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.


[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


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).

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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.