Rational cohomologies of classifying spaces for homogeneous spaces of small rank

Hirokazu Nishinobu,Toshihiro Yamaguchi

doi:10.1007/s40065-016-0156-y

Hirokazu Nishinobu, Toshihiro Yamaguchi

Open Access

https://doi.org/10.1007/s40065-016-0156-y

Copy DOI

Abstract

Let a simply-connected homogeneous space $${X}$$ satisfy the condition of $${{\rm dim} \pi_{\rm even}(X)\otimes {\mathbb{Q}}=2}$$ and $${{\rm dim} \pi_{\rm odd}(X)\otimes {\mathbb{Q}}=3}$$ (then, we say it is of (2, 3) type), which is the smallest rank in non-formal pure spaces. Then, we compute the Sullivan minimal model of the Dold–Lashof classifying space $${{\rm Baut}_1 X}$$ according to Nishinobu and Yamaguchi (Topol Appl 196:290–307, 2015) and observe whether or not its rational cohomology is a polynomial algebra, which is a necessary condition for the Serre spectral sequence of any fibration over a sphere with fibre $${X}$$ to degenerate at term $${E_2}$$ .

Highlights

A connected CW complex X is said to be rationally elliptic if the dimensions of rational cohomology and homotopy are both finite [4]
A rationally elliptic space X is said to be pure if the Sullivan model [20] is given as M(X ) = ( (x1, . . . , xm, y1, . . . , yn), d), where |xi | are even and |yi | are odd with d xi = 0 and d yi = fi ∈ Q[x1, . . . , xn] for m ≤ n [4]
In 1977, Halperin [7] conjectured that the Serre spectral sequences of all fibrations X →i E → Y of -connected CW complexes degenerate at the E2 term for any F0 space X, which is equivalent to that i∗ : H ∗(E; Q) → H ∗(X ; Q) is an epimorphism

Summary

Introduction

A connected CW complex X is said to be rationally elliptic if the dimensions of rational cohomology and homotopy are both finite [4]. On the other hand, when H ∗(Baut X ; Q) is a polynomial algebra for a space X in general, any fibration ξ : X → E → S2n+1 with fibre X over an odd-sphere S2n+1 is rationally trivial, i.e., EQ (X × S2n+1)Q. Recall an example of a rationally elliptic (non-pure) space X in [23], such that the Serre spectral sequence of any fibration with fibre X degenerates at E2 term. Question 1.2 If the Serre spectral sequence of any fibration with a rationally elliptic fibre X degenerates at E2 term, is H ∗(Baut X ; Q) a polynomial algebra? If the Serre spectral sequence of any fibration over a sphere with fibre X degenerates at E2 term, H ∗(Baut X ; Q) is a polynomial algebra.

Preliminaries

The condition to be a polynomial algebra

Proofs