On the 𝑝-compact groups corresponding to the 𝑝-adic reflection groups 𝐺(𝑞,𝑟,𝑛)

Castellana, Natàlia

doi:10.1090/S0002-9947-06-04154-7

On the $p$-compact groups corresponding to the $p$-adic reflection groups $G(q,r,n)$
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by Natàlia Castellana PDF

Trans. Amer. Math. Soc. 358 (2006), 2799-2819 Request permission

Abstract:

There exists an infinite family of $p$-compact groups whose Weyl groups correspond to the finite $p$-adic pseudoreflection groups $G(q,r,n)$ of family 2a in the Clark-Ewing list. In this paper we study these $p$-compact groups. In particular, we construct an analog of the classical Whitney sum map, a family of monomorphisms and a spherical fibration which produces an analog of the classical $J$-homomorphism. Finally, we also describe a faithful complexification homomorphism from these $p$-compact groups to the $p$-completion of unitary compact Lie groups.

References

J. Aguadé, Fiberings of spheres by spheres mod $p$, Quart. J. Math. Oxford Ser. (2) 31 (1980), no. 122, 129–137. MR 576332, DOI 10.1093/qmath/31.2.129
K. Andersen, J. Grodal, J. Møller, A. Viruel, The classification of $p$-compact groups, $p$ odd, to appear in Ann. of Math.
Paul F. Baum, On the cohomology of homogeneous spaces, Topology 7 (1968), 15–38. MR 219085, DOI 10.1016/0040-9383(86)90012-1
A. K. Bousfield and D. M. Kan, Homotopy limits, completions and localizations, Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin-New York, 1972. MR 0365573, DOI 10.1007/978-3-540-38117-4
Carlos Broto, Thom modules and mod $p$ spherical fibrations, Proc. Amer. Math. Soc. 114 (1992), no. 4, 1131–1137. MR 1065084, DOI 10.1090/S0002-9939-1992-1065084-4
Carlos Broto, Larry Smith, and Robert Stong, Thom modules and pseudoreflection groups, J. Pure Appl. Algebra 60 (1989), no. 1, 1–20. MR 1014604, DOI 10.1016/0022-4049(89)90104-7
C. Broto and S. Zarati, Nil-localization of unstable algebras over the Steenrod algebra, Math. Z. 199 (1988), no. 4, 525–537. MR 968318, DOI 10.1007/BF01161641
Allan Clark and John Ewing, The realization of polynomial algebras as cohomology rings, Pacific J. Math. 50 (1974), 425–434. MR 367979, DOI 10.2140/pjm.1974.50.425
W. G. Dwyer and C. W. Wilkerson, Homotopy fixed-point methods for Lie groups and finite loop spaces, Ann. of Math. (2) 139 (1994), no. 2, 395–442. MR 1274096, DOI 10.2307/2946585
W. G. Dwyer and C. W. Wilkerson, The elementary geometric structure of compact Lie groups, Bull. London Math. Soc. 30 (1998), no. 4, 337–364. MR 1620888, DOI 10.1112/S002460939800441X
W. G. Dwyer and C. W. Wilkerson, Kähler differentials, the $T$-functor, and a theorem of Steinberg, Trans. Amer. Math. Soc. 350 (1998), no. 12, 4919–4930. MR 1621741, DOI 10.1090/S0002-9947-98-02373-3
W. G. Dwyer and C. W. Wilkerson, The center of a $p$-compact group, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 119–157. MR 1320990, DOI 10.1090/conm/181/02032
W. Dwyer and A. Zabrodsky, Maps between classifying spaces, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 106–119. MR 928826, DOI 10.1007/BFb0083003
Samuel Eilenberg and John C. Moore, Homology and fibrations. I. Coalgebras, cotensor product and its derived functors, Comment. Math. Helv. 40 (1966), 199–236. MR 203730, DOI 10.1007/BF02564371
David Handel, Thom modules, J. Pure Appl. Algebra 36 (1985), no. 3, 237–252. MR 790616, DOI 10.1016/0022-4049(85)90076-3
Jean Lannes, Sur les espaces fonctionnels dont la source est le classifiant d’un $p$-groupe abélien élémentaire, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 135–244 (French). With an appendix by Michel Zisman. MR 1179079, DOI 10.1007/BF02699494
Jean Lannes and Saïd Zarati, Sur les ${\scr U}$-injectifs, Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 2, 303–333 (French, with English summary). MR 868302, DOI 10.24033/asens.1509
J. P. May and F. Neumann, On the cohomology of generalized homogeneous spaces, Proc. Amer. Math. Soc. 130 (2002), no. 1, 267–270. MR 1855645, DOI 10.1090/S0002-9939-01-06372-9
D. McDuff and G. Segal, Homology fibrations and the “group-completion” theorem, Invent. Math. 31 (1975/76), no. 3, 279–284. MR 402733, DOI 10.1007/BF01403148
R. James Milgram, The $\textrm {mod}\ 2$ spherical characteristic classes, Ann. of Math. (2) 92 (1970), 238–261. MR 263100, DOI 10.2307/1970836
Jesper M. Møller, Toric morphisms between $p$-compact groups, Cohomological methods in homotopy theory (Bellaterra, 1998) Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 271–306. MR 1851259
Dietrich Notbohm, Topological realization of a family of pseudoreflection groups, Fund. Math. 155 (1998), no. 1, 1–31. MR 1487985
D. Notbohm, $p$-adic lattices of pseudo reflection groups, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994) Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 337–352. MR 1397742
Volker Puppe, A remark on “homotopy fibrations”, Manuscripta Math. 12 (1974), 113–120. MR 365556, DOI 10.1007/BF01168646
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
Akihiro Tsuchiya, Characteristic classes for spherical fiber spaces, Nagoya Math. J. 43 (1971), 1–39. MR 298687, DOI 10.1017/S0027763000014355
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508, DOI 10.1007/978-1-4612-6318-0
Zdzisław Wojtkowiak, On maps from $\textrm {ho}\varinjlim F$ to $\textbf {Z}$, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 227–236. MR 928836, DOI 10.1007/BFb0083013