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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.83.

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Generalized group characters and complex oriented cohomology theories
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by Michael J. Hopkins, Nicholas J. Kuhn and Douglas C. Ravenel
J. Amer. Math. Soc. 13 (2000), 553-594
Published electronically: April 26, 2000


Let $BG$ be the classifying space of a finite group $G$. Given a multiplicative cohomology theory $E^{*}$, the assignment \[ G \longmapsto E^{*}(BG) \] 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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Bibliographic Information
  • Michael J. Hopkins
  • Affiliation: Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
  • Email:
  • Nicholas J. Kuhn
  • Affiliation: Department of Mathematics, University of Virginia, Charlottesville, Virginia 22903
  • Email:
  • Douglas C. Ravenel
  • Affiliation: Department of Mathematics, University of Rochester, Rochester, New York 14627
  • Email:
  • 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.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 13 (2000), 553-594
  • MSC (2000): Primary 55N22; Secondary 55N34, 55N91, 55R35, 57R85
  • DOI:
  • MathSciNet review: 1758754