The set of rational homotopy types with given cohomology algebra (Q1770320)

For a given graded algebra \(A^*\) over \(\mathbb Q\), let \({\mathcal M}_{A^*}\) denote the set consisting of all rational homotopy types of spaces \(X\) such that \(H^*(X;\mathbb Q)\cong A^*\). It is known that \({\mathcal M}_{A^*}\not= \emptyset\) for any \(A^*\), and it is interesting to determine the set \({\mathcal M}_{A^*}\) explicitly for a given explicit \(A^*\). For example, the authors and T. Nishimoto previously proved that there are two rational homotopy types with isomorphic cohomology algebras and isomorphic homotopy Lie algebras. In this paper, they determine the set \({\mathcal M}_{A^*}\) explicitly when \(A^*=H^*(S^m\vee S^n\vee S^{k};\mathbb Q)\) for some numbers \(m,n,k\). They obtain results by analyzing the inductive construction of minimal models.