Summary: All algebraic objects in this note will be considered over a fixed field of characteristic zero. If not stated otherwise, all operads live in the category of differential graded vector spaces over . For standard terminology concerning operads, algebras over operads, etc., see either the original paper by [“The geometry of iterated loop spaces”, Lect. Notes Math. 271 (1972; Zbl 0244.55009)], or an overview [, “La renaissance des opérads”, Sémin. Bourbaki 1994/95, Exp. No. 792, Asterisque...
It is well-known that a based space is of the weak homotopy type of a loop space iff it is a grouplike algebra over an -operad. The classical model for such an operad consists of Stasheff’s associahedra. The present paper describes a similar recognition principle for free loop spaces. Let be an operad, a -module and a -algebra. An -trace over consists of a space and a module homomorphism over the operad homomorphism given by the algebra structure on . Let be the little 1-cubes...
[For the entire collection see Zbl 0699.00032.] A fibration is called totally noncohomologuous to zero (TNCZ) with respect to the coefficient field k, if is surjective. This is equivalent to saying that acts trivially on and the Serre spectral sequence collapses at . S. Halperin conjectured that for and F a 1-connected rationally elliptic space (i.e., both and are finite dimensional) such that vanishes in odd degrees, every fibration is TNCZ. The author proves this being the case...
The cotangent cohomology of and [Trans. Am. Math. Soc. 128, 41-70 (1967; Zbl 0156.27201)] is known for its ability to control the deformation of the structure of a commutative algebra. Considering algebras in the wider sense to include coalgebras, bialgebras and similar algebraic structures such as the Drinfel’d algebras encountered in the theory of quantum groups, one can model such objects as models for an algebraic theory much in the sense of [Proc. Natl. Acad. Sci. USA 50, 869-872 (1963)]....
The author describes the moduli space of Sullivan models of 2-skeletal spaces and complements of links as quotients of spaces of derivations of finitely generated free Lie algebras by the action of a subgroup of automorphisms of . For recall, a 2-skeletal space is a path connected space satisfying and . The paper contains as an application a complete description of the Lie algebras associated to the fundamental groups of complements of two-component links in terms of their Milnor numbers....
The paper is concerned with homotopy concepts in the category of chain complexes. It is part of the author’s program to translate [ and , Homotopy invariant algebraic structures on topological spaces, Lect. Notes Math. 347, Springer-Verlag (1973; Zbl 0285.55012)] from topology to algebra.In topology the notion of operad extracts the essential algebraic information contained in the following example (endomorphism operad).The endomorphism operad of a based space consists of the family
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The article contains a list of 7 problems related to operads and configuration spaces. Problems 1-2 are about the compactification of configuration spaces (homology and Koszulness, geometric decompositions). Problems 3-4 are about configuration spaces related to knot invariants, their geometry and Koszulness. Problems 5 to 7 are related to (operadically defined) traces and cyclic homology.
The problem of the characterization of graded Lie algebras which admit a realization as the homotopy Lie algebra of a space of type is discussed. The central results are formulated in terms of varieties of structure constants, several criterions for concrete algebras are also deduced.
Distributive law is a way to compose two algebraic structures, say and , into a more complex algebraic structure . The aim of this paper is to understand distributive laws in terms of operads. The central result says that if the operads corresponding respectively to and are Koszul, then the operad corresponding to is Koszul as well. An application to the cohomology of configuration spaces is given.
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