Manifolds on which only tori can act
Authors:
Kyung Bai Lee and Frank Raymond
Journal:
Trans. Amer. Math. Soc. 304 (1987), 487499
MSC:
Primary 57S10; Secondary 57S25
MathSciNet review:
911081
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Abstract 
References 
Similar Articles 
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Abstract: A list of various types of connected, closed oriented manifolds are given. Each of the manifolds support some of the wellknown 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), 267271. MR 637181 (83a:55019)
 [AB]
A. Assadi and D. Burghelea, Examples of asymmetric differentiable manifolds, Math. Ann. 255 (1981), 423430. MR 615861 (83a:57053)
 [AH]
M. Atiyah and F. Hirzebruch, Spinmanifolds and group actions. Essay on topology and related topics, Springer, Berlin and New York, 1970, pp. 1828. MR 0278334 (43:4064)
 [B1]
E. M. Bloomberg, Manifolds with no periodic homeomorphisms, Trans. Amer. Math. Soc. 202 (1975), 6778. MR 0358842 (50:11301)
 [BDH]
G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980), 147. MR 549702 (81i:55012)
 [BH]
W. Browder and W. C. Hsiang, actions and the fundamental group, Invent. Math. 65 (1982), 411424. 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. 227264. 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. 175. MR 0358835 (50:11294)
 [CR3]
, Injective actions of toral groups, Topology 10 (1970), 283296.
 [CR4]
, Deforming homotopy equivalences to homeomorphisms in aspherical manifolds, Bull. Amer. Math. Soc. 83 (1977), 3685. 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. 124204. MR 0590802 (58:28698)
 [DS]
H. Donnelly and R. Schultz, Compact group actions and maps into aspherical manifolds, Topology 21 (1982), 443455. MR 670746 (84k:57024)
 [F]
E. Floyd, Orbits spaces of finite transformation groups. II, Duke Math. J. 22 (1955), 3338. 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), 419429. MR 766228 (86h:57034)
 [Gr]
M. Gromov, Volume and bounded cohomology, Inst. Hautes Etude Sci. Publ. Math. 56 (1982), 213307. MR 686042 (84h:53053)
 [KK]
H. T. Ku and M. C. Ku, Group actions on aspherical manifolds, Trans. Amer. Math. Soc. 278 (1983), 841859. 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), 255269. 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. 353411. MR 767121 (86h:57043)
 [LY]
H. B. Lawson and S. T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geom. 7 (1972), 211228. MR 0334083 (48:12402)
 [MY]
W. Meeks and S. T. Yau, Topology of threedimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. 112 (1980), 441484. 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. 163196. 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. 364377. MR 585669 (81m:57030)
 [Sch2]
, Group actions on hypertoral manifolds. II, J. Reine Angew. Math. 325 (1981), 7586. MR 618547 (82m:57022)
 [Sp]
E. Spanier, Algebraic topology, McGrawHill, 1966. MR 0210112 (35:1007)
 [SY]
R. Schoen and S. T. Yau, Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), 361380. 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), 5358. MR 679074 (85d:57034)
 [A]
 M. A. Armstrong, Calculating the fundamental group of an orbit space, Proc. Amer. Math. Soc. 84 (1982), 267271. MR 637181 (83a:55019)
 [AB]
 A. Assadi and D. Burghelea, Examples of asymmetric differentiable manifolds, Math. Ann. 255 (1981), 423430. MR 615861 (83a:57053)
 [AH]
 M. Atiyah and F. Hirzebruch, Spinmanifolds and group actions. Essay on topology and related topics, Springer, Berlin and New York, 1970, pp. 1828. MR 0278334 (43:4064)
 [B1]
 E. M. Bloomberg, Manifolds with no periodic homeomorphisms, Trans. Amer. Math. Soc. 202 (1975), 6778. MR 0358842 (50:11301)
 [BDH]
 G. Baumslag, E. Dyer and A. Heller, The topology of discrete groups, J. Pure Appl. Algebra 16 (1980), 147. MR 549702 (81i:55012)
 [BH]
 W. Browder and W. C. Hsiang, actions and the fundamental group, Invent. Math. 65 (1982), 411424. 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. 227264. 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. 175. MR 0358835 (50:11294)
 [CR3]
 , Injective actions of toral groups, Topology 10 (1970), 283296.
 [CR4]
 , Deforming homotopy equivalences to homeomorphisms in aspherical manifolds, Bull. Amer. Math. Soc. 83 (1977), 3685. 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. 124204. MR 0590802 (58:28698)
 [DS]
 H. Donnelly and R. Schultz, Compact group actions and maps into aspherical manifolds, Topology 21 (1982), 443455. MR 670746 (84k:57024)
 [F]
 E. Floyd, Orbits spaces of finite transformation groups. II, Duke Math. J. 22 (1955), 3338. 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), 419429. MR 766228 (86h:57034)
 [Gr]
 M. Gromov, Volume and bounded cohomology, Inst. Hautes Etude Sci. Publ. Math. 56 (1982), 213307. MR 686042 (84h:53053)
 [KK]
 H. T. Ku and M. C. Ku, Group actions on aspherical manifolds, Trans. Amer. Math. Soc. 278 (1983), 841859. 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), 255269. 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. 353411. MR 767121 (86h:57043)
 [LY]
 H. B. Lawson and S. T. Yau, Compact manifolds of nonpositive curvature, J. Differential Geom. 7 (1972), 211228. MR 0334083 (48:12402)
 [MY]
 W. Meeks and S. T. Yau, Topology of threedimensional manifolds and the embedding problems in minimal surface theory, Ann. of Math. 112 (1980), 441484. 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. 163196. 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. 364377. MR 585669 (81m:57030)
 [Sch2]
 , Group actions on hypertoral manifolds. II, J. Reine Angew. Math. 325 (1981), 7586. MR 618547 (82m:57022)
 [Sp]
 E. Spanier, Algebraic topology, McGrawHill, 1966. MR 0210112 (35:1007)
 [SY]
 R. Schoen and S. T. Yau, Compact group actions and the topology of manifolds with nonpositive curvature, Topology 18 (1979), 361380. 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), 5358. MR 679074 (85d:57034)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198709110819
PII:
S 00029947(1987)09110819
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
