Euler characteristics for spaces of string links and the modular envelope of $\mathcal{L}_{\infty}$

Tsopméné, Paul Arnaud Songhafouo; Turchin, Victor

doi:10.4310/hha.2018.v20.n2.a7

Cited by 3 publications

(4 citation statements)

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“…Theorem 1.9 realizes (over Z) the subspace Lie(m − 1) of distinctly labeled trees with m leaves in T n (m). We suspect that this identification agrees with the one given by [55], since both use graph complexes associated to configuration spaces. Some but not all trees with repeated leaf-labels could also arise from graphing braids, as explained in Conjecture 5.…”

Section: Examples and A Conjecturesupporting

confidence: 71%

“…For m ⩾ 2 and j ⩾ 1, we get k-dimensional string links in R N with k ⩾ 2 and N ⩾ 5. In this setting of equidimensional string links, we can compare our work to a result of Songhafouo Tsopméné and Turchin [56, Theorem 3.2] (further developed by these authors [55] and by Fresse, Turchin, and Willwacher [25,Section 5]). That result identifies π 0 Emb c m R k , R n+k ⊗ Q with the Q-vector space of trivalent trees with leaves labeled by {1, .…”

Section: Examples and A Conjecturementioning

confidence: 81%

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Diagrams for primitive cycles in spaces of pure braids and string links

Komendarczyk,

Koytcheff,

Volić

2024

Annales De l'Institut Fourier

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The based loop space of a configuration space of points in a Euclidean space can be viewed as a space of pure braids in a Euclidean space of one dimension higher. We continue our study of such spaces in terms of Kontsevich's CDGA of diagrams and Chen's iterated integrals. We construct a power series connection which yields a Hopf algebra isomorphism between the homology of the space of pure braids and the cobar construction on diagrams. It maps iterated Whitehead products to trivalent trees modulo the IHX relation. As an application, we establish a correspondence between Milnor invariants of Brunnian spherical links and certain Chen integrals. Finally we show that graphing induces injections of a certain submodule of the homotopy of configuration spaces into the homotopy of many spaces of string links. We conjecture that graphing is injective on all rational homotopy classes.Résumé. -L'espace des lacets basés d'un espace de configurations des points dans un espace euclidien peut être vu comme un espace de tresses pures dans un espace euclidien d'une dimension superieure. Nous continuons notre travail sur ces espaces du point de vue de l'ADGC de Kontsevich des diagrammes et des intégrales de Chen. Nous construisons une connexion série puissance qui donne une isomorphisme d'algèbre de Hopf entre l'homologie de l'espace des tresses pures et la construction cobar sur les diagrammes. Il mappe les classes primitives aux arbres trivalents modulo la relation IHX. En conséquence, nous établissons une bijection entre les invariants de Milnor des entrelacs sphériques bruniens et certaines intégrales de Chen. Enfin nous montrons que le graphe induit des injections d'un certain sous-module de l'homotopie des espaces de configurations dans l'homotopie des espaces de longs entrelacs. Nous conjecturons qu'il induit des injections de toute l'homotopie rationelle des espaces de configurations.

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Section: Examples and A Conjecturesupporting

confidence: 71%

Section: Examples and A Conjecturementioning

confidence: 81%

Diagrams for primitive cycles in spaces of pure braids and string links

Komendarczyk,

Koytcheff,

Volić

2024

Annales De l'Institut Fourier

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“…Thus for n odd (respectively even) these 3 (respectively 4) classes are linearly independent. Recall that π * (L n 1, 1 )⊗Q is known to be 3-dimensional for n odd and 4-dimensional for n even [STT18a], as discussed in the proof of Lemma 5.1 (c). Furthermore, (ι 1 ) * (K ), (ι 2 ) * (K ), and L ± L cannot be proper multiples of other classes because each one maps to a generator of π 2n−6 K n 1 .…”

Section: Configuration Space Integrals For Cohomology Of Spaces Of Lo...mentioning

confidence: 99%

“…Conjecture 5.7 concerns the left-hand group for p = 1, that is, π 2n−6 L n 1, 1 . It is known rationally [STT18a], and we expect that all the torsion comes from the graphing map from a sphere.…”

Section: Introductionmentioning

confidence: 99%

Graphing, homotopy groups of spheres, and spaces of links and knots

Koytcheff¹

2022

Preprint

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We study homotopy groups of spaces of links, focusing on long links of codimension at least three. In the case of multiple components, they admit split injections from homotopy groups of spheres. We calculate them, up to knotting, in a range depending on the dimensions of the source manifolds and target manifold which roughly generalizes the triple-point-free range for isotopy classes. At the edge of this range, joining components sends both a parametrized long Borromean rings class and a Hopf fibration to a generator of the first nontrivial homotopy group of the space of long knots. For spaces of two-component long links, we give generators of the homotopy group in this dimension in terms of this class from the Borromean rings and homotopy groups of spheres. A key ingredient in most of our results is a graphing map that increases source and target dimensions by one. CONTENTS 1. Introduction 1.1. Previous related results 1.2. Main results and organization 1.3. Acknowledgments 2. Key definitions 2.1. Spaces of embeddings, link maps, and pseudoisotopy embeddings 2.2. Maps between spaces of links 3. Injectivity of graphing 3.1. Injectivity for spaces of embeddings via the λ-invariant 3.2. Injectivity for spaces of link maps in a range via the α-invariant 4. Bijectivity of graphing in a range 4.1. Lemmas on pseudoisotopy embedding spaces 4.2. Bijectivity of graphing and homotopy groups of spaces of long links in a range 4.3. Spherical links and a further calculation for long links 5. Homotopy classes in spaces of long knots and links from joining pure braids 5.1. Configuration space integrals for cohomology of spaces of long 1-dimensional links 5.2. Some low-dimensional graph cocycles 5.3. Homology classes dual to configuration space integrals 5.4. Long Borromean rings, the Hopf map, and classes of long links and knots 5.5. Questions and future directions References

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Homology Representations of Compactified Configurations on Graphs Applied to 𝓜 _2,n

Bibby

Chan

Gadish

et al. 2023

Experimental Mathematics

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Euler characteristics for spaces of string links and the modular envelope of $\mathcal{L}_{\infty}$

Cited by 3 publications

References 27 publications

Diagrams for primitive cycles in spaces of pure braids and string links

Diagrams for primitive cycles in spaces of pure braids and string links

Graphing, homotopy groups of spheres, and spaces of links and knots

Homology Representations of Compactified Configurations on Graphs Applied to 𝓜 _2,n

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Euler characteristics for spaces of string links and the modular envelope of $\mathcal{L}_{\infty}$

Cited by 3 publications

References 27 publications

Diagrams for primitive cycles in spaces of pure braids and string links

Diagrams for primitive cycles in spaces of pure braids and string links

Graphing, homotopy groups of spheres, and spaces of links and knots

Homology Representations of Compactified Configurations on Graphs Applied to 𝓜 2,n

Contact Info

Product

Resources

About

Homology Representations of Compactified Configurations on Graphs Applied to 𝓜 _2,n