Bulletin of the American Mathematical Society

MathSciNet review: 1566978
Full text of review: PDF   This review is available free of charge.
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