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weight systems are cohomology of loop space of configuration space

Contents

Contents

Statement

For horizontal chord diagrams

Proposition

(ordinary homology of based loop space of ordered configuration space of points is universal enveloping algebra of infinitesimal braid Lie algebra)

For D,nβˆˆβ„•D, n \in \mathbb{N} natural numbers and for any ground field 𝔽\mathbb{F} (in fact over every commutative ring) the ordinary homology of the based loop space of the ordered configuration space of points in the Cartesian space/Euclidean space ℝ D\mathbb{R}^D is isomorphic, as a Hopf algebra (Remark ), to the universal enveloping algebra of the infinitesimal braid Lie algebra:

H β€’(Ξ©Conf {1,β‹―,n}(ℝ D))≃𝒰(β„’ n(D)). H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \mathcal{U} \big( \mathcal{L}_n(D) \big) \,.

This is due to Fadell-Husseini 01, Theorem 2.2, re-stated as Cohen-Gitler 01, Theorem 4.1, Cohen-Gitler 02, Theorem 2.3.

Notice also that:

Proposition

(universal enveloping of infinitesimal braids is horizontal chord diagrams)

The associative algebra

(π’œ n pb≔Span(π’Ÿ n pb)/(2T,4T),∘) \Big( \mathcal{A}_n^{pb} \;\coloneqq\; Span \big( \mathcal{D}_n^{pb} \big)/(2T, 4T) , \circ \Big)

of horizontal chord diagrams on nn strands with product given by concatenation of strands (this Def.), modulo the 2T relations and 4T relations (this Def.) is isomorphic to the universal enveloping algebra of the infinitesimal braid Lie algebra (Def. ):

(π’œ n pb,∘)≃𝒰(β„’ n(D)). \big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.

The combination of Prop. and Prop. yields:

Corollary

For D,nβˆˆβ„•D, n \in \mathbb{N} and for any ground field 𝔽\mathbb{F} (in fact over every commutative ring) the ordinary homology of the based loop space of the ordered configuration space of points in the Cartesian space/Euclidean space ℝ D\mathbb{R}^D is isomorphic, as a Hopf algebra, to the associative algebra of horizontal chord diagrams on nn strands with product given by concatenation of strands (this Def.), modulo the 2T relations and 4T relations (this Def.):

H β€’(Ξ©Conf {1,β‹―,n}(ℝ D))≃(π’œ n pb,∘). H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \big(\mathcal{A}_n^{pb}, \circ\big) \,.

For horizontal weight systems

Proposition

(integral horizontal weight systems are integral cohomology of based loop space of ordered configuration space of points in Euclidean space)

Given any ground field 𝔽\mathbb{F} (in fact any ground ring, notably the integers) there is, for each natural number nn, a canonical isomorphism of graded abelian groups between

  1. the weight systems

    𝒲 n pb≔Hom 𝔽Mod(Span(π’Ÿ n pb)/(2T,4T)βŸπ’œ n pb,𝔽) \mathcal{W}_n^{pb} \;\coloneqq\; Hom_{\mathbb{F} Mod} \big( \underset{ \mathcal{A}_n^{pb} }{ \underbrace{ Span \big( \mathcal{D}_n^{pb} \big) /(2T,4T) } } , \mathbb{F} \big)

    on horizontal chord diagrams of nn strands (elements of the set π’Ÿ pb\mathcal{D}^{pb})

  2. the ordinary cohomology of the based loop space of the ordered configuration space of n points in Euclidean space:

H β€’(Ξ©Conf {1,β‹―,n}(ℝ D))≃(𝒲 n pb) ‒≃Gr β€’(𝒱 pb). H^\bullet \big( \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^D) \big) \;\simeq\; (\mathcal{W}_n^{pb})^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}_{pb} ) \,.

(the second equivalence on the right is the fact that weight systems are associated graded of Vassiliev invariants, for D=3D =3).

This appears stated as Kohno 02, Theorem 4.1; it follows immediately by Corollary of Prop. .

For round weight systems

Proposition

(weight systems are inside real cohomology of based loop space of ordered configuration space of points in Euclidean space)

For ground field k=ℝk = \mathbb{R} the real numbers, there is a canonical injection of the real vector space 𝒲\mathcal{W} of framed weight systems (here) into the real cohomology of the based loop spaces of the ordered configuration spaces of points in 3-dimensional Euclidean space:

𝒲β†ͺHℝ β€’(βŠ”nβˆˆβ„•Ξ©Conf {1,β‹―,n}(ℝ 3)) \mathcal{W} \;\overset{\;\;\;\;}{\hookrightarrow}\; H\mathbb{R}^\bullet \Big( \underset{n \in \mathbb{N}}{\sqcup} \Omega \underset{{}^{\{1,\cdots,n\}}}{Conf} \big( \mathbb{R}^3 \big) \Big)

This is stated as Kohno 02, Theorem 4.2

Facts about chord diagrams and their weight systems:

chord diagramsweight systems
linear chord diagrams,
round chord diagrams
Jacobi diagrams,
Sullivan chord diagrams
Lie algebra weight systems,
stringy weight system,
Rozansky-Witten weight systems


knotsbraids
chord diagram,
Jacobi diagram
horizontal chord diagram
1T&4T relation2T&4T relation/
infinitesimal braid relations
weight systemhorizontal weight system
Vassiliev knot invariantVassiliev braid invariant
weight systems are associated graded of Vassiliev invariantshorizontal weight systems are cohomology of loop space of configuration space

References

The statement relating the ordinary homology of the based loop space of the ordered configuration space of points to the universal enveloping algebra of the infinitesimal braid Lie algebra:

The dual statement identifying the ordinary cohomology of the based loop space of the ordered configuration space of points with the space of weight systems on horizontal chord diagrams:

See also:

  • Toshitake Kohno, Vassiliev invariants and de Rham complex on the space of knots,

    In: Yoshiaki Maeda, Hideki Omori and Alan Weinstein (eds.), Symplectic Geometry and Quantization, Contemporary Mathematics 179 (1994): 123-123 (doi:10.1090/conm/179)

Last revised on May 10, 2021 at 14:13:36. See the history of this page for a list of all contributions to it.