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.   Do some reading! But 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)].