Book Review
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1566978
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Book Information:
Author:
Wu Yi Hsiang
Title:
Cohomology theory of topological transformation groups
Additional book information:
Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 85, Springer-Verlag, New York, Heidelberg, Berlin, 1975, x + 164 pp., $25.00.
Christopher Allday and Tor Skjelbred, The Borel formula and the topological splitting principle for torus actions on a Poincaré duality space, Ann. of Math. (2) 100 (1974), 322–325. MR 375358, DOI 10.2307/1971075
Michael Atiyah and Friedrich Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer, New York, 1970, pp. 18–28. MR 0278334
R. H. Bing, A homeomorphism between the $3$-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354–362. MR 49549, DOI 10.2307/1969804
Edward M. Bloomberg, Manifolds with no periodic homeomorphisms, Trans. Amer. Math. Soc. 202 (1975), 67–78. MR 358842, DOI 10.1090/S0002-9947-1975-0358842-7
Armand Borel, Nouvelle démonstration d’un théorème de P. A. Smith, Comment. Math. Helv. 29 (1955), 27–39 (French). MR 66649, DOI 10.1007/BF02564269
Glen E. Bredon, Examples of differentiable group actions, Topology 3 (1965), 115–122. MR 205273, DOI 10.1016/0040-9383(65)90038-8
Glen E. Bredon, The cohomology ring structure of a fixed point set, Ann. of Math. (2) 80 (1964), 524–537. MR 184225, DOI 10.2307/1970661
Glen E. Bredon, Homotopical properties of fixed point sets of circle group actions. I, Amer. J. Math. 91 (1969), 874–888. MR 259905, DOI 10.2307/2373308
Glen E. Bredon, Fixed point sets of actions on Poincaré duality spaces, Topology 12 (1973), 159–175. MR 331375, DOI 10.1016/0040-9383(73)90004-9
Glen E. Bredon, Introduction to compact transformation groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York-London, 1972. MR 0413144
Egbert Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1–14 (German). MR 206972, DOI 10.1007/BF01403388
L. E. J. Brouwer, Über die periodischen Transformationen der Kugel, Math. Ann. 80 (1919), no. 1, 39–41 (German). MR 1511946, DOI 10.1007/BF01463233
Theodore Chang and Tor Skjelbred, Group actions on Poincaré duality spaces, Bull. Amer. Math. Soc. 78 (1972), 1024–1026. MR 307226, DOI 10.1090/S0002-9904-1972-13092-1
Theodore Chang and Tor Skjelbred, The topological Schur lemma and related results, Ann. of Math. (2) 100 (1974), 307–321. MR 375357, DOI 10.2307/1971074
P. E. Conner, On the action of the circle group, Michigan Math. J. 4 (1957), 241–247. MR 96747
Pierre Conner and Deane Montgomery, An example for $\textrm {SO}(3)$, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1918–1922. MR 148795, DOI 10.1073/pnas.48.11.1918
P. E. Conner and Frank Raymond, Manifolds with few periodic homeomorphisms, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 1–75. MR 0358835
P. E. Conner, Frank Raymond, and Peter J. Weinberger, Manifolds with no periodic maps, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 299, Springer, Berlin, 1972, pp. 81–108. MR 0358836
Allan L. Edmonds and Ronnie Lee, Fixed point sets of group actions on Euclidean space, Topology 14 (1975), no. 4, 339–345. MR 413146, DOI 10.1016/0040-9383(75)90018-X
20. S. Eilenberg, Sur les transformations periodiques de la surface de sphere, Fund. Math. 22 (1934), 28-41.
E. E. Floyd, Examples of fixed point sets of periodic maps, Ann. of Math. (2) 55 (1952), 167–171. MR 45379, DOI 10.2307/1969425
E. E. Floyd, Examples of fixed point sets of periodic maps. II, Ann. of Math. (2) 64 (1956), 396–398. MR 79753, DOI 10.2307/1969982
E. E. Floyd, Fixed point sets of compact abelian Lie groups of transformations, Ann. of Math. (2) 66 (1957), 30–35. MR 88733, DOI 10.2307/1970115
E. E. Floyd and R. W. Richardson, An action of a finite group on an $n$-cell without stationary points, Bull. Amer. Math. Soc. 65 (1959), 73–76. MR 100848, DOI 10.1090/S0002-9904-1959-10282-2
J. P. E. Hodgson, Subcomplexes of Poincaré complexes, Bull. Amer. Math. Soc. 80 (1974), 1146–1150. MR 362318, DOI 10.1090/S0002-9904-1974-13650-5
Wu-chung Hsiang and Wu-yi Hsiang, On compact subgroups of the diffeomorphism groups of Kervaire spheres, Ann. of Math. (2) 85 (1967), 359–369. MR 214083, DOI 10.2307/1970349
Wu-chung Hsiang and Wu-yi Hsiang, The degree of symmetry of homotopy spheres, Ann. of Math. (2) 89 (1969), 52–67. MR 239621, DOI 10.2307/1970808
Wu Chung Hsiang and Wu-yi Hsiang, Differentiable actions of compact connected Lie groups. III, Ann. of Math. (2) 99 (1974), 220–256. MR 346828, DOI 10.2307/1970899
Wu-yi Hsiang, On the bound of the dimensions of the isometry groups of all possible riemannian metrics on an exotic sphere, Ann. of Math. (2) 85 (1967), 351–358. MR 214084, DOI 10.2307/1970446
Wu-yi Hsiang, On the splitting principle and the geometric weight system of topological transformation groups. I, Proceedings of the Second Conference on Compact Transformation Groups (Univ. Massachusetts, Amherst, Mass., 1971) Lecture Notes in Math., Vol. 298, Springer, Berlin, 1972, pp. 334–402. MR 0380845
Wu-yi Hsiang, On some fundamental theorems in cohomology theory of topological transformation groups, Taita J. Math. 2 (1970), 61–87. MR 309137
Lowell Jones, The converse to the fixed point theorem of P. A. Smith. I, Ann. of Math. (2) 94 (1971), 52–68. MR 295361, DOI 10.2307/1970734
Lowell Jones, The converse to the fixed point theorem of P. A. Smith. II, Indiana Univ. Math. J. 22 (1972/73), 309–325. MR 339249, DOI 10.1512/iumj.1972.22.22026
B. von Kerékjártó, Über die periodischen Transformationen der Kreisscheibe und der Kugelfläche, Math. Ann. 80 (1919), no. 1, 36–38 (German). MR 1511945, DOI 10.1007/BF01463232
H. Blaine Lawson Jr. and Shing Tung Yau, Scalar curvature, non-abelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helv. 49 (1974), 232–244. MR 358841, DOI 10.1007/BF02566731
John W. Milnor and John C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211–264. MR 174052, DOI 10.2307/1970615
Deane Montgomery and Leo Zippin, Periodic one-parameter groups in three-space, Trans. Amer. Math. Soc. 40 (1936), no. 1, 24–36. MR 1501864, DOI 10.1090/S0002-9947-1936-1501864-6
Deane Montgomery and Leo Zippin, Non-Abelian Compact Connected Transformation Groups of Three-Space, Amer. J. Math. 61 (1939), no. 2, 375–387. MR 1507382, DOI 10.2307/2371506
Deane Montgomery and Leo Zippin, Topological transformation groups, Interscience Publishers, New York-London, 1955. MR 0073104
Robert Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155–177. MR 375361, DOI 10.1007/BF02565743
41. R. Oliver, Projective obstructions to group actions on disks (to appear).
Robert Oliver, Smooth compact Lie group actions on disks, Math. Z. 149 (1976), no. 1, 79–96. MR 423390, DOI 10.1007/BF01301634
Volker Puppe, On a conjecture of Bredon, Manuscripta Math. 12 (1974), 11–16. MR 346778, DOI 10.1007/BF01166231
Daniel Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549–572; ibid. (2) 94 (1971), 573–602. MR 298694, DOI 10.2307/1970770
Reinhard Schultz, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology 10 (1971), 227–235. MR 283822, DOI 10.1016/0040-9383(71)90007-3
Reinhard Schultz, Semifree circle actions and the degree of symmetry of homotopy spheres, Amer. J. Math. 93 (1971), 829–839. MR 287548, DOI 10.2307/2373473
Reinhard Schultz, Circle actions on homotopy spheres not bounding spin manifolds, Trans. Amer. Math. Soc. 213 (1975), 89–98. MR 380853, DOI 10.1090/S0002-9947-1975-0380853-6
Graeme Segal, Equivariant $K$-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129–151. MR 234452
P. A. Smith, Transformations of finite period, Ann. of Math. (2) 39 (1938), no. 1, 127–164. MR 1503393, DOI 10.2307/1968718
J. C. Su, Periodic transformations on the product of two spheres, Trans. Amer. Math. Soc. 112 (1964), 369–380. MR 163309, DOI 10.1090/S0002-9947-1964-0163309-8
D. Sullivan, Differential forms and the topology of manifolds, Manifolds—Tokyo 1973 (Proc. Internat. Conf., Tokyo, 1973) Univ. Tokyo Press, Tokyo, 1975, pp. 37–49. MR 0370611
R. W. Sullivan, Linear models for compact groups acting on spheres, Topology 13 (1974), 77–87. MR 339245, DOI 10.1016/0040-9383(74)90040-8
Richard G. Swan, A new method in fixed point theory, Comment. Math. Helv. 34 (1960), 1–16. MR 115176, DOI 10.1007/BF02565923
- 1.
- C. Allday and T. Skjelbred, The Borel formula and the topological splitting principal for torus actions on a Poincaré duality space, Ann. of Math. (2) 100 (1974), 322-325. MR 51 # 11553. MR 0375358
- 2.
- M. F. Atiyah and F. Hirzebruch, Spin-manifolds and group actions, Essays on Topology and Related Topics (Mémoires dédiés à Georges de Rham), Springer-Verlag, Berlin and New York, 1970, pp. 18-28. MR 278334
- 3.
- R. H. Bing, A homeomorphism between the 3-sphere and the sum of two solid horned spheres, Ann. of Math. (2) 56 (1952), 354-362. MR 14, 192. MR 49549
- 4.
- E. M. Bloomberg, Manifolds with no periodic homeomorphisms, Trans. Amer. Math. Soc. 202 (1975), 67-78. MR 50 #11301. MR 358842
- 5.
- A. Borel, Nouvelle démonstration d'un théorème de P. A. Smith, Comment. Math. Helv. 29 (1955), 27-39. MR 16, 611. MR 66649
- 6.
- G. E. Bredon, Examples of differentiate group actions, Topology 3 (1965), 115-122. MR 34 #5105. MR 205273
- 7.
- G. E. Bredon, The cohomology ring structure of a fixed point set, Ann. of Math. (2) 80 (1964), 524-537. MR 32 #1698. MR 184225
- 8.
- G. E. Bredon, Homotopical properties of fixed point sets of circle group actions. I, Amer. J. Math. 91 (1969), 874-888. MR 41 #4534. MR 259905
- 9.
- G. E. Bredon, Fixed point sets of actions on Poincaré duality spaces, Topology 12 (1973), 159-175. MR 48 #9708. MR 331375
- 10.
- G. E. Bredon, Introduction to compact transformation groups, Academic Press, New York, 1972. MR 413144
- 11.
- E. Brieskorn, Beispiele zur Differentialtopologie von Singularitäten, Invent. Math. 2 (1966), 1-14. MR 34 #6788. MR 206972
- 12.
- L. E. J. Brouwer, Über die periodischen Transformationen der Kugel, Math. Ann. 80 (1921), 39-41. MR 1511946
- 13.
- T. Chang and T. Skjelbred, Group actions on Poincaré duality spaces, Bull. Amer. Math. Soc. 78 (1972), 1024-1026. MR 46 #6346. MR 307226
- 14.
- T. Chang and T. Skjelbred, The topological Schur lemma and related results, Ann. of Math. (2) 100 (1974), 307-321. MR 51 #11552. MR 375357
- 15.
- P. E. Conner, On the action of the circle group, Michigan Math. J. 4 (1957), 241-247. MR 20 #3230. MR 96747
- 16.
- P. E. Conner and D. Montgomery, An example for SO(3), Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 1918-1922. MR 148795
- 17.
- P. E. Conner and F. Raymond, Manifolds with few periodic homeomorphisms, Proc. Second Conf. on Compact Transformation Groups, Part II (Univ. of Massachusetts, 1971), Lecture Notes in Math., vol. 299, Springer-Verlag, Berlin and New York, 1972, pp. 1-75. MR 50 #11294. MR 358835
- 18.
- P. E. Conner, F. Raymond and P. J. Weinberger, Manifolds with no periodic maps, Proc. Second Conf. on Compact Transformation Groups, Part II (Univ. of Massachusetts, 1971), Lecture Notes in Math., vol. 299, Springer-Verlag, Berlin and New York, 1972, pp. 81-108. MR 50 #11295. MR 358836
- 19.
- A. L. Edmonds and R. Lee, Fixed point sets of group actions on euclidean space, Topology 14 (1975), 339-345. MR 413146
- 20.
- S. Eilenberg, Sur les transformations periodiques de la surface de sphere, Fund. Math. 22 (1934), 28-41.
- 21.
- E. E. Floyd, Examples of fixed point sets of periodic maps, Ann. of Math. (2) 55 (1952), 167-171. MR 13, 573. MR 45379
- 22.
- E. E. Floyd, Examples of fixed point sets of periodic maps. II, Ann. of Math. (2) 64 (1956), 396-398. MR 18, 141. MR 79753
- 23.
- E. E. Floyd, Fixed point sets of compact abelian Lie groups of transformations, Ann. of Math. (2) 66 (1957), 30-35. MR 19, 571. MR 88733
- 24.
- E. E. Floyd and R. W. Richardson, An action of a finite group on an n-cell without stationary points, Bull. Amer. Math. Soc. 65 (1959), 73-76. MR 20 #7276. MR 100848
- 25.
- J. P. E. Hodgson, Subcomplexes of Poincaré complexes, Bull. Amer. Math. Soc. 80 (1974), 1146-1150. MR 50 #14760. MR 362318
- 26.
- W. C. Hsiang and W.-Y. Hsiang, On the compact subgroups of the diffeomorphism groups of Kervaire spheres, Ann. of Math. (2) 85 (1967), 359-369. MR 35 #4934. MR 214083
- 27.
- W. C. Hsiang and W.-Y. Hsiang, The degree of symmetry of homotopy spheres, Ann. of Math. (2) 89 (1969), 52-67. MR 39 #978. MR 239621
- 28.
- W. C. Hsiang and W.-Y. Hsiang, Differentiable actions of compact connected Lie groups. III, Ann. of Math. (2) 99 (1974), 220-256. MR 49 #11550. MR 346828
- 29.
- W.-Y. Hsiang, On the bound of the dimensions of the isometry groups of all possible riemannian metrics on an exotic sphere, Ann. of Math. (2) 85 (1967), 351-358. MR 35 #4935. MR 214084
- 30.
- W.-Y. Hsiang, On the splitting principal and the geometric weight system of topological transformation groups, Proc. Second Conf. on Compact Transformation Groups, Part I (Univ. of Massachusetts, 1971), Lecture Notes in Math., vol. 298, Springer-Verlag, Berlin and New York, 1972, pp. 334-402. MR 52 #1742. MR 380845
- 31.
- W.-Y. Hsiang, On some fundamental theorems in cohomology theory of topological transformation groups, Taita J. Math. 2 (1970), 66-87. MR 46 #8247. MR 309137
- 32.
- L. Jones, The converse to the fixed point theorem of P. A. Smith. I, Ann. of Math. (2) 94 (1971), 52-68. MR 45 #4427. MR 295361
- 33.
- L. Jones, The converse to the fixed point theorem of P. A. Smith. II, Indiana Univ. Math. J. 22 (1972/73), 309-325. MR 49 #4010. MR 339249
- 34.
- R. Kerekjarto, Über die periodischen Transformationen der Kreisscheibe und der Kugelflache, Math. Ann. 80 (1921), 36-38. MR 1511945
- 35.
- B. Lawson and S. T. Yau, Scalar curvature, nonabelian group actions, and the degree of symmetry of exotic spheres, Comment. Math. Helv. 49 (1974), 232-244. MR 358841
- 36.
- J. Milnor and J. C. Moore, On the structure of Hopf algebras, Ann. of Math. (2) 81 (1965), 211-264. MR 30 #4259. MR 174052
- 37.
- D. Montgomery and L. Zippin, Periodic one-parameter groups in three-space, Trans. Amer. Math. Soc. 40 (1936), 24-36. MR 1501864
- 38.
- D. Montgomery and L. Zippin, Non-abelian, compact, connected transformation groups of three-space, Amer. J. Math. 61 (1939), 375-387. MR 1507382
- 39.
- D. Montgomery and L. Zippin, Topological transformation groups, Interscience, New York and London, 1955. MR 17, 383. MR 73104
- 40.
- R. Oliver, Fixed-point sets of group actions on finite acyclic complexes, Comment. Math. Helv. 50 (1975), 155-177. MR 51 #11556. MR 375361
- 41.
- R. Oliver, Projective obstructions to group actions on disks (to appear).
- 42.
- R. Oliver, Smooth compact Lie group actions on disks (to appear). MR 423390
- 43.
- V. Puppe, On a conjecture of Bredon, Manuscripta Math. 12 (1974), 11-16. MR 49 #11502. MR 346778
- 44.
- D. Quillen, The spectrum of an equivariant cohomology ring. I, II, Ann. of Math. (2) 94 (1971), 549-572; ibid. (2) 94 (1971), 573-602. MR 45 #7743. MR 298694
- 45.
- R. Schultz, Improved estimates for the degree of symmetry of certain homotopy spheres, Topology 10 (1971), 227-235. MR 44 # 1052. MR 283822
- 46.
- R. Schultz, Semifree circle actions and the degree of symmetry of homotopy spheres, Amer. J. Math. 93 (1971), 829-839. MR 44 #4752. MR 287548
- 47.
- R. Schultz, Circle actions on homotopy spheres not bounding spin manifolds, Trans. Amer. Math. Soc. 213 (1975), 89-98. MR 52 #1750. MR 380853
- 48.
- G. Segal, Equivariant K-theory, Inst. Hautes Études Sci. Publ. Math. 34 (1968), 129-151. MR 38 #2769. MR 234452
- 49.
- P. A. Smith, Transformations of finite period. I, II, III, IV, Ann. of Math. (2) 39 (1938), 127-164; ibid. 40 (1939), 690-711; ibid. 42 (1941), 446-458; ibid. 46 (1945), 357-364. MR 1, 30; 2, 324; 7, 136. MR 1503393
- 50.
- J. C. Su, Periodic transformations on the product of two spheres, Trans. Amer. Math. Soc. 112 (1964), 369-380. MR 29 #612. MR 163309
- 51.
- D. Sullivan, Differential forms and the topology of manifolds, Manifolds-Tokyo 1973 (Proc. Internat. Conf. on Manifolds and Related Topics in Topology), Univ. of Tokyo Press, Tokyo, 1975, pp. 37-49. MR 51 #6838. MR 370611
- 52.
- R. W. Sullivan, Linear models for compact groups acting on spheres, Topology 13 (1974), 77-87. MR 49 #4006. MR 339245
- 53.
- R. Swan, A new method in fixed point theory, Comment. Math. Helv. 34 (1960), 1-16. MR 22 #5978. MR 115176
Review Information:
Reviewer:
Glen E. Bredon
Journal:
Bull. Amer. Math. Soc.
83 (1977), 711-718
DOI:
https://doi.org/10.1090/S0002-9904-1977-14359-0