The Evens-Kahn formula for the total Stiefel-Whitney class
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- by Andrzej Kozlowski PDF
- Proc. Amer. Math. Soc. 91 (1984), 309-313 Request permission
Abstract:
Let $G\left ( X \right )$ denote the (augmented) multiplicative group of classical cohomology ring of a space $X$, with coefficients in $Z/2$. The (augmented) total Stiefel-Whitney class is a natural homomorphism $w:KO\left ( X \right ) \to G\left ( X \right )$. We show that the functor $G\left ( {} \right )$ possesses a ’transfer homomorphism’ for double coverings such that $w$ commutes with the transfer. This is related to a question of G. Segal. As a special case, we obtain a formula for the total Stiefel-Whitney class of a representation of a finite group induced from a (real) representation of a subgroup of index 2, which is analogous to the one obtained by Evens and Kahn for the total Chern class.References
- M. F. Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. 9 (1961), 23–64. MR 148722
- Leonard Evens, A generalization of the transfer map in the cohomology of groups, Trans. Amer. Math. Soc. 108 (1963), 54–65. MR 153725, DOI 10.1090/S0002-9947-1963-0153725-1
- Leonard Evens, On the Chern classes of representations of finite groups, Trans. Amer. Math. Soc. 115 (1965), 180–193. MR 212099, DOI 10.1090/S0002-9947-1965-0212099-X
- Leonard Evens and Daniel S. Kahn, Chern classes of certain representations of symmetric groups, Trans. Amer. Math. Soc. 245 (1978), 309–330. MR 511412, DOI 10.1090/S0002-9947-1978-0511412-2
- Alexander Grothendieck, La théorie des classes de Chern, Bull. Soc. Math. France 86 (1958), 137–154 (French). MR 116023
- Daniel S. Kahn and Stewart B. Priddy, Applications of the transfer to stable homotopy theory, Bull. Amer. Math. Soc. 78 (1972), 981–987. MR 309109, DOI 10.1090/S0002-9904-1972-13076-3 A. Kozlowski, The transfer in Segal’s cohomology, Illinois J. Math. (to appear) F. Roush, Thesis, Princeton University, 1971.
- Graeme Segal, The multiplicative group of classical cohomology, Quart. J. Math. Oxford Ser. (2) 26 (1975), no. 103, 289–293. MR 380770, DOI 10.1093/qmath/26.1.289
- Victor Snaith, The total Chern and Stiefel-Whitney classes are not infinite loop maps, Illinois J. Math. 21 (1977), no. 2, 300–304. MR 433446
- Daniel Quillen, The Adams conjecture, Topology 10 (1971), 67–80. MR 279804, DOI 10.1016/0040-9383(71)90018-8
Additional Information
- © Copyright 1984 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 91 (1984), 309-313
- MSC: Primary 55N15; Secondary 20J06, 55P47, 55R40, 57R20
- DOI: https://doi.org/10.1090/S0002-9939-1984-0740192-9
- MathSciNet review: 740192