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

DOI:
https://doi.org/10.1090/S0894-0347-00-00332-5

Published electronically:
April 26, 2000

MathSciNet review:
1758754

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let be the classifying space of a finite group . Given a multiplicative cohomology theory , the assignment

is a functor from groups to rings, endowed with induction (transfer) maps. In this paper we investigate these functors for

*complex oriented*cohomology theories , 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 : the abelian subgroups of serve as a detecting family for , modulo torsion dividing the order of .

When is a complete local ring, with residue field of characteristic and associated formal group of height , we construct a character ring of class functions that computes . The domain of the characters is , the set of -tuples of elements in each of which has order a power of . 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, -theory, etc.

The th Morava K-theory Euler characteristic for is computed to be the number of -orbits in . For various groups , including all symmetric groups, we prove that is concentrated in even degrees.

Our results about extend to theorems about , where is a finite -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:
https://doi.org/10.1090/S0894-0347-00-00332-5

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