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      Homology versus homotopy in fibrations and in limits

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          Abstract

          Motivated by prominent problems like the Hilali conjecture Yamaguchi--Yokura recently proposed certain estimates on the relations of the dimensions of rational homotopy and rational cohomology groups of fibre, base and total spaces in a fibration of rationally elliptic spaces. In this article we prove these estimates in the category of formal elliptic spaces and, in general, whenever the total space in addition has positive Euler characteristic or has the rational homotopy type of a homogeneous manifold (respectively of a known example) of positive sectional curvature. Additionally, we provide general estimates approximating the conjectured ones. Moreover, we suggest to study families of rationally elliptic spaces under certain asymptotics, and we discuss the conjectured estimates from this perspective for two-stage spaces.

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          Author and article information

          Journal
          05 June 2020
          Article
          2006.03390
          37c9f2d6-af9a-45f2-b683-95f5d1c6f6f4

          http://arxiv.org/licenses/nonexclusive-distrib/1.0/

          Custom metadata
          55P62 (Primary), 57N65, 53C20 (Secondary)
          math.AT math.DG

          Geometry & Topology

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