Contents

# Contents

## Idea

In mathematics, the term “configuration space” of a topological space $X$ typically refers by default to the topological space of pairwise distinct points in $X$, also called Fadell's configuration space, for emphasis.

In principle many other kinds of configurations and the spaces these form may be referred to by “configuration space”, notably in physics the usage is in a broader sense, see at configuration space (physics).

## Definition

Several variants of configuration spaces of points are of interest. They differ in whether

1. points are linearly ordered or not;

2. points are labeled in some labelling space;

3. points vanish on some subspace or if their labels are in some subspace.

Here are some of these variant definitions:

### Ordered unlabeled points

###### Definition

(ordered unlabled configurations of a fixed number of points)

Let $X$ be a closed smooth manifold. For $n \in \mathbb{N}$ write

$\underset{ {}^{\{1,\cdots, n\}} }{ Conf } \big( X \big) \;\coloneqq\; \big( X \big)^n \setminus \mathbf{\Delta}^n_X$

for the complement of the fat diagonal inside the $n$-fold Cartesian product of $X$ with itself.

This is the space of ordered but otherwise unlabeled configurations of $n$ points_ in $X$.

### Unordered unlabeled points

###### Definition

(unordered unlabled configurations of a fixed number of points)

Let $X$ be a closed smooth manifold, For $n \in \mathbb{N}$ write

(1)\begin{aligned} Conf_n \big( X \big) & \coloneqq \; \Big( \underset{{}^{1,\cdots,n}}{Conf} \big( X \big) \big) / Sym(n) \\ & =\; \Big( \big( X \big)^n \setminus \mathbf{\Delta}^n_X \Big) / Sym(n) \end{aligned}

for the quotient space of the ordered configuration space (Def. ) by the evident action of the symmetric group $Sym(n)$ via permutation of the ordering of the points.

This is the space of unordered and unlabeled configurations of $n$ points_ in $X$.

We write

(2)$Conf(X) \;\coloneqq\; \underset{n \in \mathbb{N}}{\sqcup} Conf_n\big( X\big)$

for the unordered unlabeled configuration space of any finite number of points, being the disjoint union of these spaces (1) over all natural numbers $n$.

###### Remark

(monoid-structure on configuration space of points)

For $X = \mathbb{R}^D$ a Euclidean spaces the configuration space of points $Conf\big( \mathbb{R}^D \big)$ (2) carriesthe structure of a topological monoid with product operation being the disjoint union of point configurations, after a suitable shrinking to put them next to each other (Segal 73, p. 1-2).

For emphasis, we write $B_{{}_{\sqcup}\!} Conf(\mathbb{R}^D)$ for the delooping (“classifying space”) with respect to this topological monoid-structure. The corresponding based loop space is then the group completion of the configuration space, with respect to disjoint union of points:

(3)$Conf \big( \mathbb{R}^D \big) \overset{\color{blue}\text{group completion}}{\longrightarrow} \Omega B_{{}_{\sqcup}\!} Conf(\mathbb{R}^D) \,.$
###### Remark

The configuration space of unordered unlabeled configurations of $n$ points (Def. ) is naturally a topological subspace of the space of finite subsets of cardinality $\leq n$

(4)$Conf_n(X) \hookrightarrow \exp^n(X)$
###### Proposition

Let $X$ be an non-empty regular topological space and $n \geq 2 \in \mathbb{N}$.

Then the injection (4)

(5)$Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X)$

of the unordered configuration space of n points of $X$ (Def. ) into the quotient space of the space of finite subsets of cardinality $\leq n$ by its subspace of subsets of cardinality $\leq n-1$ is an open subspace-inclusion.

Moreover, if $X$ is compact, then so is $\exp^n(X)/\exp^{n-1}(X)$ and the inclusion (5) exhibits the one-point compactification $\big( Conf_n(X) \big)^{+}$ of the configuration space:

$\big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,.$

### Unordered labeled points

###### Definition

For $X$ a smooth manifold and $k \in \mathbb{N}$, the space of unordered configurations of points in $X$ with labels in $S^k$ is

(6)$Conf_n\big(X, S^k \big) \;\coloneqq\; Conf_n\big(X\big) \underset{Sym(n)}{\times} \big( S^k \big)^n$

For $k \in \mathbb{N}$, consider the k-sphere as a pointed topological space, with the base point regarded as the “vanishing label”.

###### Definition

(unordered labeled configurations vanishing with vanishing label)

For $X$ a smooth manifold and $k \in \mathbb{N}$, the space of unordered configurations of points in $X$ with labels in $S^k$ and vanishing at vanishing label value is the quotient space

(7)$Conf \big( X, S^k \big) \;\coloneqq\; \Big( \underset{n \in \mathbb{N}}{\sqcup} Conf_n\big(X,S^k \big) \Big)/\sim$

of the disjoint union of all unordred labeled $n$-point configuration spaces (6) by the equivalence relation which regards points with vanishing label as absent.

###### Definition

(unordered labeled configurations of a fixed number of points)

Let $X$ be a manifold, possibly with boundary. For $n \in \mathbb{N}$, the configuration space of $n$ unordered points in $X$ disappearing at the boundary is the topological space

$\mathrm{Conf}_{n}(X) \;\coloneqq\; \Big( \big( X^n \setminus \mathbf{\Delta}_X^n \big) / \partial(X^n) \Big) /\Sigma(n) \,,$

where $\mathbf{\Delta}_X^n : = \{(x^i) \in X^n | \underset{i,j}{\exists} (x^i = x^j) \}$ is the fat diagonal in $X^n$ and where $\Sigma(n)$ denotes the evident action of the symmetric group by permutation of factors of $X$ inside $X^n$.

More generally, let $Y$ be another manifold, possibly with boundary. For $n \in \mathbb{N}$, the configuration space of $n$ points in $X \times Y$ vanishing at the boundary and distinct as points in $X$ is the topological space

$\mathrm{Conf}_{n}(X,Y) \;\coloneqq\; \Big( \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^n \big) /\Sigma(n) \Big) / \partial(X^n \times Y^n)$

where now $\Sigma(n)$ denotes the evident action of the symmetric group by permutation of factors of $X \times Y$ inside $X^n \times Y^n \simeq (X \times Y)^n$.

This more general definition reduces to the previous case for $Y = \ast \coloneqq \mathbb{R}^0$ being the point:

$\mathrm{Conf}_n(X) \;=\; \mathrm{Conf}_n(X,\ast) \,.$

Finally the configuration space of an arbitrary number of points in $X \times Y$ vanishing at the boundary and distinct already as points of $X$ is the quotient topological space of the disjoint union space

$Conf\left( X, Y\right) \;\coloneqq\; \left( \underset{n \in \mathbb{n}}{\sqcup} \big( ( X^n \setminus \mathbf{\Delta}_X^n ) \times Y^k \big) /\Sigma(n) \right)/\sim$

by the equivalence relation $\sim$ given by

$\big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}), (x_n, y_n) \big) \;\sim\; \big( (x_1, y_1), \cdots, (x_{n-1}, y_{n-1}) \big) \;\;\;\; \Leftrightarrow \;\;\;\; (x_n, y_n) \in \partial (X \times Y) \,.$

This is naturally a filtered topological space with filter stages

$Conf_{\leq n}\left( X, Y\right) \;\coloneqq\; \left( \underset{k \in \{1, \cdots, n\}}{\sqcup} \big( ( X^k \setminus \mathbf{\Delta}_X^k ) \times Y^k \big) /\Sigma(k) \right)/\sim \,.$

The corresponding quotient topological spaces of the filter stages reproduces the above configuration spaces of a fixed number of points:

$Conf_n(X,Y) \;\simeq\; Conf_{\leq n}(X,Y) / Conf_{\leq (n-1)}(X,Y) \,.$
###### Remark

(comparison to notation in the literature)

The above Def. is less general but possibly more suggestive than what is considered for instance in Bödigheimer 87. Concretely, we have the following translations of notation:

$\array{ \text{ here: } && \array{ \text{ Segal 73,} \\ \text{ Snaith 74}: } && \text{ Bödigheimer 87: } \\ \\ Conf(\mathbb{R}^d,Y) &=& C_d( Y/\partial Y ) &=& C( \mathbb{R}^d, \emptyset; Y ) \\ \mathrm{Conf}_n\left( \mathbb{R}^d \right) & = & F_n C_d( S^0 ) / F_{n-1} C_d( S^0 ) & = & D_n\left( \mathbb{R}^d, \emptyset; S^0 \right) \\ \mathrm{Conf}_n\left( \mathbb{R}^d, Y \right) & = & F_n C_d( Y/\partial Y ) / F_{n-1} C_d( Y/\partial Y ) & = & D_n\left( \mathbb{R}^d, \emptyset; Y/\partial Y \right) \\ \mathrm{Conf}_n( X ) && &=& D_n\left( X, \partial X; S^0 \right) \\ \mathrm{Conf}_n( X, Y ) && &=& D_n\left( X, \partial X; Y/\partial Y \right) }$

Notice here that when $Y$ happens to have empty boundary, $\partial Y = \emptyset$, then the pushout

$X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast$

is $Y$ with a disjoint basepoint attached. Notably for $Y =\ast$ the point space, we have that

$\ast/\partial \ast = S^0$

is the 0-sphere.

A slight variation of the definition is sometimes useful:

###### Definition

(configuration space of $dim(X)$-disks)

In the situation of Def. , with $X$ a manifold of dimension $dim(X) \in \mathbb{N}$

$DiskConf(X,A) \longrightarrow Conf(X,A)$

be, on the left, the labeled configuration space of joint embeddings of tuples

$\left( D^{dim(X)} \overset{ \iota_i }{\hookrightarrow} X \right)$

of $dim(X)$-dimensional disks/closed balls into $X$, with identifications as in Def. (in particular the disks centered at the basepoint are quotiented out) and with the comparison map sending each disk to its center.

This map is evidently a deformation retraction hence in particular a homotopy equivalence.

## Properties

### Ordered unlabeled configurations from unordered labeled configurations

under construction

(…)

(…)

### Cohomotopy charge map

The Cohomotopy charge map is the function that assigns to a configuration of points their total charge as measured in Cohomotopy-cohomology theory.

This is alternatively known as the “electric field map” (Salvatore 01 following Segal 73, Section 1, see also Knudsen 18, p. 49) or the “scanning map” (Kallel 98).

For $D \in \mathbb{N}$ the Cohomotopy charge map is the continuous function

(8)$Conf\big( \mathbb{R}^D \big) \overset{cc}{\longrightarrow} \mathbf{\pi}^D \Big( \big( \mathbb{R}^D \big)^{cpt} \Big) = Maps^{\ast/\!}\Big( \big(\mathbb{R}^D\big)^{cpt} , S^D\big) = \Omega^{D} S^D$

from the configuration space of points in the Euclidean space $\mathbb{R}^D$ to the $D$-Cohomotopy cocycle space vanishing at infinity on the Euclidean space, which is equivalently the space of pointed maps from the one-point compactification $S^D \simeq \big( \mathbb{R}^D \big)$ to itself, and hence equivalently the $D$-fold iterated based loop space of the D-sphere), which sends a configuration of points in $\mathbb{R}^D$, each regarded as carrying unit charge to their total charge as measured in Cohomotopy-cohomology theory (Segal 73, Section 3).

The construction has evident generalizations to other manifolds than just Euclidean spaces, to spaces of labeled configurations and to equivariant Cohomotopy. The following graphics illustrates the Cohomotopy charge map on G-space tori for $G = \mathbb{Z}_2$ with values in $\mathbb{Z}_2$-equivariant Cohomotopy:

graphics grabbed from SS 19

#### Relation to iterated loop spaces of iterated suspensions

In some situations the Cohomotopy charge map is a weak homotopy equivalence and hence exhibits, for all purposes of homotopy theory, the Cohomotopy cocycle space of Cohomotopy charges as an equivalent reflection of the configuration space of points:

###### Proposition

(group completion on configuration space of points is iterated based loop space)

$Conf \big( \mathbb{R}^D \big) \overset{ cc }{\longrightarrow} \Omega^D S^D$

from the full unordered and unlabeled configuration space (2) of Euclidean space $\mathbb{R}^D$ to the $D$-fold iterated based loop space of the D-sphere, exhibits the group completion (3) of the configuration space monoid

$\Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^D \big) \overset{ \simeq }{\longrightarrow} \Omega^D S^D$
###### Proposition

(Cohomotopy charge map is weak homotopy equivalence on sphere-labeled configuration space of points)

For $D, k \in \mathbb{N}$ with $k \geq 1$, the Cohomotopy charge map (8)

$Conf \big( \mathbb{R}^D, S^k \big) \underoverset{\simeq}{\;\;cc\;\;}{\longrightarrow} \Omega^D S^{D + k} \;\simeq\; \mathbf{\pi}^{D+ k}\Big( \big( \mathbb{R}^{D}\big)^{cpt} \Big)$
• from the configuration space (7) of unordered points with labels in $S^k$ and vanishing at the base point of the label space

• to the $D$-fold iterated loop space of the D+k-sphere

hence equivalently

This statement generalizes to equivariant homotopy theory, with equivariant configurations carrying charge in equivariant Cohomotopy:

Let $G$ be a finite group and $V \in RO(G)$ an orthogonal $G$-linear representation, with its induced pointed topological G-spaces:

1. the corresponding representation sphere $S^V \in G TopSpaces$,

2. the corresponding Euclidean G-space $\mathbb{R}^V \in G TopSpaces$.

For $X \in G TopSpaces$ any pointed topological G-space, consider

1. the equivariant $V$-suspension, given by the smash product with the $V$-representation sphere:

$\Sigma^V X \;\coloneqq\; X \wedge S^V \;\in G TopSpaces\;$

2. the equivariant $V$-iterated based loop space, given by the $G$-fixed point subspace inside the space of maps out of the representation sphere:

$\Omega^V X \;\coloneqq\; Maps^{\ast/}\big( S^V, X\big)^G$.

###### Definition

(equivariant unordered labeled configurations vanishing with vanishing label)

Write

$Conf\big( \mathbb{R}^V , X \big)^G \;\hookrightarrow\; Conf\big( \mathbb{R}^V , X \big)$

for the $G$-fixed point subspace in the unordered $X$-labelled configuration space of points (Def. ), with respect to the diagonal action on $\mathbb{R}^V \times X$.

###### Proposition

(Cohomotopy charge map-equivalence for configurations on Euclidean G-spaces)

Let

1. $G$ be a finite group,

2. $V$ an orthogonal $G$-linear representation

3. $X$ a topological G-space

If $X$ is $G$-connected, in that for all subgroups $H \subset G$ the $H$-fixed point subspace $X^H$ is a connected topological space, then the Cohomotopy charge map

$Conf \big( \mathbb{R}^V, X \big) \underoverset{\simeq}{\;cc\;}{\longrightarrow} \Omega^V \Sigma^V X \phantom{AAA} \text{if X is G-connected}$

from the equivariant un-ordered $X$-labeled configuration space of points (Def. ) in the corresponding Euclidean G-space to the based $V$-loop space of the $V$-suspension of $X$, is a weak homotopy equivalence.

If $X$ is not necessarily $G$-connected, then this still holds for the group completion of the configuration space, under disjoint union of configurations

$\Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^V, X \big) \underoverset{\simeq}{\;cc\;}{\longrightarrow} \Omega^{V+1} \Sigma^{V+1} X \,.$

More generally:

###### Proposition

(iterated loop spaces equivalent to configuration spaces of points)

For

1. $d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,

2. $Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,

the Cohomotopy charge map constitutes a homotopy equivalence

$Conf\left( \mathbb{R}^D, Y \right) \overset{cc}{\longrightarrow} \Omega^D \Sigma^D (Y/\partial Y)$

between

1. the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. )

2. the d-fold loop space of the $d$-fold reduced suspension of the quotient space $Y / \partial Y$ (regarded as a pointed topological space with basepoint $[\partial Y]$).

In particular when $Y = \mathbb{D}^k$ is the closed ball of dimension $k \geq 1$ this gives a homotopy equivalence

$Conf\left( \mathbb{R}^D, \mathbb{D}^k \right) \overset{cc}{\longrightarrow} \Omega^D S^{ D + k }$

with the d-fold loop space of the (d+k)-sphere.

###### Proposition

(stable splitting of mapping spaces out of Euclidean space/n-spheres)

For

1. $d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,

2. $Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,

there is a stable weak homotopy equivalence

$\Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$

between

1. the suspension spectrum of the configuration space of an arbitrary number of points in $\mathbb{R}^d \times Y$ vanishing at the boundary and distinct already as points of $\mathbb{R}^d$ (Def. )

2. the direct sum (hence: wedge sum) of suspension spectra of the configuration spaces of a fixed number of points in $\mathbb{R}^d \times Y$, vanishing at the boundary and distinct already as points in $\mathbb{R}^d$ (also Def. ).

Combined with the stabilization of the Cohomotopy charge map homotopy equivalence from Prop. this yields a stable weak homotopy equivalence

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) = \Omega^d \Sigma^d (Y/\partial Y) \underoverset{\Sigma^\infty cc}{\simeq}{\longrightarrow} \Sigma^\infty Conf(\mathbb{R}^d, Y) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d, Y)$

between the latter direct sum and the suspension spectrum of the mapping space of pointed continuous functions from the d-sphere to the $d$-fold reduced suspension of $Y / \partial Y$.

In fact by Bödigheimer 87, Example 5 this equivalence still holds with $Y$ treated on the same footing as $\mathbb{R}^d$, hence with $Conf_n(\mathbb{R}^d, Y)$ on the right replaced by $Conf_n(\mathbb{R}^d \times Y)$ in the well-adjusted notation of Def. :

$Maps_{cp}(\mathbb{R}^d, \Sigma^d (Y / \partial Y)) = Maps^{\ast/}( S^d, \Sigma^d (Y / \partial Y)) \overset{\simeq}{\longrightarrow} \underset{n \in \mathbb{N}}{\oplus} \Sigma^\infty Conf_n(\mathbb{R}^d \times Y)$

#### Relation to classifying space of the symmetric group

Let $X= \mathbb{R}^\infty$. Then

• the unordered configuration space of $n$ points in $\mathbb{R}^\infty$ is a model for the classifying space $B \Sigma(n)$ of the symmetric group $\Sigma(n)$;

• the ordered configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-action, is a model for the $\Sigma(n)$-universal principal bundle.

$\,$

#### Relation to James construction

The James construction of $X$ is homotopy equivalent to the configuration space of points $C(\mathbb{R}^1, X)$ of points in the real line with labels taking values in $X$.

$\,$

#### In twisted Cohomotopy

The May-Segal theorem generalizes from Euclidean space to closed smooth manifolds if at the same time one passes from plain Cohomotopy to twisted Cohomotopy, twisted, via the J-homomorphism, by the tangent bundle:

###### Proposition

Let

1. $X^n$ be a smooth closed manifold of dimension $n$;

2. $1 \leq k \in \mathbb{N}$ a positive natural number.

Then the Cohomotopy charge map constitutes a weak homotopy equivalence

$\underset{ \color{blue} { \phantom{a} \atop \text{ J-twisted Cohomotopy space}} }{ Maps_{{}_{/B O(n)}} \Big( X^n \;,\; S^{ \mathbf{n}_{def} + \mathbf{k}_{\mathrm{triv}} } \!\sslash\! O(n) \Big) } \underoverset {\simeq} { \color{blue} \text{Cohomotopy charge map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }$

between

1. the J-twisted (n+k)-Cohomotopy space of $X^n$, hence the space of sections of the $(n + k)$-spherical fibration over $X$ which is associated via the tangent bundle by the O(n)-action on $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$

2. the configuration space of points on $X^n$ with labels in $S^k$.

(Bödigheimer 87, Prop. 2, following McDuff 75)

###### Remark

In the special case that the closed manifold $X^n$ in Prop. is parallelizable, hence that its tangent bundle is trivializable, the statement of Prop. reduces to this:

Let

1. $X^n$ be a parallelizable closed manifold of dimension $n$;

2. $1 \leq k \in \mathbb{N}$ a positive natural number.

Then the Cohomotopy charge map constitutes a weak homotopy equivalence

$\underset{ \color{blue} { \phantom{a} \atop \text{ Cohomotopy space}} }{ Maps \Big( X^n \;,\; S^{ n + k } \Big) } \underoverset {\simeq} { \color{blue} \text{Cohomotopy charge map} } {\longleftarrow} \underset{ \mathclap{ \color{blue} { \phantom{a} \atop { \text{configuration space} \atop \text{of points} } } } }{ Conf \big( X^n, S^k \big) }$

between

1. $(n+k)$-Cohomotopy space of $X^n$, hence the space of maps from $X$ to the (n+k)-sphere

2. the configuration space of points on $X^n$ with labels in $S^k$.

### Knizhnik-Zamolodchicov connection

For the Definition of the Knizhnik-Zamolodchikov connection we need the following notation:

1. configuration spaces of points

For $N_{\mathrm{f}} \in \mathbb{N}$ write

(9)$\underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{R}^2) \;\coloneqq\; (\mathbb{R}^2)^n \backslash FatDiagonal$

for the ordered configuration space of n points in the plane, regarded as a smooth manifold.

Identifying the plane with the complex plane $\mathbb{C}$, we have canonical holomorphic coordinate functions

(10)$(z_1, \cdots, z_{N_{\mathrm{f}}}) \;\colon\; \underset{{}^{\{1,\cdots,n\}}}{Conf}(\mathbb{R}^2) \longrightarrow \mathbb{C}^{N_{\mathrm{f}}} \,.$
2. horizontal chord diagrams

(11)$\mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} \;\coloneqq\; Span\big(\mathcal{D}^{{}^{pb}}_{N_{\mathrm{f}}}\big)/4T$

for the quotient vector space of the linear span of horizontal chord diagrams on $n$ strands by the 4T relations (infinitesimal braid relations), regarded as an associative algebra under concatenation of strands (here).

###### Definition

(Knizhnik-Zamolodchikov form)

(12)$\omega_{KZ} \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} \big)$

given in the canonical coordinates (10) by:

(13)$\omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes t_{i j} \,,$

where

is the horizontal chord diagram with exactly one chord, which stretches between the $i$th and the $j$th strand.

Regarded as a connection form for a connection on a vector bundle, this defines the universal Knizhnik-Zamolodchikov connection $\nabla_{KZ}$, with covariant derivative

$\nabla \phi \;\coloneqq\; d \phi + \omega_{KV} \wedge \phi$

for any smooth function

$\phi \;\colon\; \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \longrightarrow \mathcal{A}^{{}^{pb}}_{N_{\mathrm{f}}} Mod$

with values in modules over the algebra of horizontal chord diagrams modulo 4T relations.

The condition of covariant constancy

$\nabla_{KZ} \phi \;=\; 0$

is called the Knizhnik-Zamolodchikov equation.

Finally, given a metric Lie algebra $\mathfrak{g}$ and a tuple of Lie algebra representations

$( V_1, \cdots, V_{N_{\mathrm{f}}} ) \;\in\; (\mathfrak{g} Rep_{/\sim})^{N_{\mathrm{f}}} \,,$

the corresponding endomorphism-valued Lie algebra weight system

$w_{V} \;\colon\; \mathcal{A}^{{}^{pf}}_{N_{\mathrm{f}}} \longrightarrow End_{\mathfrak{g}}\big( V_1 \otimes \cdots V_{N_{\mathrm{f}}} \big)$

turns the universal Knizhnik-Zamolodchikov form (12) into a endomorphism ring-valued differential form

(14)$\omega_{KZ} \;\coloneqq\; \underset{ i \lt j \in \{1, \cdots, n\} }{\sum} d_{dR} log\big( z_i - z_j \big) \otimes w_V(t_{i j}) \;\in\; \Omega \big( \underset{{}^{\{1,\cdots,N_{\mathrm{f}}\}}}{Conf}(\mathbb{C}) \,, End\big(V_1 \otimes \cdot V_{N_{\mathrm{f}}} \big) \big) \,.$

The universal formulation (12) is highlighted for instance in Bat-Natan 95, Section 4.2, Lescop 00, p. 7. Most authors state the version after evaluation in a Lie algebra weight system, e.g. Kohno 14, Section 5.

###### Proposition

(Knizhnik-Zamolodchikov connection is flat)

The Knizhnik-Zamolodchikov connection $\omega_{ZK}$ (Def. ) is flat:

$d \omega_{ZK} + \omega_{ZK} \wedge \omega_{ZK} \;=\; 0 \,.$
###### Proposition

(Kontsevich integral for braids)

The Dyson formula for the holonomy of the Knizhnik-Zamolodchikov connection (Def. ) is called the Kontsevich integral on braids.

### Action by little $n$-disk operad and by Goodwillie derivatives

Under some conditions and with suitable degrees/shifts, configuration spaces of points canonically have the structure of algebras over an operad over the little n-disk operad and the Goodwillie derivatives of the identitity functor?.

For more see there

### Homology and stabilization in homology

Let $X$ be a topological space which is the interior of a compact manifold with boundary $\overline{X}$. We may think of the boundary $\partial \overline X$ as consisting of the “points at infinity” in $X$.

In particular, there are then inclusion maps

(15)$Conf_n \big( X \big) \overset{i_n}{\longrightarrow} Conf_{n+1} \big( X \big)$

of the unordered configuration space of $n$ points in $X$ (Def. ) into that of $n + 1$ points, formalizing the idea of “adding a point at infinity” to a configuration. More formally, these maps are given by pushing configuration points away from the boundary a little and then adding a new point near to a point on the boundary of $X$.

The homotopy class of these maps depends (just) on the connected component of the boundary $\partial \overline{X}$ at which one chooses to bring in the new point. But for any choice, they have the following effect on cycles in ordinary homology:

###### Proposition

(homological stabilization for unordered configuration spaces)

Let $X$ be

Then for all $n \in \mathbb{N}$ the inclusion maps (15) are such that on ordinary homology with integer coefficients these maps induce split monomorphisms in all degrees,

$H_\bullet \big( Conf_n(X) , \mathbb{Z} \big) \overset{ H_\bullet( i_n, \mathbb{Z} ) }{\hookrightarrow} H_\bullet \big( Conf_{n+1}(X) , \mathbb{Z} \big)$

and in degrees $\leq n/2$ these are even isomorphisms

$H_p \big( Conf_n(X) , \mathbb{Z} \big) \underoverset{\simeq}{ H_p( i_n, \mathbb{Z} ) }{\hookrightarrow} H_p \big( Conf_{n+1}(X) , \mathbb{Z} \big) \phantom{AAAA} \text{for} \; p \leq n/2 \,.$

Finally, for ordinary homology with rational coefficients, these maps induce isomorphisms all the way up to degree $n$:

$H_p \big( Conf_n(X) , \mathbb{Q} \big) \underoverset{\simeq}{ H_p( i_n, \mathbb{Q} ) }{\hookrightarrow} H_p \big( Conf_{n+1}(X) , \mathbb{Q} \big) \phantom{AAAA} \text{for} \; p \leq n \,.$

### Rational homotopy type

We discuss aspects of the rational homotopy type of configuration spaces of points. See also at graph complex.

#### Rational cohomology

###### Proposition

(real cohomology of configuration spaces of ordered points in Euclidean space)

The real cohomology ring of the configuration spaces $\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big)$ (Def. ) of $n$ ordered unlabeled points in Euclidean space $\mathbb{R}^D$

is generated by elements in degree $D-1$

$\omega_{i j} \;\; \in H^2 \Big( \underset{ {}^{\{1, \cdots, n\}} }{ Conf } \big( \mathbb{R}^D \big), \mathbb{R} \Big)$

for $i, j \in \{1, \cdots, n\}$

subject to these three relations:

1. (anti-)symmetry)

$\omega_{i j} = (-1)^D \omega_{j i}$
2. nilpotency

$\omega_{i j} \wedge \omega_{i j} \;=\; 0$
3. 3-term relation

$\omega_{i j} \wedge \omega_{j k} + \omega_{j k} \wedge \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0$

Hence:

(16)$H^\bullet \Big( \underset{ {}^{\{1,\cdots,n\}} }{Conf} \big( \mathbb{R}^D \big), \mathbb{R} \Big) \;\simeq\; \mathbb{R}\Big[ \big\{\omega_{i j} \big\}_{i, j \in \{1, \cdots, n\}} \Big] \Big/ \left( \array{ \omega_{i j} = (-1)^D \omega_{j i} \\ \omega_{i j} \wedge \omega_{i j} = 0 \\ \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0 } \;\; \text{for}\; i,j \in \{1, \cdots, n\} \right)$

This is due to Cohen 76, following Arnold 69, Cohen 73. See also Félix-Tanré 03, Section 2 Lambrechts-Tourtchine 09, Section 3.

###### Remark

(real cohomology of the configuration space in terms of graph cohomology)

In the graph complex-model for the rational homotopy type of the ordered unlabled configuration space of points $\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big)$ the three relations in Prop. are incarnated as follows:

1. a graph changes sign when one of its edges is reversed (this Def.)

2. a graph with parallel edges is a vanishing graph (this Def.)

3. the graph coboundary of a single trivalent internal vertex (this Example).

#### Rational homotopy and Whitehead products

Write again

$Conf_n\big( \mathbb{R}^D \big) \;\coloneqq\; \big( \mathbb{R}^D \big)^n \setminus FatDiag$

for the configuration space of $n$ ordered points in Euclidean space.

###### Proposition
$L^n \;\coloneqq\; \pi_{\bullet+1}\Big( Conf_n\big( \mathbb{R}^D \big) \Big) \otimes_{\mathbb{Z}} \mathbb{Q}$

is generated from elements

$\omega^{i j} \;\in\; \pi_2 \Big( Conf_n\big( \mathbb{R}^D \big) \Big) \otimes_{\mathbb{Z}} \mathbb{Q} \phantom{AAA} i \neq j \in \{1, \cdots, n\} \,,$

subject to the following relations:

1. $\omega^{i j} = (-1)^D \omega^{j i}$

2. $\big[ \omega^{i j}, \omega^{k l} \big]$ $\;\;\;$ if $i,j,k,l$ are pairwise distinct;

3. $\big[ \omega^{i j}, \omega^{j k} + \omega^{k i} \big] = 0$.

This is due to Kohno 02. See also Lambrechts-Tourtchine 09, Section 3.

### Relation weight systems, chord diagrams and Vassiliev invariants

###### Proposition

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

For ground ring $R = \mathbb{Z}$ the integers, there is, for each natural number $n$, a canonical isomorphism of graded abelian groups between

1. the integral weight systems

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

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

$H \mathbb{Z}^\bullet \big( \Omega \underset{ {}^{\{1,\cdots,n\}} }{Conf} (\mathbb{R}^3) \big) \;\simeq\; (\mathcal{W}^{pb}_n)^\bullet \;\simeq\; Gr^\bullet( \mathcal{V}^{pb}_n ) \,.$

(the second equivalence on the right is the fact that weight systems are associated graded of Vassiliev invariants).

This is stated as Kohno 02, Theorem 4.1

###### 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

## Occurrences and Applications

### Compactification

The Fulton-MacPherson compactification of configuration spaces of points in $\mathbb{R}^d$ serves to exhibit them as models for the little n-disk operad.

### Stable splitting of mapping spaces

The stable splitting of mapping spaces says that suspension spectra of suitable mapping spaces are equivalently wedge sums of suspension spectra of configuration spaces of points.

### Correlators as differential forms on configuration spaces

In Euclidean field theory the correlators are often regarded as distributions of several variables with singularities on the fat diagonal. Hence they become non-singular distributions after restriction of distributions to the corresponding configuration space of points.

For more on this see at correlators as differential forms on configuration spaces of points.

## References

### General

General accounts:

In relation to the space of finite subsets:

• David Handel, Some Homotopy Properties of Spaces of Finite Subsets of Topological Spaces, Houston Journal of Mathematics, Electronic Edition Vol. 26, No. 4, 2000 (pdfhjm:Vol26-4)

• Yves Félix, Daniel Tanré Rational homotopy of symmetric products and Spaces of finite subsets, Contemp. Math 519 (2010): 77-92 (pdf)

• Martin Markl, A compactification of the real configuration space as an operadic completion, J. Algebra 215 (1999), no. 1, 185–204

### Cohomotopy charge map

The Cohomotopy charge map (“electric field map”, “scanning map”) and hence the relation of configuration spaces to Cohomotopy goes back to

Generalization to equivariant homotopy theory:

The relevant construction for the group completion of the configuration space

and from the point of view of cobordism categories:

On the homotopy type of the space of rational functions from the Riemann sphere to itself (related to the moduli space of monopoles in $\mathbb{R}^3$ and to the configuration space of points in $\mathbb{R}^2$):

• Sadok Kallel, Spaces of particles on manifolds and Generalized Poincaré Dualities, The Quarterly Journal of Mathematics, Volume 52, Issue 1, 1 March 2001 (doi:10.1093/qjmath/52.1.45)

• Shingo Okuyama, Kazuhisa Shimakawa, Interactions of strings and equivariant homology theories, (arXiv:0903.4667)

For relation to instantons via topological Yang-Mills theory:

In speculation regarding Galois theory over the sphere spectrum:

### Stable splitting of mapping spaces

The appearance of configuration spaces as summands in stable splittings of mapping spaces is originally due to

• Victor Snaith, A stable decomposition of $\Omega^n S^n X$, Journal of the London Mathematical Society 7 (1974), 577 - 583 (pdf)

An alternative proof is due to

Review and generalization is in

and the relation to the Goodwillie-Taylor tower of mapping spaces is pointed out in

### In Goodwillie-calculus

The configuration spaces of a space $X$ appear as the Goodwillie derivatives of its mapping space/nonabelian cohomology-functor $Maps(X,-)$:

• Greg Arone, A generalization of Snaith-type filtration, Transactions of the American Mathematical Society 351.3 (1999): 1123-1150. (pdf)

• Michael Ching, Calculus of Functors and Configuration Spaces, Conference on Pure and Applied Topology Isle of Skye, Scotland, 21-25 June, 2005 (pdf)

### Compactification

A compactification of configuration spaces of points was introduced in

Review includes

This underlies the models of configuration spaces by graph complexes, see there for more.

### Homology and cohomology

General discussion of ordinary homology/ordinary cohomology of configuration spaces of points:

Expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces:

• Quoc P. Ho, Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras (arXiv:2004.00252)

### Homotopy

Discussion of homotopy groups of configuration spaces:

### Rational homotopy type

Discussion of the rational homotopy type:

• Igor Kriz, On the Rational Homotopy Type of Configuration Spaces, Annals of Mathematics

Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 227-237 (jstor:2946581)

### Cohomology modeled by graph complexes

That the de Rham cohomology of (the Fulton-MacPherson compactification of) configuration spaces of points may be modeled by graph complexes (exhibiting formality of the little n-disk operad) is due to

nicely reviewed in Lambrechts-Volic 14

Further discussion of the graph complex as a model for the de Rham cohomology of configuration spaces of points is in

### Loop spaces of configuration spaces of points

Specifically on ordinary homology/ordinary cohomology of based loop spaces of configuration spaces of points and the relation to weight systems/Vassiliev invariants:

### In quantum field theory

This perspective was originally considered specifically for Chern-Simons theory in

which was re-amplified in

and highlighted as a means to obtain graph complex-models for the de Rham cohomology of configuration spaces of points in

• Maxim Kontsevich, Vassiliev’s knot invariants, Advances in Soviet Mathematics, Volume 16, Part 2, 1993 (pdf)

• Maxim Kontsevich, pages 11-12 of Feynman diagrams and low-dimensional topology, First European Congress of Mathematics, 1992, Paris, vol. II, Progress in Mathematics 120, Birkhäuser (1994), 97–121 (pdf)

with full details and proofs in

A systematic development of Euclidean perturbative quantum field theory with n-point functions considered as smooth functions on Fulton-MacPherson compactifications/wonderful compactifications of configuration spaces of points and more generally of subspace arrangements is due to

Discussion specifically in topological quantum field theory with an eye towards supersymmetric field theory, in terms of the ordinary homology of configuration spaces of points:

### As moduli of Dp-D(p+4)-brane bound states:

Discussion of configuration spaces of possibly coincident points, hence of symmetric products $X^n/Sym(n)$ as moduli spaces of D0-D4-brane bound states:

with emphasis to the resulting configuration spaces of points, as in

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