The rational homotopy type of elliptic spaces up to cohomological dimension 8 (Q337973)

A simply connected topological space is called \textit{rational} if its homotopy groups form rational vector spaces. Any simply connected space \(X\) can be approximated uniquely up to homotopy equivalence by a rational CW complex, denoted \(X_{\mathbb Q}\); the weak homotopy type of \(X_{\mathbb Q}\) is known as the \textit{rational homotopy type} of \(X\). Rational homotopy theory is the study of rational homotopy types, and the work under review is an effort toward their classification. A simply connected space \(X\) is called \textit{elliptic} provided \(\pi_* (X)\otimes{\mathbb Q}\) and \(H^*(X;{\mathbb Q})\) are finite dimensional. As the authors attest in the title and abstract of the paper, their intent was to classify the rational homotopy types of elliptic spaces of cohomological dimension less than or equal to 8. They present their classification as a table in Theorem 1, with a citation to [\textit{M. R. Hilali} et al., Adv. Pure Math. 2, No. 1, 15--21 (2012; Zbl 1242.55004)] for one of the entries. Several of the rational homotopy types are described as various combinations of spheres and James spheres (reduced product spaces created from spheres).