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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

Generalized group characters and complex oriented cohomology theories


Authors: Michael J. Hopkins, Nicholas J. Kuhn and Douglas C. Ravenel
Journal: J. Amer. Math. Soc. 13 (2000), 553-594
MSC (2000): Primary 55N22; Secondary 55N34, 55N91, 55R35, 57R85
Published electronically: April 26, 2000
MathSciNet review: 1758754
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Abstract: Let $BG$ be the classifying space of a finite group $G$. Given a multiplicative cohomology theory $E^{*}$, the assignment

\begin{displaymath}G \longmapsto E^{*}(BG) \end{displaymath}

is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for complex oriented cohomology theories $E^{*}$, using the theory of complex representations of finite groups as a model for what one would like to know.

An analogue of Artin's Theorem is proved for all complex oriented $E^*$: the abelian subgroups of $G$ serve as a detecting family for $E^*(BG)$, modulo torsion dividing the order of $G$.

When $E^*$ is a complete local ring, with residue field of characteristic $p$and associated formal group of height $n$, we construct a character ring of class functions that computes $\frac{1}{p}E^*(BG)$. The domain of the characters is $G_{n,p}$, the set of $n$-tuples of elements in $G$ each of which has order a power of $p$. A formula for induction is also found. The ideas we use are related to the Lubin-Tate theory of formal groups. The construction applies to many cohomology theories of current interest: completed versions of elliptic cohomology, $E_n^*$-theory, etc.

The $n$th Morava K-theory Euler characteristic for $BG$ is computed to be the number of $G$-orbits in $G_{n,p}$. For various groups $G$, including all symmetric groups, we prove that $K(n)^*(BG)$ is concentrated in even degrees.

Our results about $E^*(BG)$ extend to theorems about $E^*(EG\times_G X)$, where $X$ is a finite $G$-CW complex.


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Additional Information

Michael J. Hopkins
Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
Email: mjh@math.mit.edu

Nicholas J. Kuhn
Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
Email: njk4x@virginia.edu

Douglas C. Ravenel
Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
Email: drav@math.rochester.edu

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00332-5
PII: S 0894-0347(00)00332-5
Received by editor(s): July 20, 1999
Received by editor(s) in revised form: January 28, 2000
Published electronically: April 26, 2000
Additional Notes: All three authors were partially supported by the National Science Foundation.
Article copyright: © Copyright 2000 American Mathematical Society