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 Free Access

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 -dimensional generators, for small, after multiplication by the Ray elements , , land in the ideal generated by Ray elements of low dimension.

**1.**M. F. Atiyah,*Characters and cohomology of finite groups*, Inst. Hautes Études Sci. Publ. Math.**9**(1961), 23–64. MR**0148722****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.**Kenneth S. Brown,*Cohomology of groups*, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR**672956****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.**Albrecht Dold,*The fixed point transfer of fibre-preserving maps*, Math. Z.**148**(1976), no. 3, 215–244. MR**0433440****7.**Mark Feshbach,*The transfer and compact Lie groups*, Trans. Amer. Math. Soc.**251**(1979), 139–169. MR**531973**, 10.1090/S0002-9947-1979-0531973-8**8.**V. G. Gorbunov,*Symplectic cobordism of projective spaces*, Mat. Sb.**181**(1990), no. 4, 506–520 (Russian); English transl., Math. USSR-Sb.**69**(1991), no. 2, 543–557. MR**1055526****9.**Vassily Gorbunov and Nigel Ray,*Orientations of Spin bundles and symplectic cobordism*, Publ. Res. Inst. Math. Sci.**28**(1992), no. 1, 39–55. MR**1147850**, 10.2977/prims/1195168855**10.**Mitsunori Imaoka,*Symplectic Pontrjagin numbers and homotopy groups of 𝑀𝑆𝑝(𝑛)*, Hiroshima Math. J.**12**(1982), no. 1, 151–181. MR**647236****11.**Daniel S. Kahn and Stewart B. Priddy,*Applications of the transfer to stable homotopy theory*, Bull. Amer. Math. Soc.**78**(1972), 981–987. MR**0309109**, 10.1090/S0002-9904-1972-13076-3**12.**J. Milnor,*On the cobordism ring Ω* and a complex analogue. I*, Amer. J. Math.**82**(1960), 505–521. MR**0119209****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**0157381****15.**Nigel Ray,*Indecomposables in 𝑇𝑜𝑟𝑠𝑀𝑆𝑝_{∗}*, Topology**10**(1971), 261–270. MR**0300296****16.**Nigel Ray,*The symplectic bordism ring*, Proc. Cambridge Philos. Soc.**71**(1972), 271–282. MR**0290384****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, vol. 130, American Mathematical Society, Providence, RI, 1993. Translated from the Russian manuscript by M. Farber; Translation edited by David Louvish. MR**1247708****20.**Vladimir V. Vershinin and Aleksandr L. Anisimov,*A series of elements of order 4 in the symplectic cobordism ring*, Canad. Math. Bull.**38**(1995), no. 3, 373–381. MR**1347312**, 10.4153/CMB-1995-054-9

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

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.univ-montp2.fr, versh@math.nsc.ru

DOI:
http://dx.doi.org/10.1090/S0002-9939-02-06728-X

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