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)



Characteristic classes and transfer relations in cobordism

Authors: M. Bakuradze, M. Jibladze and V. V. Vershinin
Journal: Proc. Amer. Math. Soc. 131 (2003), 1935-1942
MSC (2000): Primary 55N22, 55R12
Published electronically: October 1, 2002
MathSciNet review: 1955284
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Decompositions of products of the Ray elements by free generators of small dimensions in the symplectic cobordism ring are obtained. In particular it is stated that most of the $4n$-dimensional generators, for $n$ small, after multiplication by the Ray elements $\phi_i$, $i\geqslant0$, land in the ideal generated by Ray elements of low dimension.

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

  • 1. M. F. Atiyah, Characters and cohomology of finite groups, Publ. Math. of the I.H.E.S., 9 (1961) 23-64 MR 26:6228
  • 2. J. C. Becker, P. H. Gottlieb, The transfer map and fibre bundles, Topology, 14 (1975) 1-15
  • 3. J. M. Boardman, Stable homotopy theory (mimeographed), University of Warwick (1966)
  • 4. K. Brown, Cohomology of groups. Graduate Texts in Mathematics, 87. Springer-Verlag, New York-Berlin, 1982. 306 pp. MR 83k:20002
  • 5. V. M. Buchstaber, Characteristic classes in cobordisms and topological applications of theories of one and two valued formal groups, Itogi nauki i tekniki, 10 (1977) 5-178
  • 6. A. Dold, The fixed point transfer of fibre preserving maps, Math. Z., 148 (1976) 215-244 MR 55:6416
  • 7. M. Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. (1979, July), 251, 139-169 MR 80k:55049
  • 8. V. G. Gorbunov, Symplectic cobordism of projective spaces, Math. Sbornik, 181 (1990) 506-520 MR 91i:55006
  • 9. V. Gorbunov, N. Ray, Orientations of Spin bundles and symplectic cobordism. Publ. Res. Inst. Math. Sci., 28 (1992), no. 1, 39-55 MR 93e:55008
  • 10. M. Imaoka, Symplectic Pontrjagin numbers and homotopy groups of $M{Sp}(n)$. Hiroshima Math. J., 12 (1982) no. 1, 151-181 MR 83e:57026
  • 11. D. S. Kahn, S. B. Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc., 78 (1972) 981-987 MR 46:8220
  • 12. J. Milnor, On the cobordism ring $\Omega \sp{*} $ and a complex analogue. I. Amer. J. Math., 82 (1960) 505-521 MR 22:9975
  • 13. R. Nadiradze, Characteristic classes in SC$^*$ theory and their applications, I. Baku Intern. Top. Conf. Abstracts (1987), 213; II. Preprint, Tbilisi Razmadze Math. Inst. (1991), 1-11; III. Preprint, Heidelberg, 58 (1993) 1-21
  • 14. S. P. Novikov, Homotopy properties of Thom complexes. Mat. Sb. (N.S.), 57 (99) (1962) 407-442. (Russian) MR 28:615
  • 15. N. Ray, Indecomposables in $Tors M{\mathrm{Sp}}_*$, Topology, 10 (1971) 261-270 MR 45:9342
  • 16. N. Ray, The symplectic bordism ring, Proc. Camb. Phil. Soc., 71 (1972) 271-282 MR 44:7567b
  • 17. F. W. Roush, On some torsion classes in symplectic bordism, Preprint.
  • 18. V. V. Vershinin, Computation of the symplectic cobordism ring in dimensions less than 32 and the non-triviality of the majority of the triple products of Ray's elements, Sibirsk. Math. Zh., 24 (1983) 50-63
  • 19. V. V. Vershinin, Cobordisms and spectral sequences. Translations of Mathematical Monographs, 130. AMS, Providence, RI, 1993. MR 94j:55006
  • 20. V. V. Vershinin, A. L. Anisimov, A series of elements of order $4$ in the symplectic cobordism ring. Canad. Math. Bull., 38 (1995), no. 3, 373-381 MR 96i:55011

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 55N22, 55R12

Retrieve articles in all journals with MSC (2000): 55N22, 55R12

Additional Information

M. Bakuradze
Affiliation: Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia

M. Jibladze
Affiliation: Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia

V. V. Vershinin
Affiliation: Département des sciences mathématiques, CNRS, UMR 5030 (GTA), Université Montpellier II, place Eugéne Bataillon, 34095 Montpellier Cedex 5, France – and – Institute of Mathematics, Novosibirsk 630090, Russian Federation

Received by editor(s): June 10, 2001
Received by editor(s) in revised form: January 15, 2002
Published electronically: October 1, 2002
Additional Notes: The first author was supported by the CRDF grant #GM1-2083 and by the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
The second author was supported by the TMR research network ERB FMRX CT-97-0107 and the INTAS grant #93-3218-EXT
The third author was supported in part by the French-Russian Program of Research EGIDE (dossier No 04495UL)
Communicated by: Paul Goerss
Article copyright: © Copyright 2002 American Mathematical Society