Sullivan minimal models of classifying spaces for non-formal spaces of small rank

TL;DR

This paper investigates the rational homotopy properties of classifying spaces for fibrations with specific non-formal spaces as fibers, focusing on their Sullivan minimal models and conditions for rational factorization.

Contribution

It provides new insights into the rational homotopy types of classifying spaces for non-formal spaces, including explicit minimal models for certain cases and conditions for rational factorization.

Findings

Identifies when $Baut_1X$ is a rational factor of $Baut_1(X\times S^n)$ for odd $n$.

Computes Sullivan minimal models for $Baut_1X$ with non-formal pure spaces of rank 5.

Shows that the rational cohomology of $Baut_1X$ can be non-free for certain non-formal spaces.

Abstract

We consider certain rational homotopical conditions of simly connected CW complex $X$ such that the rational cohomology of the classifying space $Baut_1X$ for fibrations with two-stage fibre $X$ is (not) free. First, we consider when is $Baut_1X$ a rational factor of $Baut_1(X\times S^n)$ for an odd-integer $n$ and observe for a non-formal elliptic space $X$ of rank 3. Second, we compute the Sullivan minimal models of $Baut_1X$ when $X$ are certain non-formal pure spaces of rank 5.