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On the Chow rings of classifying spaces for classical groups. (English) Zbl 1112.14008

The authors apply the stratification method, introduced by G. Vezzosi [J. Reine Angew. Math. 523, 1–54 (2000; Zbl 0967.14006)] in order to obtain a unified approach to the known computations of the Chow rings of the classifying spaces of the groups \(\text{GL}_n\), \(\text{SL}_n\), \(\text{Sp}_n\), \(\text{O}_n\), and \(\text{SO}_n\).

MSC:

14C15 (Equivariant) Chow groups and rings; motives
55N22 Bordism and cobordism theories and formal group laws in algebraic topology

Citations:

Zbl 0967.14006
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References:

[1] E. H. BROWN, jr., The cohomology of BSOn and BOn with integer coefficients, Proc. Amer. Math. Soc. 85, no. 2 (1982), pp. 283-288. Zbl0509.55010 MR652459 · Zbl 0509.55010 · doi:10.2307/2044298
[2] D. EDIDIN and W. GRAHAM, Characteristic classes and quadric bundles, Duke Math. J. 78, no. 2 (1995), pp. 277-299. Zbl0932.14003 MR1333501 · Zbl 0932.14003 · doi:10.1215/S0012-7094-95-07812-0
[3] D. EDIDIN and W. GRAHAM, Characteristic classes in the Chow ring, J. Algebraic Geom. 6, no. 3 (1997), pp. 431-443. Zbl0922.14003 MR1487222 · Zbl 0922.14003
[4] D. EDIDIN and W. GRAHAM, Equivariant intersection theory, Invent. Math. 131, no. 3 (1998), pp. 595-634. Zbl0940.14003 MR1614555 · Zbl 0940.14003 · doi:10.1007/s002220050214
[5] M. FESHBACH, The integral cohomology rings of the classifying spaces of O(n) and SO(n), Indiana Univ. Math. J. 32, no. 4, (1983), pp. 511-516. Zbl0507.55014 MR703281 · Zbl 0507.55014 · doi:10.1512/iumj.1983.32.32036
[6] R. E. FIELD, The Chow ring of the classifying space BSO(2n; C), arXiv: math.AG/0411424, 2004.
[7] J. GIRAUD, Cohomologie non abélienne, Springer-Verlag, Berlin, 1971, Die Grundlehren der mathematischen Wissenschaften, Band 179. Zbl0226.14011 MR344253 · Zbl 0226.14011
[8] A. GROTHENDIECK, Sur quelques propriétés fondamentales en théorie des intersections, Exposé 4, Séminaire C. Chevalley, Anneaux de Chow et Applications, Secrétariat mathématique, Paris, 1958. MR106907
[9] A. GROTHENDIECK, Torsion homologique et sections rationelles, Exposé 5, Séminaire C. Chevalley, Anneaux de Chow et Applications, Secrétariat mathématique, Paris, 1958.
[10] P. GUILLOT, Chow rings and cobordism of some Chevalley groups, Math. Proc. Cambridge Philos. Soc. 136, no. 3 (2004), pp. 625-642. Zbl1083.14008 MR2055051 · Zbl 1083.14008 · doi:10.1017/S0305004103007369
[11] P. GUILLOT, The Chow rings of G2 and Spin7, arXiv: math.AG/0508122, 2004. MR2055051
[12] P. GUILLOT, Steenrod operations on the Chow ring of a classifying space, arXiv: math.AG/0403415, to appear in Adv. Math., 2004. Zbl1082.14010 MR2166309 · Zbl 1082.14010 · doi:10.1016/j.aim.2004.07.009
[13] A. KONO and M. MIMURA, On the cohomology of the classifying spaces of PSU(4n \? 2) and PO(4n \? 2), Publ. Res. Inst. Math. Sci. 10, no. 3 (1974/ 75), 691-720. Zbl0315.55024 MR372899 · Zbl 0315.55024 · doi:10.2977/prims/1195191887
[14] A. KONO, M. MIMURA, and N. SHIMADA, Cohomology of classifying spaces of certain associative H-spaces, J. Math. Kyoto Univ. 15, no. 3 (1975), pp. 607-617. Zbl0327.55022 MR388426 · Zbl 0327.55022
[15] M. KAMEKO and Y. NOBUAKI, The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups PU(p) and exceptional Lie groups, preprint, 2005. MR2373313 · Zbl 1136.55002
[16] R. PANDHARIPANDE, Equivariant Chow rings of O(k); fSO(2k \? 1), and SO(4), J. Reine Angew. Math. 496 (1998), pp. 131-148. Zbl0905.14026 MR1605814 · Zbl 0905.14026 · doi:10.1515/crll.1998.025
[17] H. TODA, Cohomology of classifying spaces, Homotopy theory and related topics (Kyoto, 1984), Adv. Stud. Pure Math., vol. 9, North-Holland, Amsterdam, 1987, pp. 75-108. Zbl0641.55016 MR896946 · Zbl 0641.55016
[18] B. TOTARO, The Chow ring of a classifying space, Algebraic K-theory (Seattle, WA, 1997), Amer. Math. Soc., Providence, RI, 1999, pp. 249-281. Zbl0967.14005 MR1743244 · Zbl 0967.14005
[19] G. VEZZOSI, On the Chow ring of the classifying stack of PGL3;C, J. Reine Angew. Math. 523 (2000), pp. 1-54. Zbl0967.14006 MR1762954 · Zbl 0967.14006 · doi:10.1515/crll.2000.048
[20] A. VISTOLI, On the Chow ring and the cohomology of the classifying space of PGLp, arXiv: math.AG/0505052, 2005.
[21] A. VAVPETICÏ and A. VIRUEL, On the mod p cohomology of BPU(p), arXiv: math.AT/0312441, 2003.
[22] N. YAGITA, Chow rings of classifying spaces of extraspecial p-groups, Recent progress in homotopy theory (Baltimore, MD, 2000), Contemp. Math., vol. 293, Amer. Math. Soc., Providence, RI, 2002, pp. 397-409. Zbl1031.55010 MR1890746 · Zbl 1031.55010
[23] N. YAGITA, Examples for the mod p motivic cohomology of classifying spaces, Trans. Amer. Math. Soc. 355 (2003), no. 11, 4427-4450 (electronic). Zbl1077.14027 MR1990757 · Zbl 1077.14027 · doi:10.1090/S0002-9947-03-03177-5
[24] N. YAGITA, The image of cycle map of the classifying space of the exceptional group F4, J. Math. Kyoto Univ. 44, no. 1 (2004), pp. 181-191. Zbl1083.55005 MR2062713 · Zbl 1083.55005
[25] N. YAGITA, Applications of Atiyah-Hirzebruch spectral sequences for motivic cobordism, Proc. London Math. Soc. (3) 90, no. 3 (2005), pp. 783-816. Zbl1086.55005 MR2137831 · Zbl 1086.55005 · doi:10.1112/S0024611504015084
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