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Characteristic classes for permutation representations

Published online by Cambridge University Press:  24 October 2008

G. B. Segal  and
C. T. Stretch
G. B. Segal
Affiliation:
University of Oxford
C. T. Stretch
Affiliation:
University of Western Australia
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Extract

To a finite-dimensional real representation V of a finite group G there are associated its Stiefel–Whitney classes wk (V) (k = 1, 2, 3, …) in the cohomology groups Hk(G; ). ( is the field with two elements.) The total Stiefel-Whitney class

in the ring H*(G; is natural with respect to G in the obvious sense, and, in addition,

(a) exponential, i.e. w(V ⊕ W) = w(V).w(W),and

(b) stable, i.e. w(V) = 1 when F is a trivial representation.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 2 , September 1981 , pp. 265 - 272
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Laitinen, E.On the Burnside ring and stable cohomology of a finite group. Math. Scand. 44 (1979), 37–72.CrossRefGoogle Scholar
(2)Milnor, J. W. and Stasheff, J. D.Characteristic classes (Ann. of Math, studies, Princeton, no. 76, 1974).CrossRefGoogle Scholar
(3)Nakaoka, M.The homology of the infinite symmetric group. Ann. of Math. 73 (1961), 229–257.CrossRefGoogle Scholar
(3a)Quillen, D.The Adams conjecture. Topology 10 (1971), 67–80.CrossRefGoogle Scholar
(4)Segal, G. B.Equivariant stable homotopy theory. Proc. Int. Congress of Mathematicians, Nice 1970, vol. 2, 59–63.Google Scholar
(5)Segal, G. B.Operations in stable homotopy theory. New Developments in Topology, London Math. Soc. Lecture Notes in Mathematics 11 (1974), 105–110.CrossRefGoogle Scholar
(6)Stretch, C. T. The homology of the symmetric groups, (to appear.)Google Scholar
(7)Stretch, C. T.Stable cohomotopy and cobordism of abelian groups. Math. Proc. Camb. Phil. Soc. 90 (1981), 273–278.CrossRefGoogle Scholar
(8)Tom Dieck, T.Transformation groups and representation theory. Springer Lecture Notes in Mathematics 766 (1979).CrossRefGoogle Scholar