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

Moise, Edwin E.

doi:10.1090/S0002-9947-1979-0534109-2

Statically tame periodic homeomorphisms of compact connected $3$-manifolds. I. Homeomorphisms conjugate to rotations of the $3$-sphere
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by Edwin E. Moise PDF

Trans. Amer. Math. Soc. 252 (1979), 1-47 Request permission

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

R. H. Bing and R. J. Bean (eds.), Topology Seminar, Wisconsin, 1965, Annals of Mathematics Studies, No. 60, Princeton University Press, Princeton, N.J., 1966. MR 0202100
R. H. Bing, An alternative proof that $3$-manifolds can be triangulated, Ann. of Math. (2) 69 (1959), 37–65. MR 100841, DOI 10.2307/1970092
R. H. Bing, Inequivalent families of periodic homeomorphisms of $E^{3}$, Ann. of Math. (2) 80 (1964), 78–93. MR 163308, DOI 10.2307/1970492
Armand Borel, Seminar on transformation groups, Annals of Mathematics Studies, No. 46, Princeton University Press, Princeton, N.J., 1960. With contributions by G. Bredon, E. E. Floyd, D. Montgomery, R. Palais. MR 0116341
Glen E. Bredon, Orientation in generalized manifolds and applications to the theory of transformation groups, Michigan Math. J. 7 (1960), 35–64. MR 116342
C. H. Edwards Jr., Concentricity in $3$-manifolds, Trans. Amer. Math. Soc. 113 (1964), 406–423. MR 178459, DOI 10.1090/S0002-9947-1964-0178459-X
Ralph H. Fox and Emil Artin, Some wild cells and spheres in three-dimensional space, Ann. of Math. (2) 49 (1948), 979–990. MR 27512, DOI 10.2307/1969408
Edwin Moise, Periodic homeomorphisms of the $3$-sphere, Illinois J. Math. 6 (1962), 206–225. MR 150768
Edwin E. Moise, Affine structures in $3$-manifolds. IV. Piecewise linear approximations of homeomorphisms, Ann. of Math. (2) 55 (1952), 215–222. MR 46644, DOI 10.2307/1969775
Edwin E. Moise, Affine structures in $3$-manifolds. V. The triangulation theorem and Hauptvermutung, Ann. of Math. (2) 56 (1952), 96–114. MR 48805, DOI 10.2307/1969769
Edwin E. Moise, Affine structures in $3$-manifolds. VIII. Invariance of the knot-types; local tame imbedding, Ann. of Math. (2) 59 (1954), 159–170. MR 61822, DOI 10.2307/1969837
Edwin E. Moise, Geometric topology in dimensions $2$ and $3$, Graduate Texts in Mathematics, Vol. 47, Springer-Verlag, New York-Heidelberg, 1977. MR 0488059, DOI 10.1007/978-1-4612-9906-6
C. D. Papakyriakopoulos, On solid tori, Proc. London Math. Soc. (3) 7 (1957), 281–299. MR 87944, DOI 10.1112/plms/s3-7.1.281
P. A. Smith, Transformations of finite period. II, Ann. of Math. (2) 40 (1939), 690–711. MR 177, DOI 10.2307/1968950
P. A. Smith, Periodic transformations of $3$-manifolds, Illinois J. Math. 9 (1965), 343–348. MR 175126, DOI 10.1215/ijm/1256067895
John Stallings, On the loop theorem, Ann. of Math. (2) 72 (1960), 12–19. MR 121796, DOI 10.2307/1970146