## Algebraic & Geometric Topology

### Transfer and complex oriented cohomology rings

#### Abstract

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.

#### Article information

Source
Algebr. Geom. Topol., Volume 3, Number 1 (2003), 473-509.

Dates
Revised: 12 May 2003
Accepted: 5 June 2003
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882380

Digital Object Identifier
doi:10.2140/agt.2003.3.473

Mathematical Reviews number (MathSciNet)
MR1997326

Zentralblatt MATH identifier
1026.55020

#### Citation

Bakuradze, Malkhaz; Priddy, Stewart. Transfer and complex oriented cohomology rings. Algebr. Geom. Topol. 3 (2003), no. 1, 473--509. doi:10.2140/agt.2003.3.473. https://projecteuclid.org/euclid.agt/1513882380

#### References

• J.F. Adams Infinite Loop Spaces. Annals of Mathematics Studies, Princeton University Press, Princeton, (1978).
• M.F. Atiyah Characters and cohomology of finite groups, Publ. Math. of the I.H.E.S. 9, (1961), 23–64.
• K.S. Brown Cohomology of groups , Grad. Texts in Math. 87, Springer, (1982).
• M. Brunetti :Morava $K$-theory of $p$-groups with cyclic maximal subgroups and other related $p$-groups, $K$-Theory 24, (2001), 385–395.
• H. Cartan and S. Eilenberg : \em Homological Algebra, Princeton Math. Series no. 19, Princeton University Press, Princeton, (1956).
• T. tom Dieck Transformation groups and representation theory, Lecture Notes in Math. 766, (1979).
• A. Dold : The fixed point transfer of fibre-preserving maps, Math. Zeit. 148, (1976), 215–244.
• M. Hazewinkel Constructing formal groups III, Applications to complex cobordism and Brown-Peterson cohomology, J. Pure Appl. Algebra 10, (1977/78), 1–18.
• F. Hirzebruch, T. Berger, and R. Jung : \em Manifolds and Modular Forms Aspects of Mathematics, E20, Friedr. Vieweg & Sohn, Braunschweig, (1992).
• M. Hopkins, N. Kuhn, and D. Ravenel Generalized group characters and complex oriented cohomology theories , J. Amer. Math. Soc. 13 3(2000), 553–594.
• J. Hunton The Morava $K$-theories of wreath products, Math. Proc. Camb. Phil. Soc. 107, (1990), 309–318.
• J. HuntonThe complex oriented cohomology of extended powers, Ann. Inst. Fourier, Grenoble 48, 2(1998), 517–534.
• D.S. Kahn, S.B. Priddy Applications of the transfer to stable homotopy theory, Bull Amer. Math. Soc. 78, (1972), 981–987.
• I. Kriz Morava $K$-theory of classifying spaces: Some calculations, Topology 36, (1997), 1247–1273.
• J. McClure and V. Snaith :On the $K$-theory of the extended power construction, Proc. Camb. Phil. Soc. 92, (1982), 263–274.
• D. Quillen Elementary proofs of some results of cobordism theory using Steenrod operations, Advances in Math. 7, (1971), 29–56.
• D. Ravenel Complex cobordism and Stable Homotopy Groups of Spheres, Academic Press, (1986).
• D. Ravenel and S. WilsonThe Morava K-theories of Eilenbeg-MacLane spaces and the Conner-Floyd conjecture, Amer. J. of Math. 102,4(1980), 691–748.
• H. Sadofsky The root invariant and $v_i$-periodic families, Topology 31 (1991), 65–111.
• B. Schuster :On the Morava $K$-theory of some finite $2$-groups, Math. Proc. Camb. Phil. Soc. 121 (1997), 7–13.
• B. Schuster and N. Yagita :Morava $K$-theory of extraspecial $2$-groups, preprint.
• M. Tezuka and N. Yagita :Cohomology of finite groups and Brown-Peterson cohomology, Algebraic Topology Arcata 1986, Springer LNM 1370 (1989), 396–408.
• U. Würgler Commutative ring spectra in characteristic $2$, Comm. Math. Helv. 61, (1986), 33–45.
• N. Yagita :Note on $BP$-theory for extensions of cyclic groups by elementary abelian $p$-groups, Kodai Math. J. 20 2(1997), 79–84.