Sullivan minimal models of classifying spaces for non-formal spaces of small rank (Q891271)

A simply-connected CW complex \(X\) is called an elliptic space if the rational homotopy and cohomology of \(X\) are of finite dimension. Moreover, an elliptic space is said to be an \(F_0\)-space if \(\dim \pi_{odd}(X)\otimes {\mathbb Q} =\dim \pi_{even}(X)\otimes {\mathbb Q}\) and \(H^{odd}(X; {\mathbb Q})=0\). Let \(B\text{aut}_1(X)\) be the classifying space of the identity component of self-homotopy equivalences of a space \(X\). This article considers the problem of when the rational cohomology \(H^*(B\text{aut}_1(X); {\mathbb Q})\) is a finitely generated polynomial algebra for an elliptic space \(X\). The problem is closely related to Halperin's conjecture [\textit{S. Halperin}, Trans. Am. Math. Soc. 230, 173--199 (1977; Zbl 0364.55014)] which states that the Leray-Serre spectral sequences of all fibrations with \(F_0\)-spaces as the fibers collapse at the \(E_2\)-term. A main theorem (Theorem 1.10) describes equivalence conditions for \(H^*(B\text{aut}_1(X\times S^3); {\mathbb Q})\) to be a finitely generated polynomial algebra when \(X\) is elliptic. One of the conditions is that \(X\) is \textit{pure} and the rational homotopy of \(X\) concentrates in degrees \(2\) and \(3\). Another main theorem (Theorem 1.13) gives a necessary and sufficient condition for \(H^*(B\text{aut}_1(X); {\mathbb Q})\) not to be a polynomial algebra provided \(\dim \pi_{even}(X)\otimes {\mathbb Q} = 2\) and \(\dim \pi_{odd}(X)\otimes {\mathbb Q} = 3\). Tools for proving theorems are the Lie model for \(B\text{aut}_1(X)\) due to Quillen and the Cartan-Eilenberg-Chevalley construction, which converts the Lie algebra into a Sullivan model for the classifying space. The latter half of the paper determines explicitly minimal models for the classifying spaces \(B\text{aut}_1X\) and \(B\text{aut}_1(X\times S^n)\) for elliptic spaces \(X\) with \(\dim \pi_*(X)\otimes {\mathbb Q} = \dim \pi_{\text{odd}}(X)\otimes {\mathbb Q} = 3\). Moreover, one has a minimal model for \(B\text{aut}_1(SU(6)/SU(3)\times SU(3))\), which is very complicated.