Two-loop part of the rational homotopy of spaces of long embeddings

Conant, Jim; Costello, Jean; Turchin, Victor; Weed, Patrick

doi:10.1142/s0218216514500187

Cited by 13 publications

(27 citation statements)

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“…For our purposes, working with co-invariants seems to be more suitable. In fact, the generating set for H 1 (C, d 0 ) that we determine in Section 2.3 contains fewer elements than the one given in [5].…”

Section: 2mentioning

confidence: 96%

“…a certain kind of embeddings R m ֒→ R N . While the cohomology of the tree-and one-loop part of this complex was computed in [2], the cohomology of the two-loop part was established in [5]. In this section, we recall the results on the two-loop part from which we will later deduce our main theorem.…”

Section: 2mentioning

confidence: 99%

“…Remark 2.8. In [5], the authors work with invariants rather than with co-invariants to describe the cohomology. In particular, they give a generating set for H 1 (C, d 0 ) and H 2 (C, d 0 ) in terms of antisymmetric polynomials.…”

Section: 2mentioning

confidence: 99%

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Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

Felder

2018

Sel. Math. New Ser.

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We establish an isomorphism between the Grothendieck-Teichmüller Lie algebra grt 1 in depth two modulo higher depth and the cohomology of the two-loop part of the graph complex of internally connected graphs ICG(1). In particular, we recover all linear relations satisfied by the brackets of the conjectural generators σ 2k+1 modulo depth three by considering relations among two-loop graphs.The Grothendieck-Teichmüller Lie algebra is related to the zeroth cohomology of M. Kontsevich's graph complex GC 2 via T. Willwacher's isomorphism. We define a descending filtration on H 0 (GC 2 ) and show that the degree two components of the corresponding associated graded vector spaces are isomorphic under T. Willwacher's map.

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Section: 2mentioning

confidence: 96%

Section: 2mentioning

confidence: 99%

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Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

Felder

2018

Sel. Math. New Ser.

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“…This implies that the spectral sequence associated with the induced filtration in the graphcomplex GC r F has as its first term H(GC H(F ) ). Applying this general construction to HGC r,h m,n described as (16)- (19) and knowing the fact that the homology of CH h An−m−2 is concentrated only in two degrees, see Section 2.3.2, we get the statements (6) and (7) of the theorem.…”

Section: Proof Of Theoremmentioning

confidence: 96%

“…This in particular means that the splitting HGC 2,h m,n = HGC 2,h,I m,n ⊕ HGC 2,h,II m,n of Theorem 1 is trivial in homology in the sense that one of the two terms of the splitting is always acyclic. Computations made in [7], specifically its [7, Theorems 6.1 and 6.2], imply…”

Section: Hairy Graph-homology In the Loop Order R =mentioning

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

2017

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Abstract. We study the cohomology of the hairy graph complexes which compute the rational homotopy of embedding spaces, generalizing the Vassiliev invariants of knot theory. We provide spectral sequences converging to zero whose first pages contain the hairy graph cohomology. Our results yield a way to construct many hairy graph cohomology classes out of non-hairy classes by a mechanism which we call the waterfall mechanism. By this mechanism we can construct many previously unknown classes and provide a first glimpse at the tentative global structure of the hairy graph cohomology.

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Two-loop part of the rational homotopy of spaces of long embeddings

Abstract: Abstract. Arone and Turchin defined graph-complexes computing the rational homotopy of the spaces of long embeddings. The graph-complexes split into a direct sum by the number of loops in graphs. In this paper we compute the homology of its two-loop part.

Cited by 13 publications

References 7 publications

Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

Filtrations on graph complexes and the Grothendieck–Teichmüller Lie algebra in depth two

Commutative hairy graphs and representations of Out (Fr)

Differentials on graph complexes II: hairy graphs

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