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The motivic cohomology of BSOn

Published online by Cambridge University Press:  11 April 2017

MASANA HARADA
Affiliation:
Department of Mathematics, Kyoto University, Kyoto606-8502, Japan. e-mail: [email protected]
MASAYUKI NAKADA
Affiliation:
Kobe University Secondary School, 5-11-1Sumiyoshiyamate, Higashinada-ku, Kobe, Hyogo658-0063Japan. e-mail: [email protected]

Abstract

We will determine the motivic cohomology H*,* (BSOn, $\mathbb{Z}$/2) with coefficients in $\mathbb{Z}$/2 of the classifying space of special orthogonal groups SOn over the complex numbers $\mathbb{C}$.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2017 

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References

REFERENCES

[1] Edidin, D. and Graham, W. Characteristic classes and quadratic bundles. Duke Math. J. 78 (1995), 277299.Google Scholar
[2] Edidin, D. and Graham, W. Equivariant intersection theory. Invent. Math. 131 (1998), 595634.CrossRefGoogle Scholar
[3] Field, R. The Chow ring of the classifying space BSO(2n,$\mathbb{C}$). J. of Algebra 350 (2012), 330339.CrossRefGoogle Scholar
[4] Field, R. The Chow ring of the symmetric space Gl(2n,$\mathbb{C}$)/SO(2n,$\mathbb{C}$). J. of Algebra 349 (2012), 364371.Google Scholar
[5] Grothendieck, A. Torsion homologique et sections rationelles. Séminaire Claude Chevalley. tome 3, exp. 5 (1958).Google Scholar
[6] Inoue, K. and Yagita, N. The complex cobordism of BSO n. Kyoto J. Math. 50 (2010), 307324.Google Scholar
[7] Kono, A. and Yagita, N. Brown–Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups. Trans. Amer. Math. Soc. 339 (1993), 781798.CrossRefGoogle Scholar
[8] Molina, L. and Vistoli, A. On the Chow rings of classifying spaces for classical groups. Rend. Sem. Mat. Univ. Padova 116 (2006), 271298.Google Scholar
[9] Morel, F. and Voevodsky, V. $\mathbb{A}$1-homotopy theory of schemes. Publ. Math. IHES 90 (1999), 45143.CrossRefGoogle Scholar
[10] Orlov, D., Vishik, A. and Voevodsky, V. An exact sequence for K M*/2 with applications to quadratic forms. Ann. of Math. 165 (2007), 113.Google Scholar
[11] Pandraharipande, R. Equivarinat Chow rings for O(k), SO(2k+1) and SO(4). J. Reine Angew. Math. 496 (1998), 131148.Google Scholar
[12] Panin, I. Oriented cohomology theories of algebraic varieties. II (After I. Panin and A. Smirnov). Homology, Homotopy Appl. 11 (2009), 349405.Google Scholar
[13] Soulynin, A. A. Gysin homomorphism in generalised cohomology theories. St. Petersburg Math. J. 17 (2006), 511525.Google Scholar
[14] Suslin, A. and Voevodsky, V. Bloch-Kato conjecture and motivic cohomology with finite coefficients, volume 548 of NATO Sci. Ser. C Math. Phys. Sci. (Kluwer Acad. Publ, Dordrecht, 2000) 117189.Google Scholar
[15] Totaro, B. Torsion algebraic cycles and complex cobordism. J. Amer. Math. Soc. 10 (1997), 467493.Google Scholar
[16] Totaro, B. The Chow ring of a classifying space. Algebraic K-theory (Seattle, WA, 1997), Amer. Math. Soc, 67 (1999), 249281.Google Scholar
[17] Vezzosi, G. On the Chow ring of the classifying stack of ${PGL}_{3,\mathbb{C}}$. J. Reine Angew. Math. 523 (2000), 154.Google Scholar
[18] Voevodsky, V. The Milnor conjecture. Preprint (1996).Google Scholar
[19] Voevodsky, V. Motivic cohomolgy with $\mathbb{Z}$/2-coefficients. Publ. Math. IHES 98 (2003), 59104.CrossRefGoogle Scholar
[20] Voevodsky, V. Reduced power operations in motivic cohomology. Publ. Math. IHES 98 (2003), 157.Google Scholar
[21] Stephen, W. Wilson. The complex cobordism of BO(n). J. London Math. Soc. 29 (1984), 352366.Google Scholar
[22] Yagita, N. Examples for the mod p motivic cohomology of classifying spaces. Trans. AMS. 355 (2003), 44274450.Google Scholar
[23] Yagita, N. Applications of Atiyah-Hirzebruch spectral sequence for motivic cobordism. Proc. London Math. Soc. 90 (2005), 783816.CrossRefGoogle Scholar
[24] Yagita, N. Coniveau filtration of cohomology of groups. Proc. London Math. Soc. 101 (2010), 179206.Google Scholar