Characteristic classes and transfer relations in cobordism
Authors:
M. Bakuradze, M. Jibladze and V. V. Vershinin
Journal:
Proc. Amer. Math. Soc. 131 (2003), 19351942
MSC (2000):
Primary 55N22, 55R12
Published electronically:
October 1, 2002
MathSciNet review:
1955284
Fulltext PDF Free Access
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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 dimensional generators, for small, after multiplication by the Ray elements , , land in the ideal generated by Ray elements of low dimension.
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 M. F. Atiyah, Characters and cohomology of finite groups, Publ. Math. of the I.H.E.S., 9 (1961) 2364 MR 26:6228
 2.
 J. C. Becker, P. H. Gottlieb, The transfer map and fibre bundles, Topology, 14 (1975) 115
 3.
 J. M. Boardman, Stable homotopy theory (mimeographed), University of Warwick (1966)
 4.
 K. Brown, Cohomology of groups. Graduate Texts in Mathematics, 87. SpringerVerlag, New YorkBerlin, 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) 5178
 6.
 A. Dold, The fixed point transfer of fibre preserving maps, Math. Z., 148 (1976) 215244 MR 55:6416
 7.
 M. Feshbach, The transfer and compact Lie groups, Trans. Amer. Math. Soc. (1979, July), 251, 139169 MR 80k:55049
 8.
 V. G. Gorbunov, Symplectic cobordism of projective spaces, Math. Sbornik, 181 (1990) 506520 MR 91i:55006
 9.
 V. Gorbunov, N. Ray, Orientations of Spin bundles and symplectic cobordism. Publ. Res. Inst. Math. Sci., 28 (1992), no. 1, 3955 MR 93e:55008
 10.
 M. Imaoka, Symplectic Pontrjagin numbers and homotopy groups of . Hiroshima Math. J., 12 (1982) no. 1, 151181 MR 83e:57026
 11.
 D. S. Kahn, S. B. Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc., 78 (1972) 981987 MR 46:8220
 12.
 J. Milnor, On the cobordism ring and a complex analogue. I. Amer. J. Math., 82 (1960) 505521 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), 111; III. Preprint, Heidelberg, 58 (1993) 121
 14.
 S. P. Novikov, Homotopy properties of Thom complexes. Mat. Sb. (N.S.), 57 (99) (1962) 407442. (Russian) MR 28:615
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 N. Ray, Indecomposables in , Topology, 10 (1971) 261270 MR 45:9342
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 N. Ray, The symplectic bordism ring, Proc. Camb. Phil. Soc., 71 (1972) 271282 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 nontriviality of the majority of the triple products of Ray's elements, Sibirsk. Math. Zh., 24 (1983) 5063
 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 in the symplectic cobordism ring. Canad. Math. Bull., 38 (1995), no. 3, 373381 MR 96i:55011
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Additional Information
M. Bakuradze
Affiliation:
Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia
Email:
maxo@rmi.acnet.ge
M. Jibladze
Affiliation:
Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia
Email:
jib@rmi.acnet.ge
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
Email:
vershini@darboux.math.univmontp2.fr, versh@math.nsc.ru
DOI:
http://dx.doi.org/10.1090/S000299390206728X
PII:
S 00029939(02)06728X
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 #GM12083 and by the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy
The second author was supported by the TMR research network ERB FMRX CT970107 and the INTAS grant #933218EXT
The third author was supported in part by the FrenchRussian Program of Research EGIDE (dossier No 04495UL)
Communicated by:
Paul Goerss
Article copyright:
© Copyright 2002
American Mathematical Society
