Transfer and complex oriented cohomology rings

Bakuradze, Malkhaz¹ ; Priddy, Stewart²

¹Razmadze Mathematical Institute, Tbilisi 380093, Republic of Georgia
²Department of Mathematics, Northwestern University, Evanston IL 60208, USA

Algebraic and Geometric Topology, Tome 3 (2003) no. 1, pp. 473-509

Résumé

For finite coverings we elucidate the interaction between transferred Chern classes and Chern classes of transferred bundles. This involves computing the ring structure for the complex oriented cohomology of various homotopy orbit spaces. In turn these results provide universal examples for computing the stable Euler classes (ie Tr∗(1)) and transferred Chern classes for p–fold covers. Applications to the classifying spaces of p–groups are given.

Détail

DOI : 10.2140/agt.2003.3.473

Keywords: transfer, Chern class, classifying space, complex cobordism, Morava K–theory

Bibliographie

[1] J F Adams, Infinite loop spaces, Annals of Mathematics Studies 90, Princeton University Press (1978)

[2] M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. (1961) 23

[3] K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer (1982)

[4] M Brunetti, Morava $K$–theories of $p$–groups with cyclic maximal subgroups and other related groups, $K$–Theory 24 (2001) 385

[5] H Cartan, S Eilenberg, Homological algebra, Princeton University Press (1956)

[6] T Tom Dieck, Transformation groups and representation theory, Lecture Notes in Mathematics 766, Springer (1979)

[7] A Dold, The fixed point transfer of fibre-preserving maps, Math. Z. 148 (1976) 215

[8] M Hazewinkel, Constructing formal groups III: Applications to complex cobordism and Brown–Peterson cohomology, J. Pure Appl. Algebra 10 (1977/78) 1

[9] F Hirzebruch, T Berger, R Jung, Manifolds and modular forms, Aspects of Mathematics, E20, Friedr. Vieweg Sohn (1992)

[10] M J Hopkins, N J Kuhn, D C Ravenel, Generalized group characters and complex oriented cohomology theories, J. Amer. Math. Soc. 13 (2000) 553

[11] J Hunton, The Morava $K$–theories of wreath products, Math. Proc. Cambridge Philos. Soc. 107 (1990) 309

[12] J R Hunton, The complex oriented cohomology of extended powers, Ann. Inst. Fourier (Grenoble) 48 (1998) 517

[13] D S Kahn, S B Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc. 78 (1972) 981

[14] I Kriz, Morava $K$–theory of classifying spaces: some calculations, Topology 36 (1997) 1247

[15] J E Mcclure, V P Snaith, On the $K$–theory of the extended power construction, Math. Proc. Cambridge Philos. Soc. 92 (1982) 263

[16] D Quillen, Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7 (1971)

[17] D C Ravenel, Complex cobordism and stable homotopy groups of spheres, Pure and Applied Mathematics 121, Academic Press (1986)

[18] D C Ravenel, W S Wilson, The Morava $K$–theories of Eilenberg–Mac Lane spaces and the Conner–Floyd conjecture, Amer. J. Math. 102 (1980) 691

[19] H Sadofsky, The root invariant and $v_1$–periodic families, Topology 31 (1992) 65

[20] B Schuster, On the Morava $K$–theory of some finite 2–groups, Math. Proc. Cambridge Philos. Soc. 121 (1997) 7

[21] B Schuster, N Yagita, Morava $K$–theory of extraspecial 2–groups, Proc. Amer. Math. Soc. 132 (2004) 1229

[22] M Tezuka, N Yagita, Cohomology of finite groups and Brown–Peterson cohomology, from: "Algebraic topology (Arcata, CA, 1986)", Lecture Notes in Math. 1370, Springer (1989) 396

[23] U Würgler, Commutative ring-spectra of characteristic 2, Comment. Math. Helv. 61 (1986) 33

[24] N Yagita, Note on BP-theory for extensions of cyclic groups by elementary abelian $p$–groups, Kodai Math. J. 20 (1997) 79