Rational homotopy theory of mapping spaces via lie theory for $L_\infty$-algebras

Abstract

We calculate the higher homotopy groups of the Deligne-Getzler ∞-groupoid associated to a nilpotent L∞-algebra. As an application, we present a new approach to the rational homotopy theory of mapping spaces.

Algebraic Hopf invariants and rational models for mapping spaces

“…In [19], the L∞-structure from Theorem 1.2 is used to prove that the convolution algebra becomes a rational model for the mapping space M ap * (X, Y Q ). This is a generalization of a theorem by Berglund about models for mapping spaces (see Theorem 1.4 of [4] 19). Let C be a C∞-coalgebra model of finite type for a simplyconnected CW-complex X of finite Q-type, such that C is concentrated in degrees greater or equal than 2.…”

Little discs operads, graph complexes and Grothendieck-Teichmüller groups

“…The operad of little n-discs is precisely the structure defined by the collection of little n-discs spaces D n = {D n (r), r ∈ N} equipped with the unit element 1 ∈ D n (1) and the composition products (2) which satisfy the above relations (4)(5)(6). (We go back to the general definition of an operad in the next paragraph.)…”

“…In all cases, we use the notation Hom dg Λ Mod (M, N) for an enriched hom-object associated to our category of coaugmented diagrams in the category of dg-modules. The correspondence between the biderivations θ : C * CE (p n ) → W c (Pois c m ) and the collections of equivariant maps f :p n (r) ∨ →W c (Pois c m )(r), r ∈ N, which we use in the previous step is equivalent to an isomorphism of dg-modules: (6) BiDer(C * CE (p n ), W c (Pois c m )) Hom dg Λ Mod (p n ,W c (Pois m )), where we consider this enriched dg-hom object on the category of coaugmented Λ-diagrams Hom dg Λ Mod (−, −) on the right hand side.…”

Formality criteria in terms of higher Whitehead brackets

“…To achieve the mentioned result, we recall in Section 3.1 the necessary background. Then we recall in Section 3.2 that, under certain assumptions which we fix for the rest of the article, L ∞ algebras uniquely correspond to Sullivan algebras ( [10,7]). Finally, we achieve the main goal of this section by showing how some of our previous results in [5] generalize and/or complement the main result of [3].…”

“…To compare the results of the previous section with the L ∞ structures on π * (ΩX ) ⊗ Q, we need to recall when and how Sullivan algebras correspond to L ∞ structures. We refer the reader to [10] and [7] in order to find the most general results and a meticulous study of the subtleties concerning this duality. Recall that, by Theorem 1, an L ∞ structure on a graded vector space L uniquely corresponds to a codifferential δ on the coalgebra ΛsL.…”