Revisions to Finite-order self-homeomorphisms of $\mathbf{R}^n$

Removed superfluous comment.

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edited Aug 18, 2017 at 18:57

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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 Smith, Conner, Floyd. Do some reading! But hereHere 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].

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].

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 (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].

added link to the review and to the paper

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edit approved Aug 18, 2017 at 14:15

Martin Sleziak

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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:

MR0130929MR0130929 (24 #A783) Reviewed Kister, J. M. Examples of periodic maps on Euclidean spaces without fixed pointsExamples 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]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]MR0105115].

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].

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].

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created Aug 18, 2017 at 1:13

Nicholas Kuhn

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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].

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