Examples for the mod 𝑝 motivic cohomology of classifying spaces

Yagita, Nobuaki

doi:10.1090/S0002-9947-03-03177-5

Examples for the mod $p$ motivic cohomology of classifying spaces
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by Nobuaki Yagita

Trans. Amer. Math. Soc. 355 (2003), 4427-4450

DOI: https://doi.org/10.1090/S0002-9947-03-03177-5

Published electronically: July 2, 2003

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Abstract:

Let $BG$ be the classifying space of a compact Lie group $G$. Some examples of computations of the motivic cohomology $H^{*,*}(BG;\mathbb {Z}/p)$ are given, by comparing with $H^*(BG;\mathbb {Z}/p)$, $CH^*(BG)$ and $BP^*(BG)$.

References

R.Field. On the Chow ring of classifying space $BSO(2n,C)$ preprint (2000).
David J. Green and Ian J. Leary, Chern classes and extraspecial groups, Manuscripta Math. 88 (1995), no. 1, 73–84. MR 1348791, DOI 10.1007/BF02567806
Po Hu and Igor Kriz, Some remarks on Real and algebraic cobordism, $K$-Theory 22 (2001), no. 4, 335–366. MR 1847399, DOI 10.1023/A:1011196901303
Kouichi Inoue, The Brown-Peterson cohomology of $B\textrm {SO}(6)$, J. Math. Kyoto Univ. 32 (1992), no. 4, 655–666. MR 1194107, DOI 10.1215/kjm/1250519400
Akira Kono and Nobuaki Yagita, Brown-Peterson and ordinary cohomology theories of classifying spaces for compact Lie groups, Trans. Amer. Math. Soc. 339 (1993), no. 2, 781–798. MR 1139493, DOI 10.1090/S0002-9947-1993-1139493-4
I. J. Leary, The integral cohomology rings of some $p$-groups, Math. Proc. Cambridge Philos. Soc. 110 (1991), no. 1, 25–32. MR 1104598, DOI 10.1017/S0305004100070080
Marc Levine and Fabien Morel, Cobordisme algébrique. II, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 9, 815–820 (French, with English and French summaries). MR 1836092, DOI 10.1016/S0764-4442(01)01833-X
Marc Levine and Fabien Morel, Cobordisme algébrique. I, C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), no. 8, 723–728 (French, with English and French summaries). MR 1843195, DOI 10.1016/S0764-4442(01)01832-8
A. S. Merkur′ev and A. A. Suslin, $K$-cohomology of Severi-Brauer varieties and the norm residue homomorphism, Izv. Akad. Nauk SSSR Ser. Mat. 46 (1982), no. 5, 1011–1046, 1135–1136 (Russian). MR 675529
Fabien Morel and Vladimir Voevodsky, $\textbf {A}^1$-homotopy theory of schemes, Inst. Hautes Études Sci. Publ. Math. 90 (1999), 45–143 (2001). MR 1813224, DOI 10.1007/BF02698831
S. P. Novikov, Methods of algebraic topology from the point of view of cobordism theory, Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 855–951 (Russian). MR 0221509
Rahul Pandharipande, Equivariant Chow rings of $\textrm {O}(k),\ \textrm {SO}(2k+1)$, and $\textrm {SO}(4)$, J. Reine Angew. Math. 496 (1998), 131–148. MR 1605814, DOI 10.1515/crll.1998.025
Daniel Quillen, The $\textrm {mod}$ $2$ cohomology rings of extra-special $2$-groups and the spinor groups, Math. Ann. 194 (1971), 197–212. MR 290401, DOI 10.1007/BF01350050
Douglas C. Ravenel, W. Stephen Wilson, and Nobuaki Yagita, Brown-Peterson cohomology from Morava $K$-theory, $K$-Theory 15 (1998), no. 2, 147–199. MR 1648284, DOI 10.1023/A:1007776725714
Björn Schuster and Nobuaki Yagita, Transfers of Chern classes in BP-cohomology and Chow rings, Trans. Amer. Math. Soc. 353 (2001), no. 3, 1039–1054. MR 1804412, DOI 10.1090/S0002-9947-00-02647-7
Hirotaka Tamanoi, The image of the BP Thom map for Eilenberg-Mac Lane spaces, Trans. Amer. Math. Soc. 349 (1997), no. 3, 1209–1237. MR 1401530, DOI 10.1090/S0002-9947-97-01826-6
Michishige Tezuka and Nobuaki Yagita, The varieties of the mod $p$ cohomology rings of extra special $p$-groups for an odd prime $p$, Math. Proc. Cambridge Philos. Soc. 94 (1983), no. 3, 449–459. MR 720796, DOI 10.1017/S0305004100000840
M. Tezuka and N. Yagita, Cohomology of finite groups and Brown-Peterson cohomology, Algebraic topology (Arcata, CA, 1986) Lecture Notes in Math., vol. 1370, Springer, Berlin, 1989, pp. 396–408. MR 1000392, DOI 10.1007/BFb0085243
Hirosi Toda, Cohomology $\textrm {mod}$ $3$ of the classifying space $BF_{4}$ of the exceptional group $\textbf {F}_{4}$, J. Math. Kyoto Univ. 13 (1973), 97–115. MR 321086, DOI 10.1215/kjm/1250523438
Burt Totaro, Torsion algebraic cycles and complex cobordism, J. Amer. Math. Soc. 10 (1997), no. 2, 467–493. MR 1423033, DOI 10.1090/S0894-0347-97-00232-4
Burt Totaro, The Chow ring of a classifying space, Algebraic $K$-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 249–281. MR 1743244, DOI 10.1090/pspum/067/1743244
Gabriele Vezzosi, On the Chow ring of the classifying stack of $\textrm {PGL}_{3,\mathbf C}$, J. Reine Angew. Math. 523 (2000), 1–54. MR 1762954, DOI 10.1515/crll.2000.048
Gabriele Vezzosi, Brown-Peterson spectra in stable $\Bbb A^1$-homotopy theory, Rend. Sem. Mat. Univ. Padova 106 (2001), 47–64. MR 1876212
V. Voevodsky. The Milnor conjecture. Preprint (1996).
V. Voevodsky, Voevodsky’s Seattle lectures: $K$-theory and motivic cohomology, Algebraic $K$-theory (Seattle, WA, 1997) Proc. Sympos. Pure Math., vol. 67, Amer. Math. Soc., Providence, RI, 1999, pp. 283–303. Notes by C. Weibel. MR 1743245, DOI 10.1090/pspum/067/1743245
V. Voevodsky. Reduced power operations in motivic cohomology. preprint (2001).
Nobuaki Yagita, On relations between Brown-Peterson cohomology and the ordinary mod $p$ cohomology theory, Kodai Math. J. 7 (1984), no. 2, 273–285. MR 744140, DOI 10.2996/kmj/1138036912
Nobuaki Yagita, Localization of the spectral sequence converging to the cohomology of an extra special $p$-group for odd prime $p$, Osaka J. Math. 35 (1998), no. 1, 83–116. MR 1621252
N. Yagita. Chow ring of classifying spaces of extraspecial $p$ groups. Recent progress in homotopy theory, Contemp. Math. 293 (2002) 397-409.