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Quillen Grassmannians as non-modular homotopy fixed points. (English) Zbl 1065.55009
In [Ann. Math. 139, 395–442 (1994; Zbl 0801.55007)], W.G. Dwyer and C.W. Wilkerson introduced the concept of $$p$$-compact groups, and $$p$$-compact groups for an odd prime $$p$$ recently have been classified: the isomorphism classes are in 1:1-correspondence with the isomorphism classes of finite pseudo-reflection groups over the $$p$$-adic numbers. These pseudo reflection groups have been classified by D. Notbohm in [Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guixols, 1994), Prog. Math. 136, 337–352 (1996; Zbl 0855.20003)], based on the according classification of the finite pseudo reflection groups over $${\mathbb Q}\sphat_p$$ in [A. Clark and J. Ewing, Pac. J. Math. 50, 425–434 (1974; Zbl 0333.55002)]. Generalized Grassmannians are $$p$$-compact groups which correspond to the groups $$G(q,r,n)$$ in the Clark-Ewing list of pseudo reflection groups 2a in [A. Clark and J. Ewing, loc. cit.]. Here $$q,r,n$$ are integers which satisfy $$r| q| p-1$$; the corresponding $$p$$-compact groups are denoted $$X(q,r,n)$$.
In [Fundam. Math 155, 1–31 (1998; Zbl 0896.55013)], D. Notbohm showed that unstable Adams and some other related operations can be defined on the classifying spaces $$BX(q,r,n)$$. In the paper under review the author investigates the homotopy fix point sets of these actions. In particular she shows that $$BX(q,1,[n/q])$$ arises as the homotopy fix point set of the Adams operation of order $$q$$ acting on the $$p$$-completion of $$BU(n)$$. Similarly she shows how $$BX(q,1,n)$$ can arise in various ways as the homotopy fixed point set of unstable Adams type operations acting on other generalized Grassmannians. The latter then is used to describe the set of homotopy representations of elementary abelian $$p$$-groups into the $$p$$-compact group $$X(q,1,n)$$.

##### MSC:
 55R35 Classifying spaces of groups and $$H$$-spaces in algebraic topology 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 55R40 Homology of classifying spaces and characteristic classes in algebraic topology
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##### References:
 [1] Adams, F.: Vector fields on spheres. Ann. of Math. 75(3), 603-632 (1962) · Zbl 0112.38102 · doi:10.2307/1970213 [2] Andersen, K., Grodal, J., Møller, J., Viruel, A.: The classification of p-compact groups for p odd. Preprint arXiv:math.AT/0302346 · Zbl 1149.55011 [3] Baum, B.F.: On the cohomology of homogeneous spaces. Topology 7, 15-38 (1968) · Zbl 0158.42002 · doi:10.1016/0040-9383(86)90012-1 [4] Bourbaki, N.: Éléments de mathématique. Groupes et algèbres de Lie. Chapitre 9, Masson, Paris, 1982 · Zbl 0505.22006 [5] Broto, C., Møller, J.M.: Embeddings of DI2 in F4. Trans. Am. Math. Soc. 353, 4461-3379 (2001) · Zbl 1017.55011 · doi:10.1090/S0002-9947-01-02781-7 [6] Bousfield, A. K., Kan, D. M.: Homotopy limits, completions and localizations. Lecture Notes in Mathematics, Vol. 304, Springer-Verlag, Berlin, 1972 · Zbl 0259.55004 [7] Castellana, N.: On the p-compact groups corresponding to the reflection groups G(q,r,n). To appear in Trans. Am. Math. Soc. · Zbl 1145.55014 [8] Clark, A., Ewing, J.: The realization of polynomial algebras as cohomology rings. Pacific J. of Math. 5, 425-434 (1974) · Zbl 0333.55002 [9] Cooke, G.: Replacing homotopy actions by topological actions. Trans. Am. Math. Soc 237, 391-406 (1978) · Zbl 0434.55008 · doi:10.1090/S0002-9947-1978-0461544-2 [10] Dwyer, W. G., Kan, D. M.: Centric maps and the realization of diagrams in the homotopy category. Proc. Am. Math. Soc. 114(2), 575-584 (1992) · Zbl 0742.55004 · doi:10.1090/S0002-9939-1992-1070515-X [11] Dwyer, W. G., Miller, H. R., Wilkerson, C. W.: Homotopical uniqueness of classifying spaces. Topology 31(1), 29-45 (1992) · Zbl 0748.55005 · doi:10.1016/0040-9383(92)90062-M [12] Dwyer, W. G., Wilkerson, C. W.: The elementary geometric structure of compact Lie groups. Bull. London Math. Soc. 30, 337-364 (1998) · Zbl 0933.22010 · doi:10.1112/S002460939800441X [13] Dwyer, W. G., Wilkerson, C. W.: Homotopy fixed-point methods for Lie groups and finite loop spaces. Ann. Math. 139(2), 395-442 (1994) · Zbl 0801.55007 · doi:10.2307/2946585 [14] Dwyer, W.G., Zabrodsky, A.: Maps between classifying spaces. Proceedings of the 1986 Barcelona Conference on Algebraic Topology, Lecture Notes in Math. 1298, Springer Verlag, 1988 · Zbl 0646.55007 [15] tom Dieck, T.: Transformation groups, de Gruyter Studies in Mathematics, 8. Walter de Gruyter & Co, Berlin-New York, 1987 · Zbl 0611.57002 [16] Jackowski, S., McClure, J., Oliver, B.: Homotopy classification of self-maps of BG via G actions, I,II. Ann. of Math 135, 183-270 (1992) · Zbl 0758.55004 · doi:10.2307/2946568 [17] Harris, B.: On the homotopy groups of classical groups. Ann. of Math. 74, 407-413 (1961) · Zbl 0118.18501 · doi:10.2307/1970240 [18] Mac Lane, S.: Categories for the Working Mathematician. Graduate Text in Mathematics 5 (1971), Springer-Verlag · Zbl 0232.18001 [19] Møller, J.: N-determined p-compact groups. Fund. Math. 173(3), 201-300 (2002) · Zbl 1039.55011 · doi:10.4064/fm173-3-1 [20] Notbohm, D.: A topological realization of a family of pseudoreflection groups. Fund. Math. 155, 1-31 (1998) · Zbl 0896.55013 [21] Notbohm, D.: p-adic lattices of pseudo reflection groups. Proceedings of the Barcelona Conference 1994, Algebraic Topology: New trends in localization and periodicity, ed.: C. Broto, C. Casacuberta and G. Mislin, pp. 337-352 · Zbl 0855.20003 [22] Notbohm, D.: Maps between classifying spaces. Math. Z. 207, 153-168 (1991) · Zbl 0731.55011 · doi:10.1007/BF02571382 [23] Quillen, D.: On the cohomology and K-theory of the general linear groups over a finite field. Ann. of Math. 96, 552-586 (1972) · Zbl 0249.18022 · doi:10.2307/1970825 [24] Sheppard, G.C., Todd, J.A.: Finite unitary reflection groups. Canad. J. Math. 6, 274-304 (1954) · Zbl 0055.14305 · doi:10.4153/CJM-1954-028-3 [25] Smith, L.: On the charateristic zero cohomology of the free loop space. Am. J. Math. 103, 887-910 (1981) · Zbl 0475.55004 · doi:10.2307/2374251 [26] Wojtkowiak, Z.: Maps from B? into X. Quart. J. Math. Oxford 39, 117-127 (1988) · Zbl 0656.55012 · doi:10.1093/qmath/39.1.117 [27] Wojtkowiak, Z.: On maps from holim F to Z. Algebraic Topology (Barcelona 1986). Lecture Notes in Math. 1298, Springer, 227-236 (1987)
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