The Evens-Kahn formula for the total Stiefel-Whitney class

Author:
Andrzej Kozlowski

Journal:
Proc. Amer. Math. Soc. **91** (1984), 309-313

MSC:
Primary 55N15; Secondary 20J06, 55P47, 55R40, 57R20

MathSciNet review:
740192

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the (augmented) multiplicative group of classical cohomology ring of a space , with coefficients in . The (augmented) total Stiefel-Whitney class is a natural homomorphism . We show that the functor possesses a 'transfer homomorphism' for double coverings such that 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.

**[1]**M. F. Atiyah,*Characters and cohomology of finite groups*, Inst. Hautes Études Sci. Publ. Math.**9**(1961), 23–64. MR**0148722****[2]**Leonard Evens,*A generalization of the transfer map in the cohomology of groups*, Trans. Amer. Math. Soc.**108**(1963), 54–65. MR**0153725**, 10.1090/S0002-9947-1963-0153725-1**[3]**Leonard Evens,*On the Chern classes of representations of finite groups*, Trans. Amer. Math. Soc.**115**(1965), 180–193. MR**0212099**, 10.1090/S0002-9947-1965-0212099-X**[4]**Leonard Evens and Daniel S. Kahn,*Chern classes of certain representations of symmetric groups*, Trans. Amer. Math. Soc.**245**(1978), 309–330. MR**511412**, 10.1090/S0002-9947-1978-0511412-2**[5]**Alexander Grothendieck,*La théorie des classes de Chern*, Bull. Soc. Math. France**86**(1958), 137–154 (French). MR**0116023****[6]**Daniel S. Kahn and Stewart B. Priddy,*Applications of the transfer to stable homotopy theory*, Bull. Amer. Math. Soc.**78**(1972), 981–987. MR**0309109**, 10.1090/S0002-9904-1972-13076-3**[7]**A. Kozlowski,*The transfer in Segal's cohomology*, Illinois J. Math. (to appear)**[8]**F. Roush, Thesis, Princeton University, 1971.**[9]**Graeme Segal,*The multiplicative group of classical cohomology*, Quart. J. Math. Oxford Ser. (2)**26**(1975), no. 103, 289–293. MR**0380770****[10]**Victor Snaith,*The total Chern and Stiefel-Whitney classes are not infinite loop maps*, Illinois J. Math.**21**(1977), no. 2, 300–304. MR**0433446****[11]**Daniel Quillen,*The Adams conjecture*, Topology**10**(1971), 67–80. MR**0279804**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC:
55N15,
20J06,
55P47,
55R40,
57R20

Retrieve articles in all journals with MSC: 55N15, 20J06, 55P47, 55R40, 57R20

Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9939-1984-0740192-9

Keywords:
Generalized cohomology theory,
transfer homomorphism,
total Stiefel-Whitney class

Article copyright:
© Copyright 1984
American Mathematical Society