Abstract
In this paper, we study the interaction between transferred Chern classes and Chern classes of transferred bundles. We calculate the algebra \( B{P^{*}}\left( {X_{{h\varSigma p}}^p} \right) \) and show that its multiplicative structure is completely determined by the Frobenius reciprocity. We also give some tables of the initial segments of the formal group law in the Morava K-theory which are often useful in calculations.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
J. F. Adams, Infinite Loop Spaces, Ann. Math. Stud., Princeton Univ. Press, Princeton (1978).
M. F. Atiyah, “Characters and cohomology of finite groups,” Publ. Math. I.H.E.S., 9, 23–64 (1961).
M. Bakuradze, “Morava K-theory rings for the modular groups in Chern classes,” K-theory, 38, No. 2, 87–94 (2008).
M. Bakuradze and S. Priddy, “Transfer and complex oriented cohomology rings,” Alg. Geom. Topol., 3, 473–509 (2003).
M. Bakuradze and S. Priddy, “Transferred Chern classes in Morava K-theory,” Proc. Am. Math. Soc., 132, 1855–1860 (2004).
M. Bakuradze and V. Vershinin, “Morava K-theory rings for the dihedral, semidihedral, and generalized quaternion groups in Chern classes,” Proc. Am. Math. Soc., 134, 3707–3714 (2006).
M. Bakuradze, M. Jibladze, and V. Vershinin, “Characteristic classes and transfer relations in cobordisms,” Proc. Am. Math. Soc., 131 No. 6, 1935–1942 (2003).
M. Bakuradze and R. Nadiradze, “Cohomological realizations of two-valued formal groups and their application,” Proc. Tbilisi A. Razmadze Math. Inst., 94, 12–28 (1991).
J. C. Becker and D. H. Gottlieb, “The transfer map and fibre bundles,” Topology, 14, 1–12 (1975).
J. C. Becker and R. E. Schultz, “Axioms for bundle transfers and traces,” Math. Z., 227, No. 4, 583–605 (1998).
J. M. Boardman, Stable Homotopy Theory, Univ. of Warwick (1966).
M. Brunetti, “Morava K-theory of p-groups with cyclic maximal subgroups and other related p-groups,” K-Theory, 24, 385–395 (2001).
K. S. Brown, Cohomology of Groups, Grad. Texts Math., 87, Springer-Verlag, Berlin (1982).
V. M. Buchstaber, “Characteristic classes in cobordisms and topological applications of theories of one- and two-valued formal groups,” in: Itogi Nauki Tekh., 10, VINITI, Moscow (1977), pp. 5–178.
H. Cartan and S. Eilenberg, Homological Algebra, Princeton Math. Ser., 19, Princeton Univ. Press, Princeton (1956).
T. tom Dieck, “Transformation groups and representation theory,” Lect. Notes Math., 766 (1979).
A. Dold, “The fixed point transfer of fibre-preserving maps,” Math. Z., 148, 215–244 (1976).
M. Feshbach, “The transfer and compact Lie groups,” Trans. Am. Math. Soc., 251, 139–169 (1979).
V. G. Gorbunov, “Symplectic cobordism of projective spaces,” Math. Sb., 181, 506–520 (1990).
V. G Gorbunov and N. Ray, “Orientation of Spin(n) bundles and symplectic cobordism,” Publ. RIMS Kyoto Univ., 28, No. 1, 39–55 (1992).
M. Hazewinkel, “Constructing formal groups, III. Applications to complex cobordism and Brown–Peterson cohomology,” J. Pure Appl. Algebra, 10, 1–18 (1977/78).
F. Hirzebruch, T. Berger, and R. Jung, Manifolds and Modular Forms, Asp.Math., E20, Friedrich Vieweg & Sohn, Braunschweig (1992).
M. Hopkins, N. Kuhn, and D. Ravenel, “Generalized group characters and complex oriented cohomology theories,” J. Am. Math. Soc., 13, No. 3, 553–594 (2000).
J. Hunton, “The Morava K-theories of wreath products,” Math. Proc. Cambridge Philos. Soc., 107, 309–318 (1990).
J. Hunton, “The complex oriented cohomology of extended powers,” Ann. Inst. Fourier, Grenoble, 48, No. 2, 517–534 (1998).
M. Imaoka, “Symplectic Pontryagin numbers and homotopy groups of MSp(N),” Hiroshima Math. J., 12, No. 1, 151–181 (1982).
D. S. Kahn and S. B. Priddy, “Applications of the transfer to stable homotopy theory,” Bull Am. Math. Soc., 78, 981–987 (1972).
I. Kriz, “Morava K-theory of classifying spaces: Some calculations,” Topology, 36, 1247–1273 (1997).
R. Nadiradze, “Characteristic classes in the SC *-theory and their applications,” in: Proc. Int. Topol. Conf. Baku (1987); Preprint Razmadze Math. Inst., Tbilisi (1991); Preprint No. 58, Heidelberg (1993).
D. Quillen, “Elementary proofs of some results of cobordism 1theory using Steenrod operations,” Adv. Math., 7, 29–56 (1971).
D. Ravenel, Complex Cobordism and Stable Homotopy Groups of Spheres, Academic Press, Orlando, Florida (1986).
N. Ray, “Indecomposables in TorsMSp * ,” Topology, 10, 261–270 (1971).
N. Ray, “Some results in generalised homology, K-theory, and bordism,” Proc. Cambridge Philos. Soc., 71, 283–300 (1977).
F. W. Roush, “On some torsion classes in symplectic bordism,” unpublished.
B. Schuster, “On the Morava K-theory of some finite 2-groups,” Math. Proc. Cambridge Philos. Soc., 121, 7–13 (1997).
M. Tezuka and N. Yagita, “Cohomology of finite groups and Brown–Peterson cohomology, II,” in: Homotopy Theory and Related Topics (Kinosaki, 1988), Lect. Notes Math., 1418, Springer-Verlag, Berlin (1990), pp. 57–69.
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,” Sib. Mat. Zh., 24, 50–63 (1983).
U. Würgler, “Commutative ring spectra in characteristic 2,” Commun. Math. Helv., 61, 33–45 (1986).
N. Yagita, “Complex K-theory of BSL 3(Z),” K-Theory, 6, 87–95 (1992).
N. Yagita, “Note on BP-theory for extensions of cyclic groups by elementary abelian p-groups,” Kodai Math. J., 20, No. 2, 79–84 (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 77, Complex Analysis and Topology, 2012.
Rights and permissions
About this article
Cite this article
Bakuradze, M. Transferred characteristic classes and generalized cohomology rings. J Math Sci 189, 10–67 (2013). https://doi.org/10.1007/s10958-013-1172-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-013-1172-5