigma$ implies that $gv=v$ for all vertices $v\in \sigma$. By $K^G$ I denote the subcomplex formed by stationary points of the action (the set of stationary points being a subcomplex is guaranteed by admissibility of the action). Note that the admissibility condition is not really relevant, since for any action of $G$ on $K$ the action induced on the barycentric subdivision is admissible.)

Q1: Can $K^G$ be empty if $\dim(K)=3$ and $K$ is contractible?

Q2: What if $K$ is additionaly collapsible (in the sense of Whitehead's simple homotopy theory)?

Is suspect the answer is that $K^G$ can be empty. However, I wasn't able to find an example in the literature. I know that:

- If $\dim(K)=2$ and $K$ is collapsible, then $K^G$ is non-empty and collapsible [6].
- If $\dim(K)=2$ and $K$ is contractible, then it is conjectured [8,9] that $K^G$ is non-empty (this, as far as I know, is open even for infinite $K$, see [7]).
- If $\dim(K)=2$ and $K$ is acyclic, then $K^G$ may be empty, and such actions have been classified [5].
- If $\dim(K)=3$ and $K$ is collapsible, then $K^G$ is either empty or acyclic [6].
*(My question 2 is whether the case $K^G=\emptyset$ can really occur.)* - If we allow $K$ to be infinite, then for contractible $K$ of dimension $3$ there exist finite group actions with $K^G=\emptyset$. [2,Corollary II.7.4].
- There are fixed-point-free group actions on finite, contractible complexes $K$ of higher dimension (see [3] or [4]). Using [1] one can make these examples collapsible by considering products $K\times [0,1]^n$ for $n$ large enough.

## References:

[1] K. Adiprasito, B. Benedetti, Subdivisions, shellability, and collapsibility of products; also available on arXiv.

[2] A. H. Assadi, Finite Group Actions on Simply-Connected Manifolds and CW Complexes.

[3] E. E. Floyd, R. W. Richardson, An action of a finite group on an n-cell without stationary points

[4] R. Oliver, Fixed-point sets of group actions on finite acyclic complexes.

[5] R. Oliver, Y. Segev, Fixed point free actions on Z-acyclic 2-complexes.

[6] Y. Segev, Some remarks on finite 1-acyclic and collapsible complexes.

[7] J. M. Corson, On Finite Groups Acting on Contractible Complexes of Dimension Two.

[8] M. Aschbacher, Y. Segev, A Fixed Point Theorem for Groups Acting on Finite 2-Dimensional Acyclic Simplicial Complexes

[9] C. Casacuberta, W. Dicks, On finite groups acting on acyclic complexes of dimension two