Realizing spaces as classifying spaces (Q2809218)

Let \(X\) be a 1-connected CW complex of finite type. The space \(B_{\text{aut\,}X}\) is the base of the universal fibration with fixed fibre \(X\). The authors give partial answers to the following open question: Which simply connected rational homotopy type occurs as \(B_{\text{aut\,}X}\)? They prove:NEWLINENEWLINE a) There exists \(X\) of \(\pi\)-finite type such that the following spaces occur as \(B_{\text{aut\,}X}\):NEWLINENEWLINE (1) \(K(\mathbb{Q},2n+ 1)\times K(\mathbb{Q}, 4n+ 1)\), for \(n\geq 1\) and \(n\) odd,NEWLINENEWLINE (2) \(K(\mathbb{Q}, r)\times K(\mathbb{Q},r+ 4n+ 1)\), for \(r\geq 2\) and \(n\geq 1\).NEWLINENEWLINE b) There exists no \(X\) of \(\pi\)-finite type such that the spaces \(S^2\) and \(\mathbb{C} P^2\) occur as \(B_{\text{aut\,}X}\).