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 (24 #A783) Reviewed Kister, J. M. Examples of periodic maps on Euclidean spaces without fixed points. 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]. 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].
