Hostname: page-component-77c89778f8-gq7q9 Total loading time: 0 Render date: 2024-07-17T03:51:11.512Z Has data issue: false hasContentIssue false
  • English
  • Français

Surgery on -Manifolds

Published online by Cambridge University Press:  20 November 2018

J. A. Hillman  and
S. K. Roushon
J. A. Hillman
Affiliation:
School of Mathematics and Statistics, University of Sydney, Sydney, NSW 2006, Australiae-mail: jonathan.hillman@sydney.edu.au
S. K. Roushon
Affiliation:
School of Mathematics, Tata Institute, Mumbai 400-005, Indiae-mail: roushon@math.tifr.res.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We show that closed $\widetilde{\mathbb{S}\mathbb{L}}\,\times \,{{\mathbb{E}}^{n}}$-manifolds are topologically rigid if $n\,\ge \,2$, and are rigid up to $s$-cobordism, if $n\,=\,1$.

Keywords

57R6757N16topological rigiditygeometric structuresurgery groups
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 54 , Issue 2 , 01 June 2011 , pp. 283 - 287
Copyright
Copyright © Canadian Mathematical Society 2011

References

[1] Conner, P. E. and Raymond, F., Deforming homotopy equivalences to homeomorphisms in aspherical manifolds. Bull. Amer. Math. Soc. 83(1977), 36–85. doi:10.1090/S0002-9904-1977-14179-7Google Scholar
[2] Eberlein, P., A canonical form for compact nonpositively curved manifolds whose fundamental groups have nontrivial centre. Math. Ann. 260(1982), 23–29. doi:10.1007/BF01475751Google Scholar
[3] Farrell, F. T. and Hsiang, W.-C., The Whitehead group of poly-(finite or cyclic) groups. J. London Math. Soc. 24(1981), 308–324. doi:10.1112/jlms/s2-24.2.308Google Scholar
[4] Farrell, F. T. and Jones, L. E., Topological rigidity for compact nonpositively curved manifolds. In: Differential Geometry: Riemannian Geometry (Los Angeles, CA, 1990), Proc. Sympos. Pure Math. 54Part 3, Amer. Math. Soc., Providence, RI, 1993, 229–274.Google Scholar
[5] Farrell, F. T. and Jones, L. E., Compact infrasolvmanifolds are smoothly rigid. In: Geometry from the Pacific Rim (Singapore, 1994), W. de Gruyter, Berlin, 1997, 85–97.Google Scholar
[6] Hillman, J. A., Four-Manifolds, Geometries and Knots. Geometry & Topology Monographs 5, Geometry & Topology Publications, Coventry, 2002.Google Scholar
[7] Neumann, W. D., Commensurability and virtual fibration for graph manifolds. Topology 36(1997), no. 2, 355–378. doi:10.1016/0040-9383(96)00014-6Google Scholar
[8] Nicas, A. and Stark, C. W., K-Theory and surgery of codimension-two torus actions on aspherical manifolds. J. London Math. Soc. 31(1985), 173–183. doi:10.1112/jlms/s2-31.1.173Google Scholar
[9] Leeb, B., 3-manifolds with(out) metrics of nonpositive curvature. Invent. Math. 122(1995), 277–289. doi:10.1007/BF01231445Google Scholar
[10] Roushon, S. K., L-Theory of 3-manifolds with nonvanishing first Betti number. Internat. Math. Res. Notices 2000, No 3, 107–113. doi:10.1155/S1073792800000076Google Scholar
[11] Thurston, W. P., Three-Dimensional Geometry and Topology, Vol. 1. Princeton Mathematical Series 35, Princeton University Press, Princeton, NJ, 1997.Google Scholar
[12] Waldhausen, F., Algebraic K-theory of generalized free products. Ann. of Math. (2) 108(1978), 135–256. doi:10.2307/1971165Google Scholar
[13] Zieschang, H., Vogt, E. and Coldewey, H.-D., Surfaces and Planar Discontinuous Groups. Lecture Notes in Mathematics 835, Springer-Verlag, Berlin–Heideleberg–New York, 1980.Google Scholar