Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Statically tame periodic homeomorphisms of compact connected $ 3$-manifolds. I. Homeomorphisms conjugate to rotations of the $ 3$-sphere


Author: Edwin E. Moise
Journal: Trans. Amer. Math. Soc. 252 (1979), 1-47
MSC: Primary 57S17; Secondary 57Q15
DOI: https://doi.org/10.1090/S0002-9947-1979-0534109-2
MathSciNet review: 534109
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let f be a homeomorphism of the 3-sphere onto itself, of finite period n, and preserving orientation. Suppose that the fixed-point set F of f is a tame 1-sphere. It is shown that (1) the 3-sphere has a triangulation $ K({{\textbf{S}}^3})$ such that F forms a subcomplex of $ K({{\textbf{S}}^3})$ and f is simplicial relative to $ K({{\textbf{S}}^3})$. Suppose also that F is unknotted. It then follows that (2) f is conjugate to a rotation.


References [Enhancements On Off] (What's this?)

  • [B$ _{1}$] R. H. Bing and R. J. Bean (Ed.), Topology seminar, Wisconsin, 1965, Ann. of Math. Studies, No. 60, Princeton Univ. Press, Princeton, N. J., 1960, p. 82. MR 0202100 (34:1974)
  • [B$ _{2}$] R. H. Bing, An alternative proof that 3-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37-65. MR 0100841 (20:7269)
  • [B$ _{3}$] -, Inequivalent families of periodic homeomorphisms of $ {E^3}$, Ann. of Math. (2) 80 (1964), 78-93. MR 0163308 (29:611)
  • [B$ _{4}$] Armand Borel, Seminar on transformation groups, Ann. of Math. Studies, No. 46, Princeton Univ. Press, Princeton, N. J., 1960. MR 0116341 (22:7129)
  • [B$ _{5}$] Glen E. Bredon, Orientation in generalized manifolds and applications to the theory of transformation groups, Michigan Math. J. 7 (1960), 35-64. MR 0116342 (22:7130)
  • [E] C. H. Edwards, Concentricity in 3-manifolds, Trans. Amer. Math. Soc. 113 (1964), 406-423. MR 0178459 (31:2716)
  • [FA] Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979-990. MR 0027512 (10:317g)
  • [M] Edwin E. Moise, Periodic homeomorphisms of the 3-sphere, Illinois J. Math. 6 (1962), 206-225. MR 0150768 (27:755)
  • [M$ _{4}$] -, Affine structures in 3-manifolds. IV. Piecewise linear approximations of homeomorphisms, Ann. of Math. (2) 55 (1952), 215-222. MR 0046644 (13:765c)
  • [M$ _{5}$] -, V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96-114. MR 0048805 (14:72d)
  • [M$ _{8}$] -, VIII. Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159-170. MR 0061822 (15:889g)
  • [MGT] -, Geometric topology in dimensions 2 and 3, Springer-Verlag, New York, 1977. MR 0488059 (58:7631)
  • [P] C. D. Papakyriakopoulos, On solid tori, Proc. London Math. Soc. (3) 7 (1957), 248-260. MR 0087944 (19:441d)
  • [S$ _{1}$] P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690-711. MR 0000177 (1:30c)
  • [S$ _{2}$] -, Periodic transformations of 3-manifolds, Illinois J. Math. 9 (1965), 343-348. MR 0175126 (30:5311)
  • [St] John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12-19. MR 0121796 (22:12526)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 57S17, 57Q15

Retrieve articles in all journals with MSC: 57S17, 57Q15


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1979-0534109-2
Keywords: Periodic homeomorphism, 3-sphere, fixed-point set
Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society