Lie Models in Topology

Abstract

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Lie models of homotopy automorphism monoids and classifying fibrations

“…In this section we recall the basics for complete differential graded Lie algebras, from the homotopy point of view. For it, original references are [7,8,9,10] whose main results are developed in the complete and detailed monograph [11]. Sometimes we will also use classical facts from the Sullivan commutative approach to rational homotopy theory.…”

“…A morphism f : L → L ′ between cdgl's, associated to filtrations {F n } n≥1 and {G n } n≥1 respectively, is a dgl morphism which preserves the filtrations, that is, f (F n ) ⊂ G n for each n ≥ 1. We denote by cdgl the corresponding category which is complete and cocomplete [11,Proposition 3.5].…”

“…given by the global model and realization functor (see next section for a quick review or [11] for a detailed presentation).…”

The diagonal of cellular spaces and effective algebro-homotopical constructions

“…To generalize Quillen's equivalence of homotopy categories to one between (not necessarily 1-connected) simplicial sets Buijs, Félix, Murillo, and Tanré extended non-constructively in [Bui+20] the Lawrence-Sullivan structure, building natural C ∞ -coalgebra structures on the chains of standard simplices. Their construction agrees after linear dualization with the one obtained by Cheng and Getzler in [CG08], where they showed that the Kontsevich-Soibelman sum-over-trees formula defining the transfer of A ∞ -algebras through a chain contraction induces a transfer of C ∞ -algebras.…”

“…The resulting algebraic structures control much of the homotopy theory of spaces. For example, over the rationals, the quasi-isomorphism type of a C ∞ -coalgebra extension of the symmetrization of ∆ determines the Q-completion of X under certain assumptions [Qui69;Bui+20]. Similarly, over the algebraic closure of F p , the quasi-isomorphism type of the dual of an E ∞ -coalgebra extension of ∆ determines the p-completion of X provided certain finiteness assumptions are met [Man01].…”