Cohomology of minimal Sullivan algebras of non-finite type and their realizations

We prove that the morphisms from a minimal Sullivan algebra $\mathrm{\Lambda }V$ to $A_{PL}(\mathrm{\mid }\mathrm{\Lambda }V\mathrm{\mid })$, the algebra of polynomial differential forms on its realization, can be quasi-isomorphic if and only if the cohomology $H(\mathrm{\Lambda }V)$ is of finite type. Importantly, $\mathrm{\Lambda }V$ itself need not be of finite type. For example, it can be the minimal Sullivan model of the wedge sum of a circle and a sphere. This provides a negative answer to a question posed by Félix, Halperin, and Thomas. Furthermore, we study the spaces whose homotopy groups are reflected by their minimal Sullivan models as a generalization of Sullivan spaces, and explore which properties of Sullivan spaces can be broadened.