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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

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After a second thought and re-reading Segev's proof that $K^G$ is either empty or acyclic for $3$-dimensional, collapsible simplicial complexes, I conclude that Q2 is open. Details below.

Let $K$ be a collapsible, $3$-dimensional simplicial complex equipped with an admissible action of $G$. By the proof of [5, Theorem (5.1)] (reference numbering as in the question) we know that $K$ collapses to a $2$-dimensional, $G$-invariant subcomplex $L$, and $K^G$ collapses to $L^G$.

If $K^G=\emptyset$, then $L^G=\emptyset$. But $L$ is a contractible, $2$-dimensional complex.

This means that having such an action would answer in the negative the Casacuberta-Dicks-Aschbacher-Segev conjecture mentioned in the question:

If $dim(K)=2$ and $K$ is contractible, then it is conjectured [8] that $K^G$ is non-empty (this, as far as I know, is open even for infinite $K$, see [7]).

So if we know an answer to Q2, then it must be that a fixed-point-free action on a $3$-dimensional, collapsible simplicial complex does not exist. However, any paper proving this would probably cite [5], and I can't find such an article.


Also note that if a fixed-point-free action on a collapsible $3$-complex does not exist and Zeeman's collapsibility conjecture is true (which is a very strong assumption), then we have a positive solution to the Casacuberta-Dicks-Aschbacher-Segev conjecture.

Proof: Let $G$ act on a contractible $2$-complex $K$. For $n$ large enough, by Zeeman's conjecture and [1, Proposition II.2.1] the $n$-th barycentric subdivision of $K\times I$ is a collapsible $3$-complex, with $G$ acting non-trivially on the first coordinate. Thus, a fixed point of this action has to correspond to an element of $K^G$.


Update: The answer to Q1 is positive - there exist fixed point free group actions on contractible $3$-complexes.

Let $G$ be a finite group acting without fixed points on a $2$-dimensional, acyclic simplicial complex $X$. Then $G\times \mathbb{Z}_2$ acts without fixed points on the suspension $SX$, which is contractible.

A very similar idea was applied in the construction of a fixed point free action on a disk in the 1959 paper by E. Floyd and R. Richardson An action of a finite group on an n-cell without stationary points. I've learned this from section 1 of A. Adem's Finite group actions on acyclic 2 -complexes.

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