Moduli spaces and formal operads

Guillén, F.; Navarro, Vı́ctor; Pascual, P.; Roig, Agustí

doi:10.1215/s0012-7094-05-12924-6

Cited by 22 publications

(15 citation statements)

References 24 publications

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“…We do not know if the little balls operad is formal over the rational numbers, but we do think it is an interesting question. We note that a general result about descent of formality from R to Q was proved in [12], for operads without a term in degree zero. The proof does not seem to be easily adaptable to operads with a zero term.…”

Section: Formality and Splitting Of The Little Balls Operadmentioning

confidence: 83%

Calculus of functors, operad formality, and rational homology of embedding spaces

2007

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Let M be a smooth manifold and V a Euclidean space. Let Emb(M, V ) be the homotopy fiber of the map Emb(M, V ) −→ Imm(M, V ). This paper is about the rational homology of Emb(M, V ). We study it by applying embedding calculus and orthogonal calculus to the bi-functor (M, V ) → HQ ∧ Emb(M, V )+. Our main theorem states that if dim V ≥ 2 ED(M ) + 1 (where ED(M ) is the embedding dimension of M ), the Taylor tower in the sense of orthogonal calculus (henceforward called "the orthogonal tower") of this functor splits as a product of its layers. Equivalently, the rational homology spectral sequence associated with the tower collapses at E 1 . In the case of knot embeddings, this spectral sequence coincides with the Vassiliev spectral sequence. The main ingredients in the proof are embedding calculus and Kontsevich's theorem on the formality of the little balls operad.We write explicit formulas for the layers in the orthogonal tower of the functorThe formulas show, in particular, that the (rational) homotopy type of the layers of the orthogonal tower is determined by the (rational) homotopy type of M . This, together with our rational splitting theorem, implies that, under the above assumption on codimension, rational homology equivalences of manifolds induce isomorphisms between the rational homology groups of Emb(−, V ).

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Section: Formality and Splitting Of The Little Balls Operadmentioning

confidence: 83%

Calculus of functors, operad formality, and rational homology of embedding spaces

2007

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“…In general, formality of an operad is independent of formality of the individual spaces. For operads without nullary or unary operations, the paper [9] proves a descent theorem, which shows that the operad formality property is independent of the ground field in characteristic zero; it is not yet known if this remains true in the presence of nontrivial nullary or unary operations. In applications of operad formality these operations are important, so we shall take care to distinguish over which field formality takes place.…”

Section: Formality Of Spacesmentioning

confidence: 98%

“…The operad composition is obtained by gluing stable curves at marked points and tensoring the corresponding rays. This operad was introduced by Kimura-Stasheff-Voronov [14], and it is closely related to the modular operad of Deligne-Mumford compactified moduli spaces of curves whose formality is proved in [9]. Moreover f M is homotopy equivalent, as a cyclic operad, to the operad of genus zero Riemann surfaces with boundary, where the operad composition is defined by gluing along boundary components.…”

Section: Formality Of Spacesmentioning

confidence: 99%

Cyclic operad formality for compactified moduli spaces of genus zero surfaces

Giansiracusa

Salvatore

2012

Trans. Amer. Math. Soc.

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Abstract. The framed little 2-discs operad is homotopy equivalent to the Kimura-Stasheff-Voronov cyclic operad of moduli spaces of genus zero stable curves with tangent rays at the marked points and nodes. We show that this cyclic operad is formal, meaning that its chains and its homology (the Batalin-Vilkovisky operad) are quasi-isomorphic cyclic operads. To prove this we introduce a new complex of graphs in which the differential is a combination of edge deletion and contraction, and we show that this complex resolves BV as a cyclic operad.

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“…We define minimal objects in a Cartan-Eilenberg category, and call it a Sullivan category, if any object has a minimal model. As an example, we interpret some results of [23] as saying that the category of modular operads over a field of characteristic zero is a Sullivan category.…”

Section: Mscmentioning

confidence: 99%

“…We followed the same approach to study the homotopy theory of modular operads in [23]: see Theorem 4.2.9 in this paper. Theorem 2.3.4.…”

Section: Cartan-eilenberg Categoriesmentioning

confidence: 99%