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Characteristic classes of Galois representations

Published online by Cambridge University Press:  24 October 2008

A. Kozlowski
A. Kozlowski
Affiliation:
Department of Mathematics, Wayne State University, Detroit, MI 48202, U.S.A.
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Extract

Let K be a field with char if K≠2 and let Ks denote the separable closure of K and GK the Galois group of the extension Ks/K. If K⊂L is a finite extension and ρ:GL↦Or(R) a (continuous) real representation of GL we have a map ρ:BGL→BO which is used to define Stiefel–Whitney classes wi(ρ) = ρ*(wi). In general if f is any element of H*(BO; ℤ/2) we denote by f(ρ) the characteristic class ρ*(f). Now let

be a genus (see e.g. [9]), for example the total Stiefel–Whitney class w = 1+w1+w2 + … Let K⊂L and ρ be as above and let denote the multiplicative transfer (see e.g. [3, 5, 2, 14, 15]). Our principal result is a generalization of theorem 1 of [3]

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 108 , Issue 3 , November 1990 , pp. 517 - 522
Copyright
Copyright © Cambridge Philosophical Society 1990

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References

REFERENCES

[1]Atiyah, M. F. and Hirzebrtjch, F.. Cohomologie-Operationen und Charakteristische Klassen. Math. Z. 77 (1961), 149–187.CrossRefGoogle Scholar
[2]Fulton, W. and MacPherson, R.. Characteristic classes of direct image bundles for covering maps. Ann. of Math. (2) 125 (1987), 1–92.CrossRefGoogle Scholar
[3]Kahn, B.. Classes de Stiefel–Whitney de formes quadratiques et représentations Galoisiennes réelles. Invent. Math. 78 (1984), 223–256.CrossRefGoogle Scholar
[4]Kahn, D. S. and Priddy, B. S.. The transfer and stable homotopy theory. Math. Proc. Cambridge Philos. Soc. 83 (1978), 103–111.CrossRefGoogle Scholar
[5]Kozlowski, A.. The transfer in Segals cohomology. Illinois J. Math. 27 (1983), 614–623.CrossRefGoogle Scholar
[6]Kozlowski, A.. Operations in Segal's cohomology. Math. Proc. Cambridge Philos. Soc. 95 (1984), 437–441.CrossRefGoogle Scholar
[7]Kozlowski, A.. The Evens–Kahn formula for the total Stiefel–Whitney class. Proc. Amer. Math. Soc. 91 (1984), 309–314.CrossRefGoogle Scholar
[8]Kozlowski, A.. Transfers in the group of multiplicative units of the classical cohomology ring and Stiefel–Whitney classes. Publ. Res. Inst. Math. Sci. 25 (1989), 54–74.CrossRefGoogle Scholar
[9]Madsen, I. and Milgram, R. J.. The Classifying Spaces for Surgery and Cobordism of Manifolds (Princeton University Press, 1979).Google Scholar
[10]Merkurjev, A. S.. On the norm residue of degree 2. Soviet Math. Dokl. 24 (1981), 546–551.Google Scholar
[11]Milnor, J. W.. Algebraic K-theory and quadratic forms. Invent. Math. 9 (1969/1970), 318–344.CrossRefGoogle Scholar
[12]Segal, G. B.. The multiplicative group of classical cohomology. Math. Z. (2) 26 (1975), 289–293.Google Scholar
[13]Serre, J-P.. L'invariant de Witt de la forme Tr x 2. Comment. Math. Helv. 59 (1984), 651–676.CrossRefGoogle Scholar
[14]Snaith, V. P.. TopologicalMethods in Galois Representation Theory (John Wiley & Sons, 1989).Google Scholar
[15]Steiner, R.. Multiplicative transfers in classical cohomology. Proc. Edinburgh Math. Soc. 25 (1982), 113–131.CrossRefGoogle Scholar