nLab weight systems are cohomology of loop space of configuration space

Statement

For horizontal chord diagrams

Remark

(Hopf algebra-structures)

Both the ordinary homology of a based loop space as well as the universal enveloping algebra of a Lie algebra are canonically Hopf algebras, the former via the Pontrjagin ring-structure (see at homology of loop spaces).

Proposition

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

For $D, n \in \mathbb{N}$ natural numbers and for any ground field $\mathbb{F}$ (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 $\mathbb{R}^D$ is isomorphic, as a Hopf algebra (Remark ), to the universal enveloping algebra of the infinitesimal braid Lie algebra:

$$H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \mathcal{U} \big( \mathcal{L}_n(D) \big) \,.$$

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

$$\Big( \mathcal{A}_n^{pb} \;\coloneqq\; Span \big( \mathcal{D}_n^{pb} \big)/(2T, 4T) , \circ \Big)$$

of horizontal chord diagrams on $n$ 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. ):

$$\big(\mathcal{A}_n^{pb}, \circ\big) \;\simeq\; \mathcal{U}(\mathcal{L}_n(D)) \,.$$

The combination of Prop. and Prop. yields:

Corollary

For $D, n \in \mathbb{N}$ and for any ground field $\mathbb{F}$ (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 $\mathbb{R}^D$ is isomorphic, as a Hopf algebra, to the associative algebra of horizontal chord diagrams on $n$ strands with product given by concatenation of strands (this Def.), modulo the 2T relations and 4T relations (this Def.):

$$H_\bullet \big( \Omega \underset{{}^{\{1,\cdots, n\}}}{Conf}(\mathbb{R}^D) \big) \;\simeq\; \big(\mathcal{A}_n^{pb}, \circ\big) \,.$$

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

1. the weight systems

   $$\mathcal{W}_n^{pb} \;\coloneqq\; Hom_{\mathbb{F} Mod} \big( \underset{ \mathcal{A}_n^{pb} }{ \underbrace{ Span \big( \mathcal{D}_n^{pb} \big) /(2T,4T) } } , \mathbb{F} \big)$$

   on horizontal chord diagrams of $n$ strands (elements of the set $\mathcal{D}^{pb}$)

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

$$H^\bullet \big( \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^D) \big) \;\simeq\; (\mathcal{W}_n^{pb})^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}_{pb} ) \,.$$

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

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 $k = \mathbb{R}$ the real numbers, there is a canonical injection of the real vector space $\mathcal{W}$ 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:

$$\mathcal{W} \;\overset{\;\;\;\;}{\hookrightarrow}\; H\mathbb{R}^\bullet \Big( \underset{n \in \mathbb{N}}{\sqcup} \Omega \underset{{}^{\{1,\cdots,n\}}}{Conf} \big( \mathbb{R}^3 \big) \Big)$$

This is stated as Kohno 02, Theorem 4.2

Related theorems

Facts about chord diagrams and their weight systems:

• chord diagrams modulo 4T are Jacobi diagrams modulo STU

• weight systems are the associated graded objects of Vassiliev invariants

• weight systems are cohomology of loop space of configuration space

• weight systems are cohomology of knot graph complex

• stringy weight systems span classical Lie algebra weight systems

• all horizontal weight systems are partitioned Lie algebra weight systems

Related concepts

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

knots	braids
chord diagram, Jacobi diagram	horizontal chord diagram
1T&4T relation	2T&4T relation/ infinitesimal braid relations
weight system	horizontal weight system
Vassiliev knot invariant	Vassiliev braid invariant
weight systems are associated graded of Vassiliev invariants	horizontal 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:

• Edward Fadell, Sufian Husseini, Theorem 2.2 in: Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001) (MR2002k:55038, doi:10.1007/978-3-642-56446-8)

• Fred Cohen, Samuel Gitler, Theorem 4.1 of: Loop spaces of configuration spaces, braid-like groups, and knots, In: Jaume Aguadé, Carles Broto, Carles Casacuberta (eds.) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel 2001 (doi:10.1007/978-3-0348-8312-2_7)

• Fred Cohen, Samuel Gitler, Theorem 2.3 of: On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, (jstor:2693715, MR2002m:55020)

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:

• Toshitake Kohno, Theorem 4.1 in: Loop spaces of configuration spaces and finite type invariants, Geom. Topol. Monogr. 4 (2002) 143-160 (arXiv:math/0211056)

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)