Ordered and Unordered Configuration Spaces

Let $n\in \mathbb{N}$, so that $n=\{0,1,\dots ,n-1\}$ is the set of its predecessors (The natural numbers $\mathbb{N}$ (von Neumann)), and let $X$ be a topological space (Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). Write

$X^n:={\prod }_{k<n}X$

for the $n$-fold product, carrying the product topology (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), and display its points as $(x_1,\dots ,x_n)$: the label $i\in \{1,\dots ,n\}$ names the coordinate of index $i-1$ in the sense of that definition. The ordered configuration space of $n$ points in $X$ is the subspace

$F_n(X):=\{(x_1,\dots ,x_n)\in X^n:x_i\mathrm{\ne }{x}_j\text{ whenever }i\mathrm{\ne }j\}$

with the subspace topology inherited from $X^n$ (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace). Equivalently

$F_n(X)=X^n\setminus {\bigcup }_{i\mathrm{\ne }j}\{(x_1,\dots ,x_n)\in X^n:x_i=x_j\},$

since a tuple lies in $F_n(X)$ exactly when its entries are pairwise distinct: the collision diagonals $x_i=x_j$, $i\mathrm{\ne }j$, are removed from the product. Points of $F_n(X)$ are called ordered configurations of $n$ points in $X$.

The label set. The labels $1,\dots ,n$ are part of the data, and throughout this page they are identified with the set $n=\{0,1,\dots ,n-1\}$ by the bijection $\kappa (i):=i-1$. It is through $\kappa$ that the symmetric group $S_n=\mathrm{Sym}(n)$ acts on $F_n(X)$, in The symmetric group acts continuously and freely on $F_n(X)$ by permuting labels.

Elementary cases. For $n=0$ the product $X^0$ is a one-point space (The product set ${\prod }_{i\in I}{X}_i$ of functions choosing a point in each factor, the projections, the box topology, and the product topology as the initial topology of the projections; the empty product is a one-point space), and the defining condition is vacuous, so

$F_0(X)=\{\text{the empty tuple}\}$

for every $X$, including $X=\mathrm{\varnothing }$. For $n=1$ there is no pair $i\mathrm{\ne }j$, so single-coordinate evaluation $(x_1)\mapsto x_1$ is a canonical homeomorphism $F_1(X)\cong X$. For $n\ge 2$ and any $X$ one has

$F_n(X)\mathrm{\ne }\mathrm{\varnothing }⟺X\text{ has at least }n\text{ distinct points}\mathrm{.}$

if $X$ has at least $n$ points, an injection $\{1,\dots ,n\}\to X$ is exactly a tuple of pairwise distinct points of $X$, and conversely such a tuple displays $n$ distinct points. In particular $F_n(X)=\mathrm{\varnothing }$ when $X$ is empty and $n\ge 1$.

Based configurations. A base configuration in $F_n(X)$ is a point $q=(q_1,\dots ,q_n)$ of $F_n(X)$. Such a $q$ is fixed once and for all only when $F_n(X)$ is nonempty; when $n\ge 1$, $F_n(X)\mathrm{\ne }\mathrm{\varnothing }$, and $X$ is infinite, $F_n(X)$ is infinite: from any one configuration, keep coordinates $2,\dots ,n$ fixed and vary the first coordinate among the infinitely many points of $X\setminus \{q_2,\dots ,q_n\}$. The choice of $q$ is part of the data of every construction below. All base configurations on this page are chosen in the ordered space $F_n(X)$; the corresponding basepoint of the unordered quotient is its orbit (Unordered configuration spaces $C_n(X)$).

Separation of distinct coordinates. If $X$ is Hausdorff (Hausdorff space: distinct points have disjoint open neighbourhoods; every metrizable space is Hausdorff and the indiscrete topology on two points is not) and $q\in F_n(X)$, then the finitely many points $q_1,\dots ,q_n$ are pairwise distinct, and for each pair $i\mathrm{\ne }j$ Hausdorffness supplies disjoint open sets separating $q_i$ from $q_j$; a finite intersection over the finitely many pairs $j\mathrm{\ne }i$ therefore gives, for every $i$, an open neighbourhood $U_i$ of $q_i$ with $U_i\cap U_j=\mathrm{\varnothing }$ whenever $i\mathrm{\ne }j$. This is used in Disjoint coordinate neighbourhoods evenly cover the unordered configuration space.

Let $n\in \mathbb{N}$ and let $X$ be a topological space (Ordered configuration spaces $F_n(X)$). Give $S_n$ the discrete topology (The discrete, indiscrete, cofinite, cocountable, particular-point and Sierpinski topologies) and $F_n(X)$ the subspace topology of its definition. Then

$S_n\times F_n(X)⟶F_n(X),(\sigma ,x)⟼\sigma \cdot x,(\sigma \cdot x{)}_i:=x_{{\sigma }^{-1}(i-1)+1}(1\le i\le n),$

is a continuous left action of $S_n$ on $F_n(X)$, and it is free (A free group action has no nonidentity element fixing a point): $\sigma \cdot x=x$ forces $\sigma =\mathrm{id}$. The cases $n=0$ and $n=1$ are included, $S_0$ and $S_1$ being the trivial group, and so is the case $F_n(X)=\mathrm{\varnothing }$, where the action is continuous and free vacuously.

Let $n\in \mathbb{N}$ and let $X$ be a topological space (Ordered configuration spaces $F_n(X)$, Topology on a set, open and closed sets, clopen sets, the closed-set axiomatisation, and the coarser/finer comparison). The label permutation action

$S_n\times F_n(X)⟶F_n(X),(\sigma ,x)⟼\sigma \cdot x,(\sigma \cdot x{)}_i=x_{{\sigma }^{-1}(i-1)+1},$

is a continuous free left action (The symmetric group acts continuously and freely on $F_n(X)$ by permuting labels). Its set of orbits $S_n\cdot x=\{\sigma \cdot x:\sigma \in S_n\}$ (The orbit $G\cdot x$ and stabilizer $G_x$ of a point in a group action) is the unordered configuration space of $n$ points in $X$, written

$C_n(X):=F_n(X)\mathrm{/}{S}_n=\{S_n\cdot x:x\in F_n(X)\},$

and it carries the quotient topology of the canonical projection

$p_n:F_n(X)⟶C_n(X),p_n(x):=S_n\cdot x,$

in the sense of The quotient topology of a surjection, quotient maps, saturated sets, and the quotient of a space by an equivalence relation with its canonical projection: a subset $V\subseteq C_n(X)$ is open exactly when $p_n^{-1}(V)$ is open in $F_n(X)$. This projection is a quotient map, hence continuous and surjective, and points of $C_n(X)$ are written $[x]:=p_n(x)$.

Basepoint. For a base configuration $q\in F_n(X)$ in the sense of Ordered configuration spaces $F_n(X)$, the basepoint of $C_n(X)$ is the orbit

$[q]=p_n(q)=S_n\cdot q,$

and $C_n(X)$ is nonempty exactly when $F_n(X)$ is, in which case a basepoint can be fixed. The unordered space is based by the orbit of the ordered base configuration, and this is the basepoint used in every later construction on this page.

Elementary cases. $C_0(X)$ is the quotient of the one-point space $F_0(X)$ by the trivial group $S_0$, hence is a one-point space. Since $S_1$ is trivial, $p_1:F_1(X)\to C_1(X)$ is a bijective quotient map and hence a homeomorphism: for every open $U\subseteq F_1(X)$, the equality $p_1^{-1}(p_1(U))=U$ makes $p_1(U)$ open by the quotient topology. Thus $C_1(X)$ is canonically homeomorphic to $X$ by $[(x)]\mapsto x$, using the single-coordinate homeomorphism $F_1(X)\to X$. These are canonical identifications, not literal equalities of the orbit set with the original set.

Elements are $n$-element subsets, as a set. Because the coordinates of a configuration are pairwise distinct, two ordered configurations $x,y\in F_n(X)$ lie in the same orbit exactly when their underlying sets of coordinates agree: if $y=\sigma \cdot x$ then the coordinates of $y$ are those of $x$ in a different order, and conversely, if $\{x_1,\dots ,x_n\}=\{y_1,\dots ,y_n\}$, then for every label $i$ there is exactly one label $\sigma (i)$ with $y_i=x_{\sigma (i)}$, and $i\mapsto \sigma (i)$ is a bijection of $\{1,\dots ,n\}$ (Injection, surjection, bijection); composing with the identification $\kappa (i)=i-1$ of labels with $n$, the permutation $\tau =\kappa \circ {\sigma }^{-1}\circ {\kappa }^{-1}\in S_n$ takes $x$ to $y$: $(\tau \cdot x{)}_i=x_{{\tau }^{-1}(i-1)+1}=x_{\sigma (i)}=y_i$. Hence

$C_n(X)⟶\{S\subseteq X:S\text{ has exactly }n\text{ elements}\},[x]⟼\{x_1,\dots ,x_n\},$

is a bijection of sets, the inverse sending an $n$-element subset $S$ to the orbit of any enumeration of $S$ (The cardinality $\mid A\mid$ of a finite set). This identifies the elements of $C_n(X)$ with $n$-element subsets of $X$; the topology on $C_n(X)$ is the quotient topology displayed above, and no topology on a set of subsets is asserted here. In particular the quotient topology is not defined through any metric or hyperspace structure.

Let $X$ be a topological space, let $x_0,x_1\in X$ and let $c:I\to X$ be a path in $X$ from $x_0$ to $x_1$ (Paths, path-connected spaces and path components). Write $\bar{c}(s):=c(1-s)$ for the reversed path and let $\ast$ denote the first-then-second concatenation of composable paths of Paths, path-connected spaces and path components, so that for every based loop $\alpha :I\to X$ at $x_0$ (Based loops and the fundamental group) the concatenation $\bar{c}\ast \alpha \ast c$ is a loop at $x_1$; the bracketing of that triple product is immaterial up to path homotopy rel endpoints by step 1.2 below, and the bracket $(\bar{c}\ast \alpha )\ast c$ is used throughout. Then:

1. The assignment ${\phi }_c:{\pi }_1(X,x_0)⟶{\pi }_1(X,x_1),{\phi }_c([\alpha ]):=[(\bar{c}\ast \alpha )\ast c],$ is well defined: if $\alpha \simeq {\alpha }^{\mathrm{\prime }}$ rel endpoints (Homotopies of continuous maps, homotopies relative to a subspace, and path homotopies relative to the endpoints), then $(\bar{c}\ast \alpha )\ast c\simeq (\bar{c}\ast {\alpha }^{\mathrm{\prime }})\ast c$ rel endpoints.
2. ${\phi }_c$ is a group isomorphism (Group isomorphisms, automorphisms and the set $\mathrm{Aut}(G)$), and its two-sided inverse is ${\phi }_{\bar{c}}:{\pi }_1(X,x_1)⟶{\pi }_1(X,x_0),{\phi }_{\bar{c}}([\beta ]):=[(c\ast \beta )\ast \bar{c}]\mathrm{.}$

Consequently ${\pi }_1(X,x_0)\cong {\pi }_1(X,x_1)$ whenever a path from $x_0$ to $x_1$ exists, that is, whenever $x_0$ and $x_1$ lie in the same path component of $X$; the isomorphism depends on the chosen path, and no claim is made that it is independent of that choice.

Write $D^2:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }\le 1\},\mathrm{int}{D}^2:=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }<1\}$ for the closed unit disc and its interior in $\mathbb{C}$; the topological interior of $D^2$ in $\mathbb{C}$ is exactly $\mathrm{int}{D}^2$ (step 1.1), so the notation is accurate. For $n\in \mathbb{N}$ write $F_n(D^2)$, $F_n(\mathrm{int}{D}^2)$ for the ordered configuration spaces and $C_n(D^2)$, $C_n(\mathrm{int}{D}^2)$ for the unordered ones (Ordered configuration spaces $F_n(X)$, Unordered configuration spaces $C_n(X)$), with quotient maps $p_X:F_n(X)\to C_n(X)$. Let ${\iota }^F:F_n(\mathrm{int}{D}^2)\to F_n(D^2)$ be the inclusion and let ${\iota }^C:C_n(\mathrm{int}{D}^2)\to C_n(D^2)$ be the map induced by ${\iota }^F$ on the orbit quotients (constructed in step 3.2). Put ${\rho }^F(x_1,\dots ,x_n):=(\frac{x_1}{2},\dots ,\frac{x_n}{2}),H_t(x_1,\dots ,x_n):=\left(\right(1-\frac{t}{2})x_1,\dots ,(1-\frac{t}{2}\left)x_n\right)(t\in I),$ for $x\in F_n(D^2)$. Then for every $n\in \mathbb{N}$:

1. $H$ is a homotopy from ${\mathrm{id}}_{F_n(D^2)}$ to the composite ${\iota }^F\circ {\rho }^F$, and the restriction of $H$ to $F_n(\mathrm{int}{D}^2)\times I$ is a homotopy from ${\mathrm{id}}_{F_n(\mathrm{int}{D}^2)}$ to ${\rho }^F\circ {\iota }^F$; hence ${\iota }^F$ is a homotopy equivalence (Homotopy equivalences, homotopy inverses and spaces of the same homotopy type) with homotopy inverse ${\rho }^F$.
2. $H$ descends to a homotopy $H^C:C_n(D^2)\times I\to C_n(D^2)$ from ${\mathrm{id}}_{C_n(D^2)}$ to ${\iota }^C\circ {\rho }^C$, where ${\rho }^C$ is the map induced by ${\rho }^F$ (step 4.2); likewise the descended homotopy restricted to $C_n(\mathrm{int}{D}^2)$ exhibits ${\rho }^C\circ {\iota }^C\simeq {\mathrm{id}}_{C_n(\mathrm{int}{D}^2)}$ (step 6.1). Hence ${\iota }^C$ is a homotopy equivalence with homotopy inverse ${\rho }^C$, and the two equivalences are compatible with the quotient maps: $p_{D^2}\circ H_t=H_t^C\circ p_{D^2},p_{\mathrm{int}{D}^2}\circ {\rho }^F={\rho }^C\circ p_{D^2},{\iota }^C\circ p_{\mathrm{int}{D}^2}=p_{D^2}\circ {\iota }^F\mathrm{.}$
3. For every $q\in F_n(\mathrm{int}{D}^2)$ the induced homomorphisms of fundamental groups (The homomorphism on fundamental groups induced by a pointed continuous map) ${\iota }_{\ast }^F:{\pi }_1(F_n(\mathrm{int}{D}^2),q)\to {\pi }_1(F_n(D^2),q),{\iota }_{\ast }^C:{\pi }_1(C_n(\mathrm{int}{D}^2),[q])\to {\pi }_1(C_n(D^2),[q])$ are isomorphisms. In particular the closed-disc and open-disc models compute the same fundamental groups at every configuration of interior points, so no boundary basepoint change is needed when a later result is stated on either model.

The case $n=0$ is included: $F_0$ and $C_0$ of either space are one-point spaces, and the assertions are the trivial ones.