Some notes on spaces realized as classifying spaces (Q6611769)

Let \( B\mathrm{aut}_1 X\) denote the classifying space for the monoid of self homotopy equivalences of \(X\) which are homotopic to the identity on \(X\). The authors show that a certain class of rational spaces \(Y\), for which \(\pi_*(Y) \otimes \mathbb{Q}\) has dimension \(2\), cannot be realized up to rational homotopy as the classifying space \( B\mathrm{aut}_1 X\) of a space \(X\) for which \(\pi_*(X)\otimes \mathbb{Q}\) is finite dimensional. It is also shown that the Eilenberg-MacLane space \( K(\mathbb{Q}^r, n) \) (\( r\geq 2, n \geq 2\)) can be realized up to rational homotopy as the classifying space of an elliptic space, that is, a space \(X\) for which both \( \pi_*(X) \otimes \mathbb{Q}\) and \(H^*(X, \mathbb{Q})\) are finite dimensional, if and only if \( X = \prod_r S^{n-1}\), with \(n\) even.