|
Free  actions on 
Author:
Gerhard X. Ritter
Journal:
Trans. Amer. Math. Soc. 181 (1973), 195-212
MSC:
Primary 55C35; Secondary 57A10
MathSciNet review:
0321078
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: This paper is devoted to the problem of classifying periodic homeomorphisms which act freely on the  -sphere. The main result is the classification of free period eight actions and a generalization to free actions whose squares are topologically equivalent to orthogonal transformations. The result characterizes those  -manifolds which have the  -sphere as universal covering space and the cyclic group of order eight as fundamental group.
- [1]
R.
H. Bing, An alternative proof that 3-manifolds can be
triangulated, Ann. of Math. (2) 69 (1959),
37–65. MR
0100841 (20 #7269)
- [2]
Glen
E. Bredon and John
W. Wood, Non-orientable surfaces in orientable 3-manifolds,
Invent. Math. 7 (1969), 83–110. MR 0246312
(39 #7616)
- [3]
David
W. Henderson, Extensions of Dehn’s lemma and
the loop theorem, Trans. Amer. Math. Soc.
120 (1965),
448–469. MR 0187233
(32 #4686), http://dx.doi.org/10.1090/S0002-9947-1965-0187233-0
- [4]
David
W. Henderson, Relative general position, Pacific J. Math.
18 (1966), 513–523. MR 0200933
(34 #819)
- [5]
J.
F. P. Hudson, Piecewise linear topology, University of Chicago
Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees,
W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR 0248844
(40 #2094)
- [6]
G.
R. Livesay, Fixed point free involutions on the 3-sphere, Ann.
of Math. (2) 72 (1960), 603–611. MR 0116343
(22 #7131)
- [7]
K. Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102-109.
- [8]
P.
M. Rice, Free actions of 𝑍₄ on 𝑆³,
Duke Math. J. 36 (1969), 749–751. MR 0248814
(40 #2064)
- [9]
H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner Verlag, Leipzig, 1934.
- [10]
E. C. Zeeman, Seminar on combinatorial topology, Mimeographed Notes, Inst. Hautes Études Sci., Paris, 1963.
- [1]
- R. H. Bing, An alternative proof that
 -manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37-65. MR 20 #7269. MR 0100841 (20:7269)
- [2]
- G. E. Bredon and J. W. Wood, Non-orientable surfaces in orientable
 -manifolds, Invent. Math. 7 (1969), 83-110. MR 39 #7616. MR 0246312 (39:7616)
- [3]
- D. W. Henderson, Extensions of Dehn's lemma and the loop theorem, Trans. Amer. Math. Soc. 120 (1965), 448-469. MR 32 #4686. MR 0187233 (32:4686)
- [4]
- -, Relative general position, Pacific J. Math. 18 (1966), 513-523. MR 34 #819. MR 0200933 (34:819)
- [5]
- J. F. P. Hudson, Piecewise linear topology, Benjamin, New York, 1969. MR 40 #2094. MR 0248844 (40:2094)
- [6]
- G. R. Livesay, Fixed point free involutions on the
 -sphere, Ann. of Math. (2) 72 (1960), 603-611. MR 22 #7131. MR 0116343 (22:7131)
- [7]
- K. Reidemeister, Homotopieringe und Linsenräume, Abh. Math. Sem. Univ. Hamburg 11 (1935), 102-109.
- [8]
- P. M. Rice, Free actions of
 on  , Duke Math. J. 36 (1969), 749-751. MR 40 #2064. MR 0248814 (40:2064)
- [9]
- H. Seifert and W. Threlfall, Lehrbuch der Topologie, Teubner Verlag, Leipzig, 1934.
- [10]
- E. C. Zeeman, Seminar on combinatorial topology, Mimeographed Notes, Inst. Hautes Études Sci., Paris, 1963.
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC:
55C35,
57A10
Retrieve articles in all journals
with MSC:
55C35,
57A10
Additional Information
DOI:
http://dx.doi.org/10.1090/S0002-9947-1973-0321078-8
PII:
S 0002-9947(1973)0321078-8
Keywords:
Three-sphere,
free action,
orthogonal  -action,
knot,
lens space,
general position,
topologically equivalent
Article copyright:
© Copyright 1973 American Mathematical Society
|
