Differentials on graph complexes II: hairy graphs

Khoroshkin, Anton; Willwacher, Thomas; Živković, Marko

doi:10.1007/s11005-017-0964-9

Cited by 9 publications

(19 citation statements)

References 17 publications

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“…These complexes compute the rational homotopy of the spaces of embeddings of disks modulo immersions, fixed at the boundary, Emb ∂ (D m , D n ), provided that n − m ≥ 3, cf. [2,1,10,4]. In this context, Willwacher conjectured our result in [11].…”

Section: Introductionsupporting

confidence: 66%

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Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

Živković

2019

International Mathematics Research Notices

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We prove that the projection from graph complex with at least one source to oriented graph complex is a quasi-isomorphism, showing that homology of the “sourced” graph complex is also equal to the homology of standard Kontsevich’s graph complex. This result may have applications in theory of multi-vector fields $T_{\textrm{poly}}^{\geq 1}$ of degree at least one, and to the hairy graph complex that computes the rational homotopy of the space of long knots. The result is generalized to multi-directed graph complexes, showing that all such graph complexes are quasi-isomorphic. These complexes play a key role in the deformation theory of multi-oriented props recently invented by Sergei Merkulov. We also develop a theory of graph complexes with arbitrary edge types.

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Section: Introductionsupporting

confidence: 66%

“…It is SE f ix Φ i = 0, so g i : SE f ix Φ i−1 → 0 is a quasi-isomorphism. 4. The same as (3) but x = Φ + (a i ) gives the analogous result.…”

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confidence: 67%

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

Živković

2019

International Mathematics Research Notices

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“…Using this isomorphism, we easily obtain the following generalizations of statements of Theorems 3.1 and 3.2: Theorem 3.7 Let d be any even integer and v 4m+1−d be a symbol of degree 4m + 1 − d. The natural embeddings [2], [6], [8], [16], [17], [27], [28], [29] for more details about these families of graph complexes and their generalizations. For odd d, the directions on edges play a special role.…”

Section: The Version Dfgc D For An Arbitrary Even Dimension Dmentioning

confidence: 99%

“…Graph complexes provide us with a large supply of intriguing questions and conjectures 1 [3], [5], [6], [8], [9], [13], [14], [16], [17], [19], [20], [26], [27], [28], [29]. One source of the motivation for working with graph complexes comes from the study of embedding spaces [2], [4], [21], [25], [26].…”

Section: Introductionmentioning

confidence: 99%

The Cohomology of the Full Directed Graph Complex

Dolgushev

Rogers

2019

Algebr Represent Theor

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In his seminal paper "Formality conjecture", M. Kontsevich introduced a graph complex GC1ve closely connected with the problem of constructing a formality quasi-isomorphism for Hochschild cochains. In this paper, we express the cohomology of the full directed graph complex dfGC explicitly in terms of the cohomology of GC1ve. Applications of our results include a recent work by the first author which completely characterizes homotopy classes of formality quasi-isomorphisms for Hochschild cochains in the stable setting. [29] T. Willwacher and M.Živković, Multiple edges in M. Kontsevich's graph complexes and computations of the dimensions and Euler characteristics, Adv. Math. 272 (2015) 553-578; arXiv:1401.4974.

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“…is given by the operator (12). Finally, the odd number d matters only in providing a global degree shift.…”

Section: Evenmentioning

confidence: 99%

Commutative hairy graphs and representations of Out (Fr)

Turchin

Willwacher

2017

Journal of Topology

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We express the hairy graph complexes computing the rational homotopy groups of long embeddings (modulo immersions) of R m in R n as 'decorated' graph complexes associated to certain representations of the outer automorphism groups of free groups. This interpretation gives rise to a natural spectral sequence, which allows us to shed some light on the structure of the hairy graph cohomology. We also explain briefly the connection to the deformation theory of the little disks operads and some conclusions that this brings. . † For a weaker range of dimensions this result has been shown earlier in [2,21]. To recall, the space Embc(R m , R n ) of long embeddings modulo immersions is the homotopy fiber of the inclusion Embc(R m , R n ) → Immc(R m , R n ) of the space of long embeddings R m → R n (that is, embeddings coinciding with the fixed linear inclusion R m ⊂ R n outside a compact subset of R m ) into the space of long immersions (that is, immersions with the same behavior at infinity). The rational homotopy groups of Embc(R m , R n ) and those of Embc(R m , R n ) are the same up to a finite dimensional correction, which was computed in [2].

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Differentials on graph complexes II: hairy graphs

Cited by 9 publications

References 17 publications

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

Multi-directed Graph Complexes and Quasi-isomorphisms Between Them II: Sourced Graphs

The Cohomology of the Full Directed Graph Complex

Commutative hairy graphs and representations of Out (Fr)

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