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Deconstructing Hopf spaces. (English) Zbl 1109.55005
This paper represents one further step in the programme of understanding larger and larger classes of Hopf spaces (commonly known as \(H\)-spaces) in terms of familiar examples. Perhaps the definitive first step of this programme is J. R. Hubbuck’s theorem that a finite, connected, homotopy-commutative \(H\)-space is homotopy equivalent to a product of circles [Topology 8, 119–126 (1969; Zbl 0176.21301)]. In this latest work the authors relax the finiteness condition to include any \(H\)-space whose cohomology is finitely generated as an algebra over the Steenrod algebra. Such a space, if connected (but not necessarily homotopy-commutative), is shown to form the total space of an \(H\)-fibration whose base space is an \(H\)-space with finite mod \(p\) cohomology and whose fibre is a \(p\)-torsion \(H\)-Postnikov piece whose homotopy groups are finite direct sums of copies of cyclic groups \({\mathbb Z}/p^r\) and Prüfer groups \({\mathbb Z}/p^{\infty}\). In the commutative setting this implies that such an \(H\)-space is a product of circles and a connected \(H\)-Postnikov piece.

MSC:
55P45 \(H\)-spaces and duals
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[1] Adams, J.F.: On the non-existence of elements of Hopf invariant one. Ann. Math. 72, 20–104 (1960) · Zbl 0096.17404 · doi:10.2307/1970147
[2] Aguadé, J., Smith, L.: On the mod p torus theorem of John Hubbuck. Math. Z. 191, 325–326 (1986) · Zbl 0591.55003 · doi:10.1007/BF01164037
[3] Andersen, K.S., Bauer, T., Grodal, J., Pedersen, E.K.: A finite loop space not rationally equivalent to a compact lie group. Invent. Math. 157, 1–10 (2004) · Zbl 1054.55009 · doi:10.1007/s00222-003-0341-4
[4] Bauer, T., Kitchloo, N., Notbohm, D., Pedersen, E.K.: Finite loop spaces are manifolds. Acta Math. 192, 5–31 (2004) · Zbl 1055.55009 · doi:10.1007/BF02441084
[5] Bousfield, A.K.: Constructions of factorization systems in categories. J. Pure Appl. Algebra 9, 207–220 (1976/77)
[6] Bousfield, A.K.: Localization and periodicity in unstable homotopy theory. J. Am. Math. Soc. 7, 831–873 (1994) · Zbl 0839.55008 · doi:10.1090/S0894-0347-1994-1257059-7
[7] Bousfield, A.K.: Homotopical localizations of spaces. Am. J. Math. 119, 1321–1354 (1997) · Zbl 0886.55011 · doi:10.1353/ajm.1997.0036
[8] Bousfield, A.K., Friedlander, E.M.: Homotopy theory of \(\Gamma\)-spaces, spectra, and bisimplicial sets. Geometric applications of homotopy theory (Proc. Conf., Evanston, Ill., 1977), II. Lect. Notes Math., vol. 658, pp. 80–130. Berlin: Springer 1978 · Zbl 0405.55021
[9] Bousfield, A.K., Kan, D.M.: Homotopy limits, completions and localizations. Lect. Notes Math., vol. 304. Berlin: Springer 1972 · Zbl 0259.55004
[10] Broto, C., Crespo, J.A.: H-spaces with Noetherian mod two cohomology algebra. Topology 38, 353–386 (1999) · Zbl 0927.55017 · doi:10.1016/S0040-9383(98)00017-2
[11] Broto, C., Crespo, J.A., Saumell, L.: Non-simply connected H-spaces with finiteness conditions. Math. Proc. Camb. Philos. Soc. 130, 475–488 (2001) · Zbl 1002.55007 · doi:10.1017/S0305004101005023
[12] Cartan, H., Eilenberg, S.: Homological algebra. Princeton Landmarks in Mathematics. With an appendix by David A. Buchsbaum. Reprint of the 1956 original. Princeton, NJ: Princeton University Press 1999
[13] Castellana, N., Crespo, J.A., Scherer, J.: Postnikov pieces and B Z/p-homotopy theory. To appear in Trans. Am. Math. Soc. · Zbl 1110.55006
[14] Castellana, N., Crespo, J.A., Scherer, J.: Relating Postnikov pieces with the Krull filtration: A spin-off of Serre’s theorem. To appear in Forum Math. · Zbl 1138.55008
[15] Crespo, J.A.: Structure of mod p H-spaces with finiteness conditions, Cohomological methods in homotopy theory (Bellaterra, 1998). Prog. Math., vol. 196, pp. 103–130. Basel: Birkhäuser 2001
[16] Dwyer, W.G.: Strong convergence of the Eilenberg-Moore spectral sequence. Topology 13, 255–265 (1974) · Zbl 0303.55012 · doi:10.1016/0040-9383(74)90018-4
[17] Dwyer, W.G.: The centralizer decomposition of BG. Algebraic topology: new trends in localization and periodicity (Sant Feliu de Guíxols, 1994). Prog. Math., vol. 136, pp. 167–184. Basel: Birkhäuser 1996
[18] Dwyer, W.G., Wilkerson, C.W.: A new finite loop space at the prime two. J. Am. Math. Soc. 6, 37–64 (1993) · Zbl 0769.55007 · doi:10.1090/S0894-0347-1993-1161306-9
[19] Dwyer, W.G., Wilkerson, C.W.: Spaces of null homotopic maps. Astérisque 6, 97–108 (1990), International Conference on Homotopy Theory (Marseille-Luminy, 1988)
[20] Dror Farjoun, E.: Cellular spaces, null spaces and homotopy localization. Lect. Notes Math., vol. 1622. Berlin: Springer 1996 · Zbl 0842.55001
[21] Goerss, P.G.: On the André-Quillen cohomology of commutative F 2-algebras. Astérisque 186, 1990
[22] Hilton, P., Roitberg, J.: On principal S 3-bundles over spheres. Ann. Math. 90, 91–107 (1969) · Zbl 0176.21404 · doi:10.2307/1970683
[23] Hubbuck, J.R.: On homotopy commutative H-spaces. Topology 8, 119–126 (1969) · Zbl 0176.21301 · doi:10.1016/0040-9383(69)90004-4
[24] Kuhn, N.J.: On topologically realizing modules over the Steenrod algebra. Ann. Math. 141, 321–347 (1995) · Zbl 0849.55022 · doi:10.2307/2118523
[25] Lannes, J.: Sur les espaces fonctionnels dont la source est le classifiant d’un p-groupe abélien élémentaire. With an appendix by Michel Zisman. Inst. Hautes Études Sci. Publ. Math. 75, 135–244 (1992) · Zbl 0857.55011 · doi:10.1007/BF02699494
[26] Lannes, J., Schwartz, L.: Sur la structure des \(\mathcal{A}\) -modules instables injectifs. Topology 28, 153–169 (1989) · Zbl 0683.55016 · doi:10.1016/0040-9383(89)90018-9
[27] Lin, J.P., Williams, F.: Homotopy-commutative H-spaces. Proc. Am. Math. Soc. 113, 857–865 (1991) · Zbl 0758.55008
[28] McGibbon, C.A.: Infinite loop spaces and Neisendorfer localization. Proc. Am. Math. Soc. 125, 309–313 (1997) · Zbl 0864.55007 · doi:10.1090/S0002-9939-97-03744-1
[29] Miller, H.: The Sullivan conjecture on maps from classifying spaces. Ann. Math. 120, 39–87 (1984) · Zbl 0552.55014 · doi:10.2307/2007071
[30] Milnor, J.W., Moore, J.C.: On the structure of Hopf algebras. Ann. Math. 81, 211–264 (1965) · Zbl 0163.28202 · doi:10.2307/1970615
[31] Mislin, G.: Localization with respect to K-theory. J. Pure Appl. Algebra 10, 201–213 (1977/78)
[32] Rotman, J.J.: An introduction to the theory of groups, fourth ed. Graduate Texts in Mathematics, vol. 148. New York: Springer 1995 · Zbl 0810.20001
[33] Schwartz, L.: Unstable modules over the Steenrod algebra and Sullivan’s fixed point set conjecture. Chicago Lectures in Mathematics. Chicago, IL: University of Chicago Press 1994 · Zbl 0871.55001
[34] Schwartz, L.: À propos de la conjecture de non-réalisation due à N. Kuhn. Invent. Math. 134, 211–227 (1998) · Zbl 0919.55007 · doi:10.1007/s002220050263
[35] Schwartz, L.: La filtration de Krull de la catégorie \(\mathcal{U}\) et la cohomologie des espaces. Algebr. Geom. Topol. 1, 519–548 (2001) (electronic) · Zbl 1007.55014 · doi:10.2140/agt.2001.1.519
[36] Segal, G.: Categories and cohomology theories. Topology 13, 293–312 (1974) · Zbl 0284.55016 · doi:10.1016/0040-9383(74)90022-6
[37] Slack, M.: A classification theorem for homotopy commutative H-spaces with finitely generated mod 2 cohomology rings. Mem. Am. Math. Soc. 92, iv+116 (1991) · Zbl 0755.55003
[38] Smith, L.: The cohomology of stable two stage Postnikov systems. Ill. J. Math. 11, 310–329 (1967) · Zbl 0171.21803
[39] Smith, L.: Lectures on the Eilenberg-Moore spectral sequence. Lect. Notes Math., vol. 134. Berlin: Springer 1970 · Zbl 0197.19702
[40] Zabrodsky, A.: Hopf spaces. North-Holland Mathematics Studies, vol. 22, Notas de Matemática, No. 59. Amsterdam: North-Holland Publishing Co. 1976 · Zbl 0339.55009
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