Realizing spaces as classifying spaces

Lupton, Gregory; Smith, Samuel

doi:10.1090/proc/13022

Realizing spaces as classifying spaces
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by Gregory Lupton and Samuel Bruce Smith PDF

Proc. Amer. Math. Soc. 144 (2016), 3619-3633 Request permission

Abstract:

Which spaces occur as a classifying space for fibrations with a given fibre? We address this question in the context of rational homotopy theory. We construct an infinite family of finite complexes realized (up to rational homotopy) as classifying spaces. We also give several non-realization results, including the following: the rational homotopy types of $\mathbb {C} P^2$ and $S^4$ are not realized as the classifying space of any simply connected, rational space with finite-dimensional homotopy groups.

References

Richard Body and Roy Douglas, Unique factorization of rational homotopy types, Pacific J. Math. 90 (1980), no. 1, 21–26. MR 599316, DOI 10.2140/pjm.1980.90.21
Richard Body, Mamoru Mimura, Hiroo Shiga, and Dennis Sullivan, $p$-universal spaces and rational homotopy types, Comment. Math. Helv. 73 (1998), no. 3, 427–442. MR 1633367, DOI 10.1007/s000140050063
Albrecht Dold, Halbexakte Homotopiefunktoren, Lecture Notes in Mathematics, vol. 12, Springer-Verlag, Berlin-New York, 1966 (German). MR 0198464, DOI 10.1007/BFb0084856
Albrecht Dold and Richard Lashof, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285–305. MR 101521
Yves Félix, Stephen Halperin, and Jean-Claude Thomas, Rational homotopy theory, Graduate Texts in Mathematics, vol. 205, Springer-Verlag, New York, 2001. MR 1802847, DOI 10.1007/978-1-4613-0105-9
Yves Félix, Gregory Lupton, and Samuel B. Smith, The rational homotopy type of the space of self-equivalences of a fibration, Homology Homotopy Appl. 12 (2010), no. 2, 371–400. MR 2771595, DOI 10.4310/HHA.2010.v12.n2.a13
Martin Fuchs, A modified Dold-Lashof construction that does classify $H$-principal fibrations, Math. Ann. 192 (1971), 328–340. MR 292080, DOI 10.1007/BF02075360
J. B. Gatsinzi, LS-category of classifying spaces, Bull. Belg. Math. Soc. Simon Stevin 2 (1995), no. 2, 121–126. MR 1332389, DOI 10.36045/bbms/1103408749
J.-B. Gatsinzi, LS-category of classifying spaces. II, Bull. Belg. Math. Soc. Simon Stevin 3 (1996), no. 2, 243–248. MR 1389618
J.-B. Gatsinzi, Products and Lusternik-Schnirelmann category of classifying spaces, Indag. Math. (N.S.) 9 (1998), no. 3, 351–357. MR 1692177, DOI 10.1016/S0019-3577(98)80004-9
J.-B. Gatsinzi, Lusternik-Schnirelmann category of classifying spaces, Bull. Belg. Math. Soc. Simon Stevin 8 (2001), no. 3, 405–409. MR 1860871
Gregory Lupton and Samuel Bruce Smith, Rationalized evaluation subgroups of a map. I. Sullivan models, derivations and $G$-sequences, J. Pure Appl. Algebra 209 (2007), no. 1, 159–171. MR 2292124, DOI 10.1016/j.jpaa.2006.05.018
J. Peter May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975), no. 1, 155, xiii+98. MR 370579, DOI 10.1090/memo/0155
W. Meier, Rational universal fibrations and flag manifolds, Math. Ann. 258 (1981/82), no. 3, 329–340. MR 649203, DOI 10.1007/BF01450686
Mamoru Mimura, Ronald C. O’Neill, and Hirosi Toda, On $p$-equivalence in the sense of Serre, Jpn. J. Math. 40 (1971), 1–10. MR 305399, DOI 10.4099/jjm1924.40.0_{1}
Mike Schlessinger and James Stasheff, Deformation theory and rational homotopy type, http://arxiv.org/pdf/1211.1647.
Samuel Bruce Smith, Rational type of classifying spaces for fibrations, Groups of homotopy self-equivalences and related topics (Gargnano, 1999) Contemp. Math., vol. 274, Amer. Math. Soc., Providence, RI, 2001, pp. 299–307. MR 1817019, DOI 10.1090/conm/274/04472
James Stasheff, A classification theorem for fibre spaces, Topology 2 (1963), 239–246. MR 154286, DOI 10.1016/0040-9383(63)90006-5
Dennis Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269–331 (1978). MR 646078, DOI 10.1007/BF02684341
Daniel Tanré, Homotopie rationnelle: modèles de Chen, Quillen, Sullivan, Lecture Notes in Mathematics, vol. 1025, Springer-Verlag, Berlin, 1983 (French). MR 764769, DOI 10.1007/BFb0071482
Toshihiro Yamaguchi, When is the classifying space for elliptic fibrations rank one?, Bull. Korean Math. Soc. 42 (2005), no. 3, 521–525. MR 2162207, DOI 10.4134/BKMS.2005.42.3.521