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On the transfer in the homology of symmetric groups

Published online by Cambridge University Press:  24 October 2008

Daniel S. Kahn  and
Stewart B. Priddy
Daniel S. Kahn
Affiliation:
Northwestern University, Evanston, Illinois, U.S.A.
Stewart B. Priddy
Affiliation:
Northwestern University, Evanston, Illinois, U.S.A.
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Extract

The transfer has long been a fundamental tool in the study of group cohomology. In recent years, symmetric groups and a geometric version of the transfer have begun to play an important role in stable homotopy theory (2, 5). Thus, motivated by geometric considerations, we have been led to investigate the transfer homomorphism

in group homology, where n is the nth symmetric group, (n, p) is a p-Sylow sub-group and simple coefficients are taken in /p (the integers modulo a prime p). In this paper, we obtain an explicit characterization (Theorem 3·8) of this homomorphism. Roughly speaking, elements in H*(n) are expressible in terms of the wreath product k ∫ l → n (n = kl) and the ordinary product k × n−k→ n. We show that tr* preserves the form of these elements.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 1 , January 1978 , pp. 91 - 101
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

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