Homotopy idempotent functors on classifying spaces

Castellana, Natàlia; Flores, Ramón

doi:10.1090/S0002-9947-2014-06132-1

Homotopy idempotent functors on classifying spaces
HTML articles powered by AMS MathViewer

by Natàlia Castellana and Ramón Flores PDF

Trans. Amer. Math. Soc. 367 (2015), 1217-1245 Request permission

Abstract:

Fix a prime $p$. Since their definition in the context of localization theory, the homotopy functors $P_{B\mathbb {Z}/p}$ and $CW_{B\mathbb {Z}/p}$ have shown to be powerful tools used to understand and describe the mod $p$ structure of a space. In this paper, we study the effect of these functors on a wide class of spaces which includes classifying spaces of compact Lie groups and their homotopical analogues. Moreover, we investigate their relationship in this context with other relevant functors in the analysis of the mod $p$ homotopy, such as Bousfield-Kan completion and Bousfield homological localization.

References

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
Carles Broto and Nitu Kitchloo, Classifying spaces of Kac-Moody groups, Math. Z. 240 (2002), no. 3, 621–649. MR 1924024, DOI 10.1007/s002090100391
Carles Broto, Ran Levi, and Bob Oliver, The homotopy theory of fusion systems, J. Amer. Math. Soc. 16 (2003), no. 4, 779–856. MR 1992826, DOI 10.1090/S0894-0347-03-00434-X
Carles Broto, Ran Levi, and Bob Oliver, Discrete models for the $p$-local homotopy theory of compact Lie groups and $p$-compact groups, Geom. Topol. 11 (2007), 315–427. MR 2302494, DOI 10.2140/gt.2007.11.315
A. K. Bousfield, The localization of spaces with respect to homology, Topology 14 (1975), 133–150. MR 380779, DOI 10.1016/0040-9383(75)90023-3
A. K. Bousfield, Localization and periodicity in unstable homotopy theory, J. Amer. Math. Soc. 7 (1994), no. 4, 831–873. MR 1257059, DOI 10.1090/S0894-0347-1994-1257059-7
A. K. Bousfield, Homotopical localizations of spaces, Amer. J. Math. 119 (1997), no. 6, 1321–1354. MR 1481817
Kenneth S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 672956
Natàlia Castellana, Juan A. Crespo, and Jérôme Scherer, Deconstructing Hopf spaces, Invent. Math. 167 (2007), no. 1, 1–18. MR 2264802, DOI 10.1007/s00222-006-0518-8
Natàlia Castellana, Juan A. Crespo, and Jérôme Scherer, Postnikov pieces and $B\Bbb Z/p$-homotopy theory, Trans. Amer. Math. Soc. 359 (2007), no. 3, 1099–1113. MR 2262843, DOI 10.1090/S0002-9947-06-03957-2
Natàlia Castellana, Ran Levi, and Dietrich Notbohm, Homology decompositions for $p$-compact groups, Adv. Math. 216 (2007), no. 2, 491–534. MR 2351369, DOI 10.1016/j.aim.2007.05.004
Wojciech Chachólski, On the functors $CW_A$ and $P_A$, Duke Math. J. 84 (1996), no. 3, 599–631. MR 1408539, DOI 10.1215/S0012-7094-96-08419-7
William G. Dwyer, Haynes R. Miller, and Clarence W. Wilkerson, The homotopic uniqueness of $BS^3$, Algebraic topology, Barcelona, 1986, Lecture Notes in Math., vol. 1298, Springer, Berlin, 1987, pp. 90–105. MR 928825, DOI 10.1007/BFb0083002
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, The centralizer decomposition of $BG$, Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994) Progr. Math., vol. 136, Birkhäuser, Basel, 1996, pp. 167–184. MR 1397728
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
Emmanuel Dror Farjoun, Cellular spaces, null spaces and homotopy localization, Lecture Notes in Mathematics, vol. 1622, Springer-Verlag, Berlin, 1996. MR 1392221, DOI 10.1007/BFb0094429
Ramón J. Flores and Richard M. Foote, The cellular structure of the classifying spaces of finite groups, Israel J. Math. 184 (2011), 129–156. MR 2823972, DOI 10.1007/s11856-011-0062-0
Emmanuel Dror Farjoun, Rüdiger Göbel, and Yoav Segev, Cellular covers of groups, J. Pure Appl. Algebra 208 (2007), no. 1, 61–76. MR 2269828, DOI 10.1016/j.jpaa.2005.11.004
Ramón J. Flores, Nullification and cellularization of classifying spaces of finite groups, Trans. Amer. Math. Soc. 359 (2007), no. 4, 1791–1816. MR 2272149, DOI 10.1090/S0002-9947-06-03926-2
Ramón J. Flores and Jérôme Scherer, Cellularization of classifying spaces and fusion properties of finite groups, J. Lond. Math. Soc. (2) 76 (2007), no. 1, 41–56. MR 2351607, DOI 10.1112/jlms/jdm031
A. González, The structure of $p$-local compact groups. PhD thesis (2010)
Stefan Jackowski, James E. McClure, and Bob Oliver, Self-maps of classifying spaces of compact simple Lie groups, Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 65–72. MR 1010725, DOI 10.1090/S0273-0979-1990-15841-0
Ran Levi, On finite groups and homotopy theory, Mem. Amer. Math. Soc. 118 (1995), no. 567, xiv+100. MR 1308466, DOI 10.1090/memo/0567
C. A. McGibbon, Infinite loop spaces and Neisendorfer localization, Proc. Amer. Math. Soc. 125 (1997), no. 1, 309–313. MR 1371135, DOI 10.1090/S0002-9939-97-03744-1
Haynes Miller, The Sullivan conjecture on maps from classifying spaces, Ann. of Math. (2) 120 (1984), no. 1, 39–87. MR 750716, DOI 10.2307/2007071
Guido Mislin, Cohomologically central elements and fusion in groups, Algebraic topology (San Feliu de Guíxols, 1990) Lecture Notes in Math., vol. 1509, Springer, Berlin, 1992, pp. 294–300. MR 1185979, DOI 10.1007/BFb0087519
Joseph A. Neisendorfer, Localization and connected covers of finite complexes, The Čech centennial (Boston, MA, 1993) Contemp. Math., vol. 181, Amer. Math. Soc., Providence, RI, 1995, pp. 385–390. MR 1321002, DOI 10.1090/conm/181/02044
José L. Rodríguez and Dirk Scevenels, Iterating series of localization functors, Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 1999) Contemp. Math., vol. 265, Amer. Math. Soc., Providence, RI, 2000, pp. 211–221. MR 1804680, DOI 10.1090/conm/265/04251
J. L. Rodríguez and J. Scherer, Cellular approximations using Moore spaces. In: Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math. 196, Amer. Math. Soc., Basel, 357–374 (2001)
Edwin H. Spanier, Algebraic topology, McGraw-Hill Book Co., New York-Toronto, Ont.-London, 1966. MR 0210112