The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups 𝑃𝑈(𝑝) and exceptional Lie groups

Kameko, Masaki; Yagita, Nobuaki

doi:10.1090/S0002-9947-07-04425-X

The Brown-Peterson cohomology of the classifying spaces of the projective unitary groups $PU(p)$ and exceptional Lie groups
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Trans. Amer. Math. Soc. 360 (2008), 2265-2284 Request permission

Abstract:

For a fixed prime $p$, we compute the Brown-Peterson cohomologies of classifying spaces of $PU(p)$ and exceptional Lie groups by using the Adams spectral sequence. In particular, we see that $BP^*(BPU(p))$ and $K(n)^*(BPU(p))$ are even dimensionally generated.

References

K. Andersen, J. Grodal, M. Møller and A. Viruel. The classification of $p$-compact groups for $p$ odd. preprint arXiv:math. AT/0302346 vl 27 Feb. (2003).
J.M. Boardman. Operations on the Adams spectral sequence for Brown-Peterson homology and cohomology. preprint (1988).
J. Michael Boardman, Conditionally convergent spectral sequences, Homotopy invariant algebraic structures (Baltimore, MD, 1998) Contemp. Math., vol. 239, Amer. Math. Soc., Providence, RI, 1999, pp. 49–84. MR 1718076, DOI 10.1090/conm/239/03597
J. Michael Boardman, David Copeland Johnson, and W. Stephen Wilson, Unstable operations in generalized cohomology, Handbook of algebraic topology, North-Holland, Amsterdam, 1995, pp. 687–828. MR 1361900, DOI 10.1016/B978-044481779-2/50016-X
M. Kameko and M. Mimura. Mui invariant and Milnor operations. preprint (2003).
M. Kameko. Poincaré series of cotorsion products. Preprint (2005).
Akira Kono and Mamoru Mimura, Cohomology mod $3$ of the classifying space of the Lie group $E_{6}$, Math. Scand. 46 (1980), no. 2, 223–235. MR 591603, DOI 10.7146/math.scand.a-11865
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
Luis Alberto Molina Rojas and Angelo Vistoli, On the Chow rings of classifying spaces for classical groups, Rend. Sem. Mat. Univ. Padova 116 (2006), 271–298. MR 2287351
Huỳnh Mui, Modular invariant theory and cohomology algebras of symmetric groups, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 22 (1975), no. 3, 319–369. MR 422451
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
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
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, 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
Aleš Vavpetič and Antonio Viruel, On the mod $p$ cohomology of $B\textrm {PU}(p)$, Trans. Amer. Math. Soc. 357 (2005), no. 11, 4517–4532. MR 2156719, DOI 10.1090/S0002-9947-05-03983-8
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
A. Vistoli. On the cohomology and the Chow ring of the classifying space of $PGL_p$. arXiv:math.AG/0505052 (2005).
V. Voevodsky. The Milnor conjecture. www.math.unic.edu/K-theory/0170 (1996)
Vladimir Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57. MR 2031198, DOI 10.1007/s10240-003-0009-z
Vladimir Voevodsky, Reduced power operations in motivic cohomology, Publ. Math. Inst. Hautes Études Sci. 98 (2003), 1–57. MR 2031198, DOI 10.1007/s10240-003-0009-z
Nobuaki Yagita, Applications of Atiyah-Hirzebruch spectral sequences for motivic cobordism, Proc. London Math. Soc. (3) 90 (2005), no. 3, 783–816. MR 2137831, DOI 10.1112/S0024611504015084