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Axiomatic Homotopy Theory for Operads
 Comment. Math. Helv
, 2002
"... We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced. ..."
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Cited by 102 (8 self)
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We give sufficient conditions for the existence of a model structure on operads in an arbitrary symmetric monoidal model category. General invariance properties for homotopy algebras over operads are deduced.
Noncommutative differential calculus, homotopy . . .
, 2000
"... We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures. ..."
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Cited by 57 (1 self)
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We define a notion of a strong homotopy BV algebra and apply it to deformation theory problems. Formality conjectures for Hochschild cochains are formulated. We prove several results supporting these conjectures.
Manifoldtheoretic compactifications of configuration spaces
 Selecta Math. (N.S
"... Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to FultonMacPherson and AxelrodSinger in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elemen ..."
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Cited by 41 (6 self)
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Abstract. We present new definitions for and give a comprehensive treatment of the canonical compactification of configuration spaces due to FultonMacPherson and AxelrodSinger in the setting of smooth manifolds, as well as a simplicial variant of this compactification. Our constructions are elementary and give simple global coordinates for the compactified configuration space of a general manifold embedded in Euclidean space. We stratify the canonical compactification, identifying the diffeomorphism types of the strata in terms of spaces of configurations in the tangent bundle, and give completely explicit local coordinates around the strata as needed to define a manifold with corners. We analyze the quotient map from the canonical to the simplicial compactification, showing it is a homotopy equivalence. We define projection maps and diagonal maps, which for the simplicial variant satisfy cosimplicial identities.
Operads and knot spaces
 J. Amer. Math. Soc
"... Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent ..."
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Cited by 40 (3 self)
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Let Em denote the space of embeddings of the interval I = [−1, 1] in the cube I m with endpoints and tangent vectors at those endpoints fixed on opposite faces of the cube, equipped with a homotopy through immersions to the unknot – see Definition 5.1. By Proposition 5.17, Em is homotopy equivalent to Emb(I, I m) × ΩImm(I, I m). In [28], McClure and Smith define a cosimplicial object O • associated
The BoardmanVogt resolution of operads in monoidal model categories, in preparation
"... Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain ..."
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Cited by 32 (10 self)
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Abstract. We extend the Wconstruction of Boardman and Vogt to operads of an arbitrary monoidal model category with suitable interval, and show that it provides a cofibrant resolution for wellpointed Σcofibrant operads. The standard simplicial resolution of Godement as well as the cobarbar chain resolution are shown to be particular instances of this generalised Wconstruction.
The Hopf algebra of rooted trees in EpsteinGlaser renormalization
, 2005
"... We show how the Hopf algebra of rooted trees encodes the combinatorics of EpsteinGlaser renormalization and coordinate space renormalization in general. In particular, we prove that the EpsteinGlaser timeordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping t ..."
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Cited by 27 (15 self)
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We show how the Hopf algebra of rooted trees encodes the combinatorics of EpsteinGlaser renormalization and coordinate space renormalization in general. In particular, we prove that the EpsteinGlaser timeordered products can be obtained from the Hopf algebra by suitable Feynman rules, mapping trees to operatorvalued distributions. Twisting the antipode with a renormalization map formally solves the EpsteinGlaser recursion and provides local counterterms due to the Hochschild 1closedness of the grafting operator B+.
The symmetrisation of noperads and compactification of real configuration spaces
 Adv. Math
"... It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s ..."
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Cited by 19 (5 self)
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It is well known that the forgetful functor from symmetric operads to nonsymmetric operads has a left adjoint Sym1 given by product with the symmetric group operad. It is also well known that this functor does not affect the category of algebras of the operad. From the point of view of the author’s theory of higher operads, the nonsymmmetric operads are 1operads and Sym1 is the first term of the infinite series of left adjoint functors Symn, called symmetrisation functors, from noperads to symmetric operads with the property that the category of one object, one arrow,..., one (n − 1)arrow algebras of an noperad A is isomorphic to the category of algebras of Symn(A). In this paper we consider some geometrical and homotopical aspects of the symmetrisation of noperads. We follow Getzler and Jones and consider their decomposition of the FultonMacpherson operad of compactified real configuration spaces. We construct an noperadic counterpart of
The minimal model for the Batalin–Vilkovisky operad
 SELECTA MATHEMATICA NEW SERIES
, 2012
"... The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov–Kontsevich and Manin, which states that the underlying homology groups of some Batalin–Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model ..."
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Cited by 16 (6 self)
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The purpose of this paper is to explain and to generalize, in a homotopical way, the result of Barannikov–Kontsevich and Manin, which states that the underlying homology groups of some Batalin–Vilkovisky algebras carry a Frobenius manifold structure. To this extent, we first make the minimal model for the operad encoding BValgebras explicit. Then, we prove a homotopy transfer theorem for the associated notion of homotopy BValgebra. The final result provides an extension of the action of the homology of the Deligne–Mumford–Knudsen moduli space of genus 0 curves on the homology of some BValgebras to an action via higher homotopical operations organized by the cohomology of the open moduli space of genus zero curves. Applications in Poisson geometry and Lie algebra cohomology and to the Mirror Symmetry conjecture are given.
Realizing Commutative Ring Spectra as E∞ Ring Spectra
, 1999
"... We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for comput ..."
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Cited by 9 (2 self)
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We outline an obstruction theory for deciding when a homotopy commutative and associative ring spectrum is actually an E∞ ring spectrum. The obstruction groups are AndréQuillen cohomology groups of an algebra over an E∞ operad. The same cohomology theory is part of a spectral sequence for computing the homotopy type of mapping spaces between E∞ ring spectrum. The obstruction theory arises out of techniques of Dwyer, Kan, and Stover, and the main application here is to prove an analog of a theorem of Haynes Miller and the second author: the LubinTate spectra En are E∞ and the space of E∞ selfmaps has weakly contractible components.