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](https://en.wikipedia.org/wiki/Collapse_(topology)) (in the sense of Whitehead's simple homotopy theory)?
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> 
> 


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:
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[1] K. Adiprasito, B. Benedetti, [Subdivisions, shellability, and
collapsibility of products](https://link.springer.com/article/10.1007/s00493-016-3149-8); also available on [arXiv](https://arxiv.org/abs/1202.6606).


[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](https://projecteuclid.org/euclid.bams/1183523039)


[4] R. Oliver, [Fixed-point sets of group actions on finite acyclic complexes](https://link.springer.com/article/10.1007%2FBF02565743?LI=true).


[5] R. Oliver, Y. Segev, [Fixed point free actions on Z-acyclic 2-complexes](https://projecteuclid.org/euclid.acta/1485891524).


[6] Y. Segev, [Some remarks on finite 1-acyclic and collapsible complexes](http://www.sciencedirect.com/science/article/pii/0097316594900418).


[7] J. M. Corson, [On Finite Groups Acting on Contractible Complexes of Dimension Two](https://link.springer.com/article/10.1023/A:1012062101922).


[8] M. Aschbacher, Y. Segev, [A Fixed Point Theorem for Groups Acting on Finite 2-Dimensional Acyclic Simplicial Complexes](http://onlinelibrary.wiley.com/doi/10.1112/plms/s3-67.2.329/full)


[9] C. Casacuberta, W. Dicks, [On finite groups acting on acyclic complexes of dimension two](http://dmle.icmat.es/pdf/PUBLICACIONSMATEMATIQUES_1992_36_2A_11.pdf)



Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex
**EDIT (2021-02-26)**: The authors of <https://arxiv.org/abs/2102.11458>, <https://arxiv.org/abs/2102.11459> claim that they've proven that every action of a finite group on a finite and contractible 2-complex has a fixed point. It follows (see the reasoning below) that **the answer to Q2 is negative** (i.e. every action on a finite, collapsible 3-complex has a fixed point).




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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.




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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 $K^G$.




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**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](https://mathoverflow.net/a/73702/2578).


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](https://projecteuclid.org/euclid.bams/1183523039). I've learned this from section 1 of A. Adem's [Finite group actions on acyclic
2
-complexes](http://www.math.ubc.ca/%7Eadem/bourbaki-talk.pdf).


