Manifolds on which only tori can act

Authors:
Kyung Bai Lee and Frank Raymond

Journal:
Trans. Amer. Math. Soc. **304** (1987), 487-499

MSC:
Primary 57S10; Secondary 57S25

DOI:
https://doi.org/10.1090/S0002-9947-1987-0911081-9

MathSciNet review:
911081

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the well-known compact transformation group properties enjoyed by aspherical manifolds. We list and describe these classes and their transformation group properties in increasing generality. We show by various examples that these implications can never be reversed. This establishes a hierarchy in terms of spaces in one direction and the properties they enjoy in the opposite direction.

**[A]**M. A. Armstrong,*Calculating the fundamental group of an orbit space*, Proc. Amer. Math. Soc.**84**(1982), 267-271. MR**637181 (83a:55019)****[AB]**A. Assadi and D. Burghelea,*Examples of asymmetric differentiable manifolds*, Math. Ann.**255**(1981), 423-430. MR**615861 (83a:57053)****[AH]**M. Atiyah and F. Hirzebruch,*Spin-manifolds and group actions. Essay on topology and related topics*, Springer, Berlin and New York, 1970, pp. 18-28. MR**0278334 (43:4064)****[B1]**E. M. Bloomberg,*Manifolds with no periodic homeomorphisms*, Trans. Amer. Math. Soc.**202**(1975), 67-78. MR**0358842 (50:11301)****[BDH]**G. Baumslag, E. Dyer and A. Heller,*The topology of discrete groups*, J. Pure Appl. Algebra**16**(1980), 1-47. MR**549702 (81i:55012)****[BH]**W. Browder and W. C. Hsiang, -*actions and the fundamental group*, Invent. Math.**65**(1982), 411-424. MR**643560 (83d:57026)****[C]**P. E. Conner,*Differentiable periodic maps*, 2nd ed., Lecture Notes in Math., vol. 738, Springer, 1979. MR**548463 (81f:57018)****[CR1]**P. E. Conner and Frank Raymond,*Actions of compact Lie groups on aspherical manifolds*, Topology of Manifolds (Proc. Inst., Univ. of Georgia, Athens, 1969), Markham, Chicago, Ill., 1970, pp. 227-264. MR**0271958 (42:6839)****[CR2]**-,*Manifolds with few periodic homeomorphisms*, Proc. Second Conference on Compact Transformation Groups, Part II, Lecture Notes in Math., vol. 299, Springer, 1972, pp. 1-75. MR**0358835 (50:11294)****[CR3]**-,*Injective actions of toral groups*, Topology**10**(1970), 283-296.**[CR4]**-,*Deforming homotopy equivalences to homeomorphisms in aspherical manifolds*, Bull. Amer. Math. Soc.**83**(1977), 36-85. MR**0467777 (57:7629)****[CR5]**-,*Holomorphic Seifert fiberings*, Proc. Second Conference on Compact Transformation Groups, Part II, Lecture Notes in Math., vol. 299, Springer, 1972, pp. 124-204. MR**0590802 (58:28698)****[DS]**H. Donnelly and R. Schultz,*Compact group actions and maps into aspherical manifolds*, Topology**21**(1982), 443-455. MR**670746 (84k:57024)****[F]**E. Floyd,*Orbits spaces of finite transformation groups*. II, Duke Math. J.**22**(1955), 33-38. MR**0066642 (16:610b)****[GLO]**D. Gottlieb, K. B. Lee and M. Ozaydin,*Compact group actions and maps into*-*spaces*, Trans. Amer. Math. Soc.**287**(1985), 419-429. MR**766228 (86h:57034)****[Gr]**M. Gromov,*Volume and bounded cohomology*, Inst. Hautes Etude Sci. Publ. Math.**56**(1982), 213-307. MR**686042 (84h:53053)****[KK]**H. T. Ku and M. C. Ku,*Group actions on aspherical*-*manifolds*, Trans. Amer. Math. Soc.**278**(1983), 841-859. MR**701526 (85b:57042)****[LR1]**K. B. Lee and F. Raymond,*Topological, affine and isometric actions on flat Riemannian manifolds*, J. Differential Geom.**16**(1982), 255-269. MR**638791 (84k:57027)****[LR2]**-,*Geometric realization of group extensions by the Seifert construction*, Contemporary Math., vol. 33, Amer. Math. Soc., Providence, R. I., 1984, pp. 353-411. MR**767121 (86h:57043)****[LY]**H. B. Lawson and S. T. Yau,*Compact manifolds of non-positive curvature*, J. Differential Geom.**7**(1972), 211-228. MR**0334083 (48:12402)****[MY]**W. Meeks and S. T. Yau,*Topology of three-dimensional manifolds and the embedding problems in minimal surface theory*, Ann. of Math.**112**(1980), 441-484. MR**595203 (83d:53045)****[NR]**W. Neumann and F. Raymond,*Seifert manifolds, plumbing*, -*invariant and orientation reversing maps*, Alg. and Geom. Topology (Proc. Santa Barbara, 1977), Lecture Notes in Math., vol. 664, Springer, 1978, pp. 163-196. MR**518415 (80e:57008)****[Sch1]**R. Schultz,*Group actions on hypertoral manifolds*. I, Topology Symposium (Siegen 1979), Lecture Notes in Math., vol. 788, Springer, pp. 364-377. MR**585669 (81m:57030)****[Sch2]**-,*Group actions on hypertoral manifolds*. II, J. Reine Angew. Math.**325**(1981), 75-86. MR**618547 (82m:57022)****[Sp]**E. Spanier,*Algebraic topology*, McGraw-Hill, 1966. MR**0210112 (35:1007)****[SY]**R. Schoen and S. T. Yau,*Compact group actions and the topology of manifolds with non-positive curvature*, Topology**18**(1979), 361-380. MR**551017 (81a:53044)****[WW]**R. Washiyama and T. Watabe,*On the degree of symmetry of a certain manifold*, J. Math. Soc. Japan**35**(1983), 53-58. MR**679074 (85d:57034)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
57S10,
57S25

Retrieve articles in all journals with MSC: 57S10, 57S25

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1987-0911081-9

Keywords:
Compact transformation group,
aspherical manifold,
covering space,
ends,
-manifold,
,
essential manifold,
admissible,
injective action,
inner action,
compact Lie group,
hyperaspherical manifold,
lens space,
spherical space form,
toral action

Article copyright:
© Copyright 1987
American Mathematical Society