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On the homotopy groups of \(p\)-completed classifying spaces. (English) Zbl 1086.55007
Working one prime at a time has been a successful route into the structure of the homotopy groups of a space. As a consequence of Haynes Miller’s solution to the Sullivan conjecture, McGibbon and Neisendorfer proved that a simply-connected CW-complex \(X\) with finite mod 2 cohomology has infinitely many homotopy groups \(\pi_i(X)\) with nontrivial 2-torsion. Generalizations of the algebraic condition, \(H^*(X; {\mathbb F}_2)\) finite, followed with the development of ideas like Lannes’s \(T\) functor, \(p\)-local groups, topped off by a theorem of Dwyer and Wilkerson that a 2-connected CW-complex of finite type with nontrivial mod \(p\) cohomology and for which the indecomposable module of \(H^*(X;{\mathbb F}_p)\) is locally finite as a module over the Steenrod algebra has nontrivial \(p\)-torsion in infinitely many homotopy groups. The present paper concerns the gap between connected and 2-connected. The authors prove that a \(p\)-complete space \(X\) for which \(H^*(X;{\mathbb F}_p)\) is of finite type and for which the indecomposable module of \(H^*(X;{\mathbb F}_p)\) is locally finite as a module over the Steenrod algebra must satisfy either \(X\) is aspherical, \(X\) is a finite product of Eilenberg-Mac Lane spaces of the type \(K({\mathbb Z}_p^{^2})\), or \(X\) has infinitely many homotopy groups with nontrivial \(p\)-torsion. The authors further prove the same results for the particular case of \(X = BG\) for \(G\) a finite group, a \(p\)-local finite group, a finitely generated virtually nilpotent group, or a quasi-\(p\)-perfect group of finite virtual mod \(p\) cohomology.

MSC:
55P60 Localization and completion in homotopy theory
55Q05 Homotopy groups, general; sets of homotopy classes
55R35 Classifying spaces of groups and \(H\)-spaces in algebraic topology
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[1] Bastardas, G.: Localitzacions i complecions d’espais anesfèrics. Ph.D. thesis, Universitat Autònoma de Barcelona, 2003
[2] Bastardas, G., Descheemaeker, A.: On the homotopy type of p-completions of infra-nilmanifolds. Math. Z. 241 (4), 685–696 (2002) · Zbl 1038.55009 · doi:10.1007/s00209-002-0440-8
[3] Bousfield, A.K.: The localization of spaces with respect to homology. Topology 14, 133–150 (1975) · Zbl 0309.55013 · doi:10.1016/0040-9383(75)90023-3
[4] Bousfield, A.K.: On the p-adic completions of nonnilpotent spaces. Trans. Amer. Math. Soc. 331 (1), 335–359 (1992) · Zbl 0763.55008 · doi:10.2307/2154012
[5] Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. Springer-Verlag, Berlin, 1972, Lecture Notes in Mathematics, Vol. 304 · Zbl 0259.55004
[6] Broto, C., Castellana, N., Grodal, J., Levi, R., Oliver, B.: Extensions of p-local finite groups. Preprint 2005, available at: http://front.math.ucdavis.edu/math.AT/0502359 · Zbl 1145.55013
[7] Broto, C., Kitchloo, N.: Classifying spaces of Kac-Moody groups. Math. Z. 240 (3), 621–649 (2002) · Zbl 1031.55009 · doi:10.1007/s002090100391
[8] Broto, C., Levi, R., Oliver, B.: The homotopy theory of fusion systems. J. Amer. Math. Soc. 16 (4), 779–856 (2003) (electronic) · Zbl 1033.55010 · doi:10.1090/S0894-0347-03-00434-X
[9] Brown, K.S.: Cohomology of groups. Graduate Texts in Mathematics, vol. 87, Springer-Verlag, New York, 1982 · Zbl 0584.20036
[10] Dekimpe, K.: Almost-Bieberbach groups: affine and polynomial structures. Lecture Notes in Mathematics, vol. 1639, Springer-Verlag, Berlin, 1996 · Zbl 0865.20001
[11] Dror, E., Dwyer, W.G., Kan, D.M.: An arithmetic square for virtually nilpotent spaces. Illinois J. Math. 21 (2), 242–254 (1977) · Zbl 0355.55015
[12] Dwyer, W.G., Wilkerson, C.W.: Spaces of null homotopic maps. Astérisque 191 (6), 97–108 (1990) International Conference on Homotopy Theory (Marseille-Luminy, 1988)
[13] Grodal, J.: The transcendence degree of the mod p cohomology of finite Postnikov systems. Stable and unstable homotopy (Toronto, ON, 1996), Fields Inst. Commun., vol. 19, Amer. Math. Soc., Providence, RI, 1998, pp. 111–130 · Zbl 0905.55012
[14] Grodal, J.: Serre’s theorem and the filtration of Lionel Schwartz. Cohomological methods in homotopy theory (Bellaterra, 1998), Progr. Math., vol. 196, Birkhäuser, Basel, 2001, pp. 177–183 · Zbl 0992.55012
[15] Lannes, J.: Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire. Inst. Hautes Études Sci. Publ. Math. 75, 135–244 (1992) with an appendix by Michel Zisman · Zbl 0857.55011 · doi:10.1007/BF02699494
[16] Lannes, J., Schwartz, L.: À propos de conjectures de Serre et Sullivan. Invent. Math. 83 (3), 593–603 (1986) · Zbl 0563.55011 · doi:10.1007/BF01394425
[17] Levi, R.: On finite groups and homotopy theory. Mem. Amer. Math. Soc. 118 (567), (1995), xiv+100 · Zbl 0861.55002
[18] Levi, R.: On p-completed classifying spaces of discrete groups and finite complexes. J. London Math. Soc. (2) 59 (3), 1064–1080 (1999) · Zbl 0946.55008
[19] Loday, J.-L.: K-théorie algébrique et représentations de groupes. Ann. Sci. École Norm. Sup. (4) 9 (3), 309–377 (1976)
[20] McGibbon, C.A., Neisendorfer, J.A.: On the homotopy groups of a finite-dimensional space. Comment. Math. Helv. 59 (2), 253–257 (1984) · Zbl 0538.55010 · doi:10.1007/BF02566349
[21] Miller, H.: The Sullivan conjecture on maps from classifying spaces. Ann. of Math. (2) 120 (1), 39–87 (1984) · Zbl 0552.55014
[22] Ragnarsson, K.: Classifying spectra of saturated fusion systems. Preprint 2005, available at: http://front.math.ucdavis.edu/math.AT/0502092 · Zbl 1098.55012
[23] Schwartz, L.: Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture. Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994 · Zbl 0871.55001
[24] Serre, J.-P.: Cohomologie modulo 2 des complexes d’Eilenberg-MacLane. Comment. Math. Helv. 27, 198–232 (1953) · Zbl 0052.19501 · doi:10.1007/BF02564562
[25] Wagoner, J.B.: Delooping classifying spaces in algebraic K-theory. Topology 11, 349–370 (1972) · Zbl 0276.18012 · doi:10.1016/0040-9383(72)90031-6
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