Koszul spaces

Abstract

We prove that a nilpotent space is both formal and coformal if and only if it is rationally homotopy equivalent to the derived spatial realization of a graded commutative Koszul algebra. We call such spaces Koszul spaces and we show that the rational homotopy groups and the rational homology of iterated loop spaces of Koszul spaces can be computed by applying certain Koszul duality constructions to the cohomology algebra.

Koszulity of cohomology = K(π,1)-ness + quasi-formality

“…[5]). One can say that this is an instance of intrinsic quasi-formality, i. e., a situation when any augmented DG-algebra with a given cohomology algebra is quasi-formal.…”

Free loop space homology of highly connected manifolds

“…In this section we will review the notions of formality, coformality and Koszul algebras and the "2-out-of-3" property for these notions [Ber14a], along the way introducing notation and definitions that we will need in later sections.…”

“…Being simultaneously formal and coformal is a rather restrictive constraint, but there are several interesting examples of spaces that fulfill it, see [Ber14a]. We will see in Section 4.1 below that highly connected manifolds are formal and coformal over any field.…”

Formality criteria in terms of higher Whitehead brackets

“…From a classical point of view, one might see coformal spaces as building blocks for rational homotopy types, since every rational simply connected homotopy type can be realized as a perturbation of the corresponding coformal model ( [25]). More recently, a series of works embracing [26,6,8,9] prove very interesting results, combining Koszul duality and rational homotopy theory methods, which allow for effectively computing new results on (free and based) loop space homology and other interesting topics. In these, coformality plays a distinguished role.…”