Consider the $n$-dimensional euclidean space $\mathbf{R}^n$. A self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ is said to be *of finite order* if $\phi^m = \mathrm{id}\_{\mathbf{R}^n}$ for some positive integer $m$. 


**Question**: Does every finite order self-homeomorphism $\phi:\mathbf{R}^n\to \mathbf{R}^n$ have a fixed point?


What I know about it:


If for every divisor $d | m$, the fixed-point set $\left(\mathbf{R}^n\right)^{\phi^d}$ of $\phi^d$ has its cohomology groups $H^\*\_c\left(\left(\mathbf{R}^n\right)^{\phi^d}, \mathbf{Z}\right)$ finitely generated over $\mathbf{Z}$, then the theorem of "Verdier, *Caractéristique d'Euler-Poincaré*, 1973" will be applicable. In particular, all the self-homeomorphisms of $\mathbf{R}^n$ of prime order has a fixed-point. 


In that article Verdier derived a formula of the finite group representation on the alternating sum of the cohomology group with $\mathbf{Q}$-coefficients. This in particular implies a version of Lefschetz trace formula for finite-order self-homeomorphisms.


Unfortunately, I don't know if there can be some self-homeomorphism of non-prime order such that a certain power of it has its fixed-point set very complicated. 



Finite-order self-homeomorphisms of $\mathbf{R}^n$
There is, naturally, a huge history regarding a basic question like this. This particular problem was figured out between 1930 and the early 1960's. The main names are P.A. Smith, Conner, Floyd. Here is a math review to get you going:


[MR0130929](http://www.ams.org/mathscinet-getitem?mr=0130929) (24 #A783) Reviewed
Kister, J. M.
[Examples of periodic maps on Euclidean spaces without fixed points](https://doi.org/10.1090/S0002-9904-1961-10641-1).
Bull. Amer. Math. Soc. 67 1961 471–474.
54.80 (57.47)


Let T be a homeomorphism of period r on Euclidean space En. The classical result of P. A. Smith is that T has a fixed point if r is prime [Ann. of Math. (2) 35 (1934), 572–578] or a prime power [Amer. J. Math. 63 (1941), 1–8; [MR0003199](http://www.ams.org/mathscinet-getitem?mr=0003199)]. The author settles a question of long standing with the following theorem: If r is not a prime power, then there exists a triangulation μ of $E^{9r}$ and a homeomorphism T of period r on $E^{9r}$ without a fixed point, such that T is simplicial over μ. The construction is a modification of an example due to Conner and Floyd [Proc. Amer. Math. Soc. 10 (1959), 354–360; [MR0105115](http://www.ams.org/mathscinet-getitem?mr=0105115)]. 


