Cyclic<i>A</i><sub>∞</sub>structures and Deligne’s conjecture

Ward, Benjamin C.

doi:10.2140/agt.2012.12.1487

Cited by 9 publications

(6 citation statements)

References 37 publications

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“…For the definition of the filtration F a O M and verification of (i), see Remark 7.6. To verify (ii), it is sufficient to check that the generators of the operad T S ∞ (see [27] Section 4.1) preserve the length filtration, and this is straightforward from (i). The isomorphism H * (L a M ) ∼ = H * (F a O M ) in (iii) will be defined in Section 8.2.…”

Section: Resultsmentioning

confidence: 99%

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A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

Irie

2017

International Mathematics Research Notices

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The aim of this paper is to define a chain level refinement of the Batalin-Vilkovisky (BV) algebra structure on the homology of the free loop space of a closed, oriented C ∞ -manifold. For this purpose, we define a (nonsymmetric) cyclic dg operad which consists of "de Rham chains" of free loops with marked points. A notion of de Rham chains, which is a certain hybrid of the notions of singular chains and differential forms, is a key ingredient in our construction. Combined with a generalization of cyclic Deligne's conjecture, this dg operad produces a chain model of the free loop space which admits an action of a chain model of the framed little disks operad, recovering the string topology BV algebra structure on the homology level.2010 Mathematics Subject Classification. 55P50, 57N75, 18D50. 1 well-known fact: the cup product on (singular) cohomology can be defined at the chain level, and (part of) extra information is used to define the classical Massey products; see [22] Section 9.4.5. For (a), what is called Deligne's conjecture claims that a certain chain model of the little disks operad acts on the Hochschild cochain complex. Various affirmative solutions to this conjecture and its variations are known; see [23] Part I Section 1.19, [22] Section 13.3.15, and the references therein. The aim of this paper is to propose a chain level algebraic structure which lifts (b) (the Batalin-Vilkovisky (BV) algebra structure in string topology), and compare it with a solution to Deligne's conjecture via a chain map which is a chain level lift of (c).Our construction is based on generalizations of Deligne's conjecture in operadic contexts. More specifically, we use results in a recent paper [28], which we describe here very briefly. For any (nonsymmetric) dg operad O = (O(k)) k≥0 and a Maurer-Cartan element ζ = (ζ k ) k≥2 of O, one can assign a chain complex (Õ, ∂ ζ ) (here we use our notation in Section 2.5, which is different from the notation in [28]). Then Theorem A in [28] asserts that the chain complexÕ admits an action of a certain chain model of the little disks operad. Moreover, if O has a unital cyclic structure and ζ is cyclically invariant, Theorem B in [28] asserts that this action extends to an action of a certain chain model of the framed little disks operad (strictly speaking, in Theorem B one has to consider a quasi-isomorphic subcomplexÕ nm ⊂Õ which consists of "normalized chains"). Actually, in this paper we consider only the simple case that ζ k = 0 for every k = 2. A Maurer-Cartan element satisfying this condition is equivalent to a "multiplication" of the operad (see Definition 2.6). Let us briefly describe our main result. For any closed oriented C ∞ -manifold M , we define a nonsymmetric cyclic dg operad O M with a multiplication and a unit. Applying Theorem B in [28], the associated chain complex O M nm admits an action of a chain model of the framed little disks operad. In particular, the homology H * ( O M nm ) ∼ = H * ( O M ) has a BV algebra structure. Then we show that there exists an isomor...

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Section: Resultsmentioning

confidence: 99%

“…The operad f P in our statement is the operad T S ∞ , which is introduced in [27] Section 4. The inclusion of operads P → f P follows from Lemma 5.9 in [27]. The sign for the operator ∆ in the statement (i) will be discussed in Section 2.5.4.…”

Section: Now Let Us State a Version Ofmentioning

confidence: 99%

A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

Irie

2017

International Mathematics Research Notices

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“…The added difficulty, is that for cacti one is dealing with a bi-crossed product and not just a semi-direct product. The cyclic A ∞ version again based on cacti is contained in [47]. 5.1.…”

Section: Discussion: Relations To Other E 2 Operads Application and mentioning

confidence: 99%

Permutohedral structures on E2E_{2}-operads

Kaufmann

Zhang

2016

Forum Mathematicum

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Abstract. There are basically two interesting breeds of E 2 operads, those that detect loop spaces and those that solve Deligne's conjecture. The former deformation retract to Milgram's space obtained by gluing together permutahedra at their faces. We show how the second breed can be covered by permutahedra as well. Even more is true, the quotient is actually already an operad up to homotopy, which induces the operad structure on cellular chains adapted to prove Deligne's conjecture, while no such structure is known on Milgram's space. We show, explicitely, that these two quotients are homotopy equivalent. This gives a new topological proof that operads of this type are indeed of the right homotopy type. It also furnishes a very nice clean description in terms of polyhedra, and with it PL topology, for the whole story. The permutahedra and partial orders play a central role. This, in turn, provides direct links to other fields of mathematics. We for instance find a new cellular decomposition of permutahedra using partial orders and that the permutahedra give the cells for the Dyer-Lashof operations.

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“…In [33] we showed that one can relax the condition of A being associative to A 1 for the brace operations, aka. A 1 -Deligne conjecture, and in [51] the same was done tor the BV operators, aka. cyclic A 1 conjecture.…”

Section: 23mentioning

confidence: 99%

A detailed look on actions on Hochschild complexes especially the degree 1 coproduct and actions on loop spaces

Kaufmann¹

2022

J. Noncommut. Geom.

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We explain our previous results about Hochschild actions (2007/2008) pertaining, in particular, to the coproduct, which appeared in a different form in a work by Goresky and Hingston (2009), and provide a fresh look at the results. We recall the general action, specialize to the aforementioned coproduct and prove that the assumption of commutativity, made for convenience in a previous article (2008), is not needed. We give detailed background material on loop spaces, Hochschild complexes and dualizations, and discuss details and extensions of these techniques which work for all operations given in two previous articles (2007/2008).With respect to loop spaces, we show that the coproduct is well defined modulo constant loops and going one step further that in the case of a graded Gorenstein Frobenius algebra, the coproduct is well defined on the reduced normalized Hochschild complex. We discuss several other aspects such as "time reversal" duality and several homotopies of operations induced by it. This provides a cohomology operation which is a homotopy of the anti-symmetrization of the coproduct. The obstruction again vanishes on the reduced normalized Hochschild complex if the Frobenius algebra is graded Gorenstein. Further structures such as "animation", the BV structure, a coloring for operations on chains and cochains, and a Gerstenhaber double bracket are briefly treated.

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CyclicA_∞structures and Deligne’s conjecture

Cited by 9 publications

References 37 publications

A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

Permutohedral structures on E2E_{2}-operads

A detailed look on actions on Hochschild complexes especially the degree 1 coproduct and actions on loop spaces

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CyclicA∞structures and Deligne’s conjecture

Cited by 9 publications

References 37 publications

A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

A Chain Level Batalin–Vilkovisky Structure in String Topology Via de Rham Chains

Permutohedral structures on E2E_{2}-operads

A detailed look on actions on Hochschild complexes especially the degree 1 coproduct and actions on loop spaces

Contact Info

Product

Resources

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CyclicA_∞structures and Deligne’s conjecture