# Fixed-point-free group action on a finite, contractible, 3-dimensional simplicial complex

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

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

## 1 Answer

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.