Let \(X\) be a simply connected CW complex of finite type and \(\text{aut}_1(X)\) the identity component of the space of self-equivalences of \(X\). The Dold-Lashof classifying space for this monoid, \(\text{Baut}_1(X),\) is the classifying space for orientable fibrations with fibre the homotopy type of \(X\). The first systematic calculations of the cohomology of this space were made, for \(X\) a sphere, by J. Milnor in [Comment. Math. Helv. 43, 51–73 (1968; Zbl 0153.53403)]. Milnor’s results in the rational case imply \(\text{Baut}_1(S^m) \simeq_\mathbb Q K(\mathbb Q,n)\) where \(n =m+1\) for \(m\) odd and \(m = 2n\) for \(m\) even.
The author observes that the rational homotopy groups of \(\text{Baut}_1(X)\) are also of rank one when \(X\) is the total space of a particular spherical fibration of the form \(S^{4m+1} \to X \to (S^{2m+1})^{2n}.\) The main result of the paper establishes that these are the only elliptic spaces \(X\) (spaces with finite-dimensional rational homotopy and cohomology) for which \(\dim \pi_*(\text{Baut}_1(X)) \otimes \mathbb Q = 1\). The author proves this result by a direct algebraic argument using the identification \(\pi_{*+1}(\text{Baut}_1(X)) \otimes \mathbb Q \cong H_*(\text{Der}(M(X)), \partial)\) where \(M(X)\) denotes the Sullivan model of \(X\) and \((\text{Der}(M(X)), \partial)\) the differential graded space of degree lowering derivations of \(M(X)\).