nLab
configuration space of points

Contents

Context

Topology

topology (point-set topology, point-free topology)

see also differential topology, algebraic topology, functional analysis and topological homotopy theory

Introduction

Basic concepts

Universal constructions

Extra stuff, structure, properties

Examples

Basic statements

Theorems

Analysis Theorems

topological homotopy theory

Contents

Idea

In mathematics, the term “configuration space” of a topological space XX typically refers by default to the topological space of pairwise distinct points in XX, 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 XX be a closed smooth manifold. For nn \in \mathbb{N} write

Conf {1,,n}(X)(X) nΔ X n \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 nn-fold Cartesian product of XX with itself.

This is the space of ordered but otherwise unlabeled configurations of nn points_ in XX.


Unordered unlabeled points

Definition

(unordered unlabled configurations of a fixed number of points)

Let XX be a closed smooth manifold, For nn \in \mathbb{N} write

(1)Conf n(X) (Conf 1,,n(X))/Sym(n) =((X) nΔ X n)/Sym(n) \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)Sym(n) via permutation of the ordering of the points.

This is the space of unordered and unlabeled configurations of nn points_ in XX.

We write

(2)Conf(X)nConf n(X) 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 nn.

Remark

(monoid-structure on configuration space of points)

For X= DX = \mathbb{R}^D a Euclidean spaces the configuration space of points Conf( D)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 Conf( D)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( D)group completionΩB Conf( D). 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 nn points (Def. ) is naturally a topological subspace of the space of finite subsets of cardinality n\leq n

(4)Conf n(X)exp n(X) Conf_n(X) \hookrightarrow \exp^n(X)
Proposition

Let XX be an non-empty regular topological space and n2n \geq 2 \in \mathbb{N}.

Then the injection (4)

(5)Conf n(X)exp n(X)/exp n1(X) Conf_n(X) \hookrightarrow \exp^n(X)/\exp^{n-1}(X)

of the unordered configuration space of n points of XX (Def. ) into the quotient space of the space of finite subsets of cardinality n\leq n by its subspace of subsets of cardinality n1\leq n-1 is an open subspace-inclusion.

Moreover, if XX is compact, then so is exp n(X)/exp n1(X)\exp^n(X)/\exp^{n-1}(X) and the inclusion (5) exhibits the one-point compactification (Conf n(X)) +\big( Conf_n(X) \big)^{+} of the configuration space:

(Conf n(X)) +exp n(X)/exp n1(X). \big( Conf_n(X) \big)^{+} \;\simeq\; \exp^n(X)/\exp^{n-1}(X) \,.

(Handel 00, Prop. 2.23, see also Félix-Tanré 10)

Unordered labeled points

Definition

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

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

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

(7)Conf(X,S k)(nConf n(X,S k))/ 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 nn-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 XX be a manifold, possibly with boundary. For nn \in \mathbb{N}, the configuration space of nn unordered points in XX disappearing at the boundary is the topological space

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

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

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

Conf n(X,Y)(((X nΔ X n)×Y n)/Σ(n))/(X n×Y n) \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 Σ(n)\Sigma(n) denotes the evident action of the symmetric group by permutation of factors of X×YX \times Y inside X n×Y n(X×Y) nX^n \times Y^n \simeq (X \times Y)^n.

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

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

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

Conf(X,Y)(n𝕟((X nΔ X n)×Y k)/Σ(n))/ 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

((x 1,y 1),,(x n1,y n1),(x n,y n))((x 1,y 1),,(x n1,y n1))(x n,y n)(X×Y). \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 n(X,Y)(k{1,,n}((X kΔ X k)×Y k)/Σ(k))/. 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)Conf n(X,Y)/Conf (n1)(X,Y). 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:

here: Segal 73, Snaith 74: Bödigheimer 87: Conf( d,Y) = C d(Y/Y) = C( d,;Y) Conf n( d) = F nC d(S 0)/F n1C d(S 0) = D n( d,;S 0) Conf n( d,Y) = F nC d(Y/Y)/F n1C d(Y/Y) = D n( d,;Y/Y) Conf n(X) = D n(X,X;S 0) Conf n(X,Y) = D n(X,X;Y/Y) \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 YY happens to have empty boundary, Y=\partial Y = \emptyset, then the pushout

X/YYY* X / \partial Y \coloneqq Y \underset{\partial Y}{\sqcup} \ast

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

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

is the 0-sphere.

A slight variation of the definition is sometimes useful:

Definition

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

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

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

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

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

of dim(X)dim(X)-dimensional disks/closed balls into XX, 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 DD \in \mathbb{N} the Cohomotopy charge map is the continuous function

(8)Conf( D)ccπ D(( D) cpt)=Maps */(( D) cpt,S D)=Ω DS D 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 D\mathbb{R}^D to the DD-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( D)S^D \simeq \big( \mathbb{R}^D \big) to itself, and hence equivalently the DD-fold iterated based loop space of the D-sphere), which sends a configuration of points in D\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= 2G = \mathbb{Z}_2 with values in 2\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)

The Cohomotopy charge map (8)

Conf( D)ccΩ DS D 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 D\mathbb{R}^D to the DD-fold iterated based loop space of the D-sphere, exhibits the group completion (3) of the configuration space monoid

ΩB Conf( D)Ω DS D \Omega B_{{}_{\sqcup}\!} Conf \big( \mathbb{R}^D \big) \overset{ \simeq }{\longrightarrow} \Omega^D S^D

(Segal 73, Theorem 1)

Proposition

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

For D,kD, k \in \mathbb{N} with k1k \geq 1, the Cohomotopy charge map (8)

Conf( D,S k)ccΩ DS D+kπ D+k(( D) cpt) 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)

is a weak homotopy equivalence

  • from the configuration space (7) of unordered points with labels in S kS^k and vanishing at the base point of the label space

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

hence equivalently

(Segal 73, Theorem 3)

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

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

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

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

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

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

    Σ VXXS VGTopSpaces\Sigma^V X \;\coloneqq\; X \wedge S^V \;\in G TopSpaces\;

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

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

Definition

(equivariant unordered labeled configurations vanishing with vanishing label)

Write

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

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

Proposition

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

Let

  1. GG be a finite group,

  2. VV an orthogonal GG-linear representation

  3. XX a topological G-space

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

Conf( V,X)ccΩ VΣ VXAAAif X is G-connected 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 XX-labeled configuration space of points (Def. ) in the corresponding Euclidean G-space to the based VV-loop space of the VV-suspension of XX, is a weak homotopy equivalence.

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

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

(Rourke-Sanderson 00, Theorem 1, Theorem 2)

More generally:

Proposition

(iterated loop spaces equivalent to configuration spaces of points)

For

  1. dd \in \mathbb{N}, d1d \geq 1 a natural number with d\mathbb{R}^d denoting the Cartesian space/Euclidean space of that dimension,

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

the Cohomotopy charge map constitutes a homotopy equivalence

Conf( D,Y)ccΩ DΣ D(Y/Y) 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 d×Y\mathbb{R}^d \times Y vanishing at the boundary (Def. )

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

In particular when Y=𝔻 kY = \mathbb{D}^k is the closed ball of dimension k1k \geq 1 this gives a homotopy equivalence

Conf( D,𝔻 k)ccΩ DS D+k 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.

(May 72, Theorem 2.7, Segal 73, Theorem 3, see Bödigheimer 87, Example 13)

Proposition

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

For

  1. dd \in \mathbb{N}, d1d \geq 1 a natural number with d\mathbb{R}^d denoting the Cartesian space/Euclidean space of that dimension,

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

there is a stable weak homotopy equivalence

Σ Conf( d,Y)nΣ Conf n( d,Y) \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 d×Y\mathbb{R}^d \times Y vanishing at the boundary and distinct already as points of d\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 d×Y\mathbb{R}^d \times Y, vanishing at the boundary and distinct already as points in d\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( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))=Ω dΣ d(Y/Y)Σ ccΣ Conf( d,Y)nΣ Conf n( d,Y) 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 dd-fold reduced suspension of Y/YY / \partial Y.

(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)

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

Maps cp( d,Σ d(Y/Y))=Maps */(S d,Σ d(Y/Y))nΣ Conf n( d×Y) 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= X= \mathbb{R}^\infty. Then

\,

Relation to James construction

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

(e.g. Bödigheimer 87, Example 9)

\,

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 nX^n be a smooth closed manifold of dimension nn;

  2. 1k1 \leq k \in \mathbb{N} a positive natural number.

Then the Cohomotopy charge map constitutes a weak homotopy equivalence

Maps /BO(n)(X n,S n def+k trivO(n))a J-twisted Cohomotopy spaceCohomotopy charge mapConf(X n,S k)aconfiguration spaceof points \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 nX^n, hence the space of sections of the (n+k)(n + k)-spherical fibration over XX which is associated via the tangent bundle by the O(n)-action on S n+k=S( n× k+1)S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})

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

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

Remark

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

Let

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

  2. 1k1 \leq k \in \mathbb{N} a positive natural number.

Then the Cohomotopy charge map constitutes a weak homotopy equivalence

Maps(X n,S n+k)a Cohomotopy spaceCohomotopy charge mapConf(X n,S k)aconfiguration spaceof points \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)(n+k)-Cohomotopy space of X nX^n, hence the space of maps from XX to the (n+k)-sphere

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

Action by little nn-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 XX be a topological space which is the interior of a compact manifold with boundary X¯\overline{X}. We may think of the boundary X¯\partial \overline X as consisting of the “points at infinity” in XX.

In particular, there are then inclusion maps

(9)Conf n(X)i nConf n+1(X) Conf_n \big( X \big) \overset{i_n}{\longrightarrow} Conf_{n+1} \big( X \big)

of the unordered configuration space of nn points in XX (Def. ) into that of n+1n + 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 XX.

(Randal-Williams 13, Section 4)

The homotopy class of these maps depends (just) on the connected component of the boundary X¯\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 XX be

Then for all nn \in \mathbb{N} the inclusion maps (9) are such that on ordinary homology with integer coefficients these maps induce split monomorphisms in all degrees,

H (Conf n(X),)H (i n,)H (Conf n+1(X),) 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 n/2\leq n/2 these are even isomorphisms

H p(Conf n(X),)H p(i n,)H p(Conf n+1(X),)AAAAforpn/2. 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 nn:

H p(Conf n(X),)H p(i n,)H p(Conf n+1(X),)AAAAforpn. 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 \,.

(Randal-Williams 13, Theorem A and Threorem B)


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 Conf {1,,n}( D)\underset{{}^{\{1,\cdots,n\}}}{Conf}\big( \mathbb{R}^D\big) (Def. ) of nn ordered unlabeled points in Euclidean space D\mathbb{R}^D

is generated by elements in degree D1D-1

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

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

subject to these three relations:

  1. (anti-)symmetry)

    ω ij=(1) Dω ji\omega_{i j} = (-1)^D \omega_{j i}
  2. nilpotency

    ω ijω ij=0\omega_{i j} \wedge \omega_{i j} \;=\; 0
  3. 3-term relation

    ω ijω jk+ω jkω ki+ω kiω ij=0 \omega_{i j} \wedge \omega_{j k} + \omega_{j k} \wedge \omega_{k i} + \omega_{k i} \wedge \omega_{i j} = 0

Hence:

(10)H (Conf {1,,n}( D),)[{ω ij} i,j{1,,n}]/(ω ij=(1) Dω ji ω ijω ij=0 ω ijω jk+ω jkω ki+ω kiω ij=0fori,j{1,,n}) 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.

See also at Fulton-MacPherson compactification the section de Rham cohomology.

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 Conf {1,,n}( D)\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( D)( D) nFatDiag Conf_n\big( \mathbb{R}^D \big) \;\coloneqq\; \big( \mathbb{R}^D \big)^n \setminus FatDiag

for the configuration space of nn ordered points in Euclidean space.

Proposition

The Whitehead product super Lie algebra of rationalized homotopy groups

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

is generated from elements

ω ijπ 2(Conf n( D)) AAAij{1,,n}, \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. ω ij=(1) Dω ji\omega^{i j} = (-1)^D \omega^{j i}

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

  3. [ω ij,ω jk+ω ki]=0\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.



Occurrences and Applications

Compactification

The Fulton-MacPherson compactification of configuration spaces of points in d\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)

The algebra-structure of configuration spaces over little n-disk operads/Fulton-MacPherson operads:

  • 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

and generalization to equivariant homotopy theory is discussed in

The relevant construction for the group completion of the configuration space

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

See also

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 Ω nS nX\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 XX appear as the Goodwillie derivatives of its mapping space/nonabelian cohomology-functor Maps(X,)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

and an operad-structure defined on it (Fulton-MacPherson operad) 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:

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

On loop spaces of configuration spaces of points:

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

Discussion of configuration spaces of points as moduli spaces of D0-D4-brane bound states

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

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