Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On $\mathit{h}$-cobordisms of spherical space forms


Authors: Slawomir Kwasik and Reinhard Schultz
Journal: Proc. Amer. Math. Soc. 127 (1999), 1525-1532
MSC (1991): Primary 57R80, 57S25
DOI: https://doi.org/10.1090/S0002-9939-99-04637-7
Published electronically: January 29, 1999
MathSciNet review: 1473672
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Given a manifold $M$ of dimension at least 4 whose universal covering is homeomorphic to a sphere, the main result states that a compact manifold $W$ is isomorphic to a cylinder $M\times [0,1]$ if and only if $W$ is homotopy equivalent to this cylinder and the boundary is isomorphic to two copies of $M$; this holds in the smooth, PL and topological categories. The result yields a classification of smooth, finite group actions on homotopy spheres (in dimensions $\geq 5$) with exactly two singular points.


References [Enhancements On Off] (What's this?)

  • [Ba] D. Barden, On the structure and classification of differential manifolds, Ph. D. Thesis, Cambridge University, 1965.
  • [BQ] W. Browder and F. Quinn, A surgery theory for $G$-manifolds and stratified sets, in ``Manifolds-Tokyo, 1973 (Conf. Proc. , Univ. of Tokyo, 1973), University of Tokyo Press, 1975, pp. 27-36. MR 51:11543
  • [CS1] S. Cappell and J. Shaneson, On $4$-dimensional $s$-cobordisms, J. Diff. Geom. 22 (1985), 97-115. MR 87j:57034a
  • [CS2] -, On $4$-dimensional $s$-cobordisms II, Comment. Math. Helv. 64 (1989), 338-347. MR 90i:57011
  • [DM] J. Davis and R. Milgram, ``A Survey of the Spherical Space Form Problem", Mathematical Reports Vol. 2 Part 2, Harwood Academic Publishers, London, 1985, pp. 223-283. MR 87e:57001
  • [tD] T. tom Dieck, ``Transformation Groups and Representation Theory", Lecture Notes in Math. Vol. 766, Springer, Berlin-Heidelberg-New York, 1979.
  • [Dr] A. Dress, Induction and structure theorems for orthogonal representations of finite groups, Ann. of Math. 102 (1975), 291-325. MR 52:8235
  • [H] J.-C. Hausmann, $h$-cobordismes entre variétés homéomorphes, Comment. Math. Helv. 50 (1975), 9-13. MR 51:4279
  • [KS1] S. Kwasik and R. Schultz, Desuspension of group actions and the Ribbon Theorem, Topology 27 (1988), 443-457. MR 89m:57043
  • [KS2] -, On $s$-cobordisms of metacyclic prism manifolds, Invent. Math. 97 (1989), 523-552. MR 90e:57063
  • [KS3] -, Vanishing of Whitehead torsion in dimension four, Topology 31 (1992), 735-756. MR 93j:57013
  • [KS4] -, Fake spherical space forms of constant positive scalar curvature, Comment. Math. Helv. 71 (1996), 1-40. MR 97a:53057
  • [KS5] -, Visible surgery, $4$-dimensional $s$-cobordisms and related questions in geometric topology, $K$-Theory 9 (1995), 323-352. MR 96g:57038
  • [MS] M. Masuda and R. Schultz, Invariants of Atiyah-Singer type, classifications up to finite ambiguity, and equivariant inertia groups, Indiana Univ. Math. J. 45 (1996), 545-581. MR 97j:57032
  • [Mi] J. Milnor, Whitehead torsion, Bull. Amer. Math. Soc. 72 (1966), 358-426. MR 33:4922
  • [O1] R. Oliver, $SK_{1}$ for finite group rings I, Invent. Math. 57 (1980), 183-204. MR 82j:18019
  • [O2] -, $SK_{1}$ for finite group rings III, in Algebraic $K$-Theory, Evanston 1980, Lecture Notes in Mathematics Vol. 854, Springer, Berlin-Heidelberg-New York, 1981, pp. 299-337. MR 82m:18006b
  • [Sc] R. Schultz, Differentiable group actions on homotopy spheres II: Ultrasemifree actions, Trans. Amer. Math. Soc. 268 (1981), 255-297. MR 83a:57055
  • [Se] M. Sebastiani, Actions de groupes finis sur les sphères, Comment. Math. Helv. 45 (1970), 405-439. MR 43:4067
  • [Sh] J. Shaneson, Non-simply connected surgery and some results in low-dimensional topology, Comment. Math. Helv. 45 (1970), 333-352. MR 43:1200
  • [TW] C. B. Thomas and C.T.C. Wall, On the structure of finite groups with periodic cohomology, preprint, University of Liverpool, 1979.
  • [U1] F. Ushitaki, $SK_{1}(\mathbf{Z}[G])$ of finite solvable groups which linearly and freely on spheres, Osaka J. Math. 28 (1991), 117-127. MR 92d:19001
  • [U2] -, On $G$-$h$- cobordisms between $G$-homotopy spheres, Osaka J. Math. 31 (1994), 597-612. MR 96b:57042
  • [U3] -, A generalization of theorem of Milnor, Osaka J. Math., to appear. MR 95i:19002
  • [W1] C.T.C. Wall, ``Surgery on Compact Manifolds", London Math. Soc. Monographs No.1, Academic Press, New York and London, 1970. MR 55:4217
  • [W2] -, Norms of units in group rings, Proc. London Math. Soc. (3) 29 (1974), 593-632. MR 51:12921
  • [W3] -, Formulæ for surgery obstructions, Topology 15 (1976), 189-210. Correction, Topology 16 (1977), 495-496. MR 58:7663

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 57R80, 57S25

Retrieve articles in all journals with MSC (1991): 57R80, 57S25


Additional Information

Slawomir Kwasik
Affiliation: Department of Mathematics, Tulane University, New Orleans, Louisiana 70118
Email: kwasik@math.tulane.edu

Reinhard Schultz
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Address at time of publication: Department of Mathematics, University of California, Riverside, California 92521
Email: schultz@math.ucr.edu

DOI: https://doi.org/10.1090/S0002-9939-99-04637-7
Received by editor(s): June 23, 1997
Received by editor(s) in revised form: September 2, 1997
Published electronically: January 29, 1999
Additional Notes: The first author was partially supported by NSF Grant DMS 91-01575 and by a COR grant from Tulane University. The second author was partially supported by NSF grant DMS 91-02711.
Communicated by: Thomas Goodwillie
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society