The Milnor-Moore theorem for $$L_\infty $$ algebras in rational homotopy theory

We give a construction of the universal enveloping $$A_\infty $$ A ∞ algebra of a given $$L_\infty $$ L ∞ algebra, alternative to the already existing versions. As applications, we derive a higher homotopy algebras version of the classical Milnor-Moore theorem. This proposes a new $$A_\infty $$ A ∞ model for simply connected rational homotopy types, and uncovers a relationship between the higher order rational Whitehead products in homotopy groups and the Pontryagin-Massey products in the rational loop space homology algebra.

Koszul A∞-algebras and free loop space homology

We introduce a notion of Koszul A∞-algebra that generalizes Priddy's notion of a Koszul algebra and we use it to construct small A∞-algebra models for Hochschild cochains. As an application, this yields new techniques for computing free loop space homology algebras of manifolds that are either formal or coformal (over a field or over the integers). We illustrate these techniques in two examples.

METRIC AND TOPOLOGICAL FREEDOM FOR SEQUENTIAL OPERATOR SPACES

In 2002 Anselm Lambert in his PhD thesis [1] introduced the definition of sequential operator space and managed to establish a considerable amount of analogs of corresponding results in operator space theory. Informally speaking, the category of sequential operator spaces is situated ”between” the categories of normed and operator spaces. This article aims to describe free and cofree objects for different versions of sequential operator space homology. First of all, we will show that duality theory in above-mentioned category is in many respects analogous to that in the category of normed spaces. Then, based on those results, we will give a full characterization of both metric and topological free and cofree objects.

Free loop space homology of highly connected manifolds

We calculate the homology of the free loop space of