Homology versus homotopy in fibrations and in limits

TL;DR

This paper proves estimates relating rational homotopy and cohomology dimensions in fibrations of formal elliptic spaces, especially under conditions like positive Euler characteristic or homogeneous manifold type, and explores asymptotic behaviors.

Contribution

It establishes the conjectured estimates in specific classes of rationally elliptic spaces and proposes studying asymptotic properties of these spaces.

Findings

Proved estimates for formal elliptic spaces with positive Euler characteristic.

Extended estimates to spaces with rational homotopy types of homogeneous manifolds.

Discussed asymptotic behavior of rationally elliptic spaces in the context of the conjecture.

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.