Koszul spaces

TL;DR

This paper characterizes Koszul spaces as nilpotent spaces that are both formal and coformal, showing they are rationally homotopy equivalent to derived realizations of Koszul algebras, and provides methods to compute their homotopy and homology.

Contribution

It establishes a new characterization of Koszul spaces using rational homotopy theory and Koszul duality, linking algebraic and topological properties.

Findings

Koszul spaces are characterized as formal and coformal nilpotent spaces.

Rational homotopy groups and homology of loop spaces are computable via Koszul duality.

Provides a bridge between algebraic Koszul duality and topological space classification.

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.