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

TL;DR

This paper investigates the relationship between minimal Sullivan algebras and their realizations, showing that quasi-isomorphisms depend on the finite type of cohomology, and extends the concept of Sullivan spaces to broader classes.

Contribution

It demonstrates that quasi-isomorphisms between minimal Sullivan algebras and their polynomial forms depend on cohomology being finite type, even if the algebra itself is not, and generalizes Sullivan spaces.

Findings

Quasi-isomorphisms occur iff cohomology is finite type.

Minimal Sullivan algebras can be non-finite type yet still relate to their realizations.

Extended the concept of Sullivan spaces to broader classes.

Abstract

We prove that the morphisms from a minimal Sullivan algebra $\Lambda V$ to $A_{PL}(|\Lambda V|)$, the algebra of polynomial differential forms on its realization, can be quasi-isomorphic if and only if the cohomology $H(\Lambda V)$ is of finite type. Importantly, $\Lambda V$ 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 F\'elix, 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.