WHEN IS THE CLASSIFYING SPACE FOR ELLIPTIC FIBRATIONS RANK ONE?

Abstract

We give a necessary and sufficient condition of a rationally elliptic space X such that the Dold-Lashof classifying space Baut1X for fibrations with the fiber X is rank one. It is only when X has the rational homotopy type of a sphere or the total space of a spherical fibration over a product of spheres.

Keywords

• elliptic space;
• minimal model;
• derivation;
• classifying space for fibrations

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References

1. J. B. Gatsinzi, The homotopy Lie algebra of classifying spaces, J. Pure Appl. Algebra 120 (1997), 281-289 https://doi.org/10.1016/S0022-4049(96)00037-0
2. A. Dold and R. Lashof, Principal quasi-fibrations and fibre homotopy equivalence of bundles, Illinois J. Math. 3 (1959), 285-305
3. Y. Felix, S. Halperin, and J. C. Thomas, Rational homotopy theory, Springer G.T.M. 205 (2001)
4. J. B. Gatsinzi, On the genus of elliptic fibrations, Proc. Amer. Math. Soc. 132 (2004), 597-606
5. S. B. Smith, Rational type of classifying space for fibrations, Contemp. Math. 274 (2001), 299-307 https://doi.org/10.1090/conm/274/04472
6. D. Sullivan, Infinitesimal computations in topology, Publ. I.H.E.S. 47 (1977), 269- 332 https://doi.org/10.1007/BF02684341
7. S. Halperin, Finiteness in the minimal models of Sullivan, Trans. Amer. Math. Soc. 230 (1977) 173-199 https://doi.org/10.2307/1997716

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1. Rational cohomologies of classifying spaces for homogeneous spaces of small rank vol.5, pp.4, 2016, https://doi.org/10.1007/s40065-016-0156-y
2. Sullivan minimal models of classifying spaces for non-formal spaces of small rank vol.196, 2015, https://doi.org/10.1016/j.topol.2015.10.003