topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
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fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
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Examples
Basic statements
closed subspaces of compact Hausdorff spaces are equivalently compact subspaces
open subspaces of compact Hausdorff spaces are locally compact
compact spaces equivalently have converging subnet of every net
continuous metric space valued function on compact metric space is uniformly continuous
paracompact Hausdorff spaces equivalently admit subordinate partitions of unity
injective proper maps to locally compact spaces are equivalently the closed embeddings
locally compact and second-countable spaces are sigma-compact
Theorems
Analysis Theorems
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).
Several variants of configuration spaces of points are of interest. They differ in whether
points are linearly ordered or not;
points are labeled in some labelling space;
points vanish on some subspace or if their labels are in some subspace.
Here are some of these variant definitions:
(ordered unlabled configurations of a fixed number of points)
Let $X$ be a closed smooth manifold. For $n \in \mathbb{N}$ write
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 unlabled configurations of a fixed number of points)
Let $X$ be a closed smooth manifold, For $n \in \mathbb{N}$ write
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
for the unordered unlabeled configuration space of any finite number of points, being the disjoint union of these spaces (1) over all natural numbers $n$.
(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) carries the 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:
The configuration space of unordered unlabeled configurations of $n$ points (Def. ) is naturally a topological subspace of the space of finite subsets of cardinality $\leq n$
Let $X$ be an non-empty regular topological space and $n \geq 2 \in \mathbb{N}$.
Then the injection (4)
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:
(Handel 00, Prop. 2.23, see also Félix-Tanré 10)
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
For $k \in \mathbb{N}$, consider the k-sphere as a pointed topological space, with the base point regarded as the “vanishing label”.
(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
of the disjoint union of all unordered labeled $n$-point configuration spaces (6) by the equivalence relation which regards points with vanishing label as absent.
(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
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
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:
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
by the equivalence relation $\sim$ given by
This is naturally a filtered topological space with filter stages
The corresponding quotient topological spaces of the filter stages reproduces the above configuration spaces of a fixed number of points:
(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:
Notice here that when $Y$ happens to have empty boundary, $\partial Y = \emptyset$, then the pushout
is $Y$ with a disjoint basepoint attached. Notably for $Y =\ast$ the point space, we have that
is the 0-sphere.
A slight variation of the definition is sometimes useful:
(configuration space of $dim(X)$-disks)
In the situation of Def. , with $X$ a manifold of dimension $dim(X) \in \mathbb{N}$
be, on the left, the labeled configuration space of joint embeddings of tuples
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.
(forgetting points is a fibration)
Let $X$ be a topological manifold. For $n, N \,\in\, \mathbb{N}$, the continuous map
which forgets the first $n$ points is a locally trivial fiber bundle with typical fiber $X \setminus \{x_1, \cdots, x_N\}$. In particular it is a Hurewicz fibration.
Using this one may deduce that:
(configuration space of points in plane is EM-space of braid group)
The homotopy type of a configuration space of points in the plane is that of an Eilenberg-MacLane space of the braid group $Br(n)$ in degree 1:
under construction
(…)
(…)
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
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
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:
(group completion on configuration space of points is iterated based loop space)
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
(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)
is a weak homotopy equivalence
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:
the corresponding representation sphere $S^V \in G TopSpaces$,
the corresponding Euclidean G-space $\mathbb{R}^V \in G TopSpaces$.
For $X \in G TopSpaces$ any pointed topological G-space, consider
the equivariant $V$-suspension, given by the smash product with the $V$-representation sphere:
$\Sigma^V X \;\coloneqq\; X \wedge S^V \;\in G TopSpaces\;$
the equivariant $V$-iterated based loop space, given by the $G$-fixed point subspace inside the space of maps out of the representation sphere:
$\Omega^V X \;\coloneqq\; Maps^{\ast/}\big( S^V, X\big)^G$.
(equivariant unordered labeled configurations vanishing with vanishing label)
Write
for the $G$-fixed point subspace in the unordered $X$-labelled configuration space of points (Def. ), with respect to the diagonal action on $\mathbb{R}^V \times X$.
(Cohomotopy charge map-equivalence for configurations on Euclidean G-spaces)
Let
$G$ be a finite group,
$V$ an orthogonal $G$-linear representation
$X$ 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
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
(Rourke-Sanderson 00, Theorem 1, Theorem 2)
More generally:
(iterated loop spaces equivalent to configuration spaces of points)
For
$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,
$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,
the Cohomotopy charge map constitutes a homotopy equivalence
between
the configuration space of arbitrary points in $\mathbb{R}^d \times Y$ vanishing at the boundary (Def. )
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
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)
(stable splitting of mapping spaces out of Euclidean space/n-spheres)
For
$d \in \mathbb{N}$, $d \geq 1$ a natural number with $\mathbb{R}^d$ denoting the Cartesian space/Euclidean space of that dimension,
$Y$ a manifold, with non-empty boundary so that $Y / \partial Y$ is connected,
there is a stable weak homotopy equivalence
between
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. )
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
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 $Y / \partial Y$.
(Snaith 74, theorem 1.1, Bödigheimer 87, Example 2)
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. :
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 $\Sigma(n)$;
(e.g. Bödigheimer 87, Example 10)
the ordered configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-action, is a model for the $\Sigma(n)$-universal principal bundle.
$\,$
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 $X$.
(e.g. Bödigheimer 87, Example 9)
$\,$
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:
Let
$X^n$ be a smooth closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the Cohomotopy charge map constitutes a weak homotopy equivalence
between
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 $S^{n+k} = S(\mathbb{R}^{n} \times \mathbb{R}^{k+1})$
the configuration space of points on $X^n$ with labels in $S^k$.
(Bödigheimer 87, Prop. 2, following McDuff 75)
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
$X^n$ be a parallelizable closed manifold of dimension $n$;
$1 \leq k \in \mathbb{N}$ a positive natural number.
Then the Cohomotopy charge map constitutes a weak homotopy equivalence
between
$(n+k)$-Cohomotopy space of $X^n$, hence the space of maps from $X$ to the (n+k)-sphere
the configuration space of points on $X^n$ with labels in $S^k$.
A similar relation holds for mapping spaces not to spheres, but to complex projective spaces:
The homotopy type of the space of rational maps from the Riemann sphere to complex projective $n$-space $\mathbb{C}P^n$ of algebraic degree $d$ is that of the configuration space of at most $d$ points in $\mathbb{R}^2$ with labels in $S^{2n-1}$:
(Cohen & Shimamoto 91, Theorem 1)
For the Definition of the Knizhnik-Zamolodchikov connection we need the following notation:
configuration spaces of points
For $N_{\mathrm{f}} \in \mathbb{N}$ write
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
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).
The universal Knizhnik-Zamolodchikov form is the horizontal chord diagram-algebra valued differential form (12) on the configuration space of points (10)
given in the canonical coordinates (11) by:
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
for any smooth function
with values in modules over the algebra of horizontal chord diagrams modulo 4T relations.
The condition of covariant constancy
is called the Knizhnik-Zamolodchikov equation.
Finally, given a metric Lie algebra $\mathfrak{g}$ and a tuple of Lie algebra representations
the corresponding endomorphism-valued Lie algebra weight system
turns the universal Knizhnik-Zamolodchikov form (13) into a endomorphism ring-valued differential form
The universal formulation (13) 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.
(Knizhnik-Zamolodchikov connection is flat)
The Knizhnik-Zamolodchikov connection $\omega_{ZK}$ (Def. ) is flat:
(Kontsevich integral for braids)
The Dyson formula for the holonomy of the Knizhnik-Zamolodchikov connection (Def. ) is called the Kontsevich integral on braids.
(e.g. Lescop 00, side-remark 1.14)
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
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
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 $X$.
(Randal-Williams 13, Section 4)
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:
(homological stabilization for unordered configuration spaces)
Let $X$ be
which is the interior of a compact manifold with boundary
of dimension $dim(X) \geq 2$.
Then for all $n \in \mathbb{N}$ the inclusion maps (16) are such that on ordinary homology with integer coefficients these maps induce split monomorphisms in all degrees,
and in degrees $\leq n/2$ these are even isomorphisms
Finally, for ordinary homology with rational coefficients, these maps induce isomorphisms all the way up to degree $n$:
(Randal-Williams 13, Theorem A and Threorem B)
On Gauss-Manin connections over configuration spaces of points:
Daniel C. Cohen, Peter Orlik, Gauss-Manin Connections for Arrangements, I Eigenvalues, Compositio Math. 136 (2003) 299-316 $[$arXiiv:math/0105063, doi:10.1023/A:1023262022279$]$
Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, II Nonresonant weights, Amer. J. Math. 127 (2005) 569-594 $[$arXiv:math/0207114, jstor:40067930$]$
Daniel C. Cohen, Peter Orlik, Gauss-Manin connections for arrangements, III Formal connections, Trans. Amer. Math. Soc. 357 (2005) 3031-3050 $[$arXiv:math/0307210, doi:10.1090/S0002-9947-04-03621-9$]$
and review in the context of hypergeometric solutions to the Knizhnik-Zamolodchikov equation:
We discuss aspects of the rational homotopy type of configuration spaces of points. See also at graph complex.
(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 $D-1$
for $i, j \in \{1, \cdots, n\}$
subject to these three relations:
(anti-)symmetry)
nilpotency
3-term relation
Hence:
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.
(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:
a graph changes sign when one of its edges is reversed (this Def.)
a graph with parallel edges is a vanishing graph (this Def.)
the graph coboundary of a single trivalent internal vertex (this Example).
Write again
for the configuration space of $n$ ordered points in Euclidean space.
The Whitehead product super Lie algebra of rationalized homotopy groups
is generated from elements
subject to the following relations:
$\omega^{i j} = (-1)^D \omega^{j i}$
$\big[ \omega^{i j}, \omega^{k l} \big]$ $\;\;\;$ if $i,j,k,l$ are pairwise distinct;
$\big[ \omega^{i j}, \omega^{j k} + \omega^{k i} \big] = 0$.
This is due to Kohno 02. See also Lambrechts-Tourtchine 09, Section 3.
weight systems are cohomology of loop space of configuration space:
(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
the integral weight systems
on horizontal chord diagrams of $n$ strands (elements of the set $\mathcal{D}^{pb}$)
the integral cohomology of the based loop space of the ordered configuration space of n points in 3d Euclidean space:
(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
(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:
This is stated as Kohno 02, Theorem 4.2
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 |
In the course of providing a geometric-proof of the spin-statistics theorem, Berry & Robbins 1997 asked, at each natural number $n \in \mathbb{N}$, for a continuous and $Sym(n)$-equivariant function
from the configuration space of $n$ points (ordered and unlabeled) in Euclidean space $\mathbb{R}^3$
to the coset space of the unitary group $\mathrm{U}(n)$ by its maximal torus, hence the complete flag manifold of flags in $\mathbb{C}^n$,
both equipped with the evident group action by the symmetric group $Sym(n)$.
For the first non-empty case $n = 2$ this readily reduces to asking for a continuous map of the form $\mathbb{R}^3 \setminus \{0\} \xrightarrow{\;\;} \mathbb{C}P^1 \simeq S^2$ which is equivariant with respect to passage to antipodal points. This is immediately seen to be given by the radial projection. But this special case turns out not to be representative of the general case, as this simple construction idea does not generalize to $n \gt 2$.
That a continuous and $Sym(n)$-equivariant Berry-Robbins map (18) indeed exists for all $n$ was proven in Atiyah 2000.
In this article, Atiyah turned attention to the stronger question asking for a smooth and $Sym(n) \times$$SO(3)$-equivariant function (18) and provided an elegant proof strategy for this stronger statement, which however hinges on some conjectural positivity properties of a certain determinant (discussed in more detail and with first numerical evidence in Atiyah 2001), interpreted as the electrostatic energy of $n$-particles in $\mathbb{R}^3$.
Extensive numerical checks of this stronger but conjectural construction was recorded, up to $n \lt 30$ , in Atiyah & Sutcliffe 2002, together with a refined formulation of the conjecture, whence it came to be known as the Atiyah-Sutcliffe conjecture.
The Atiyah-Sutcliffe conjecture has been proven for $n = 3$ in Atiyah 2000/01 and for $n = 4$ by Eastwood & Norbury 01. A general proof is claimed in Atiyah & Malkoun 18.
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.
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.
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.
The braid group regarded as the fundamental group of a configuration space of points is considered (neither of them under these names, though) already in:
there regarded as acting on Riemann surfaces forming branched covers, by movement of the branch points.
The concept of configuration spaces is then re-discovered/re-vived by:
Edward Fadell, Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111-118 $[$doi:10.7146/math.scand.a-10517, MR141126$]$
Ralph H. Fox, Lee Neuwirth, The braid groups, Math. Scand. 10 (1962) 119-126
$[$doi:10.7146/math.scand.a-10518, pdf, MR150755$]$
(focus on the resulting braid groups)
See also early occurences in physics, listed below.
General accounts:
Edward Fadell, Sufian Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001) [doi:10.1007/978-3-642-56446-8, MR2002k:55038] xvi+313
Frederick R. Cohen, Introduction to configuration spaces and their applications, in Braids Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore 19 (2009) 183-261 [doi:10.1142/9789814291415_0003, doi:10.1142/7550, pdf]
Craig Westerland, Configuration spaces in geometry and topology, 2011 (pdf)
Ben Knudsen, Configuration spaces in algebraic topology (arXiv:1803.11165)
(in algebraic topology)
Lucas Williams, Configuration Spaces for the Working Undergraduate, Rose-Hulman Undergraduate Mathematics Journal, 21 1 (2020) Article 8. (arXiv:1911.11186, rhumj:vol21/iss1/8)
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:
The Cohomotopy charge map (“electric field map”, “scanning map”) and hence the relation of configuration spaces to Cohomotopy goes back to
Peter May, The geometry of iterated loop spaces, Springer 1972 (pdf)
Graeme Segal, Configuration-spaces and iterated loop-spaces, Invent. Math. 21 (1973), 213–221. (doi:10.1007/BF01390197, pdf, MR 0331377)
Generalization of these constructions and results is due to
Dusa McDuff, Configuration spaces of positive and negative particles, Topology Volume 14, Issue 1, March 1975, Pages 91-107 (doi:10.1016/0040-9383(75)90038-5)
Carl-Friedrich Bödigheimer, Stable splittings of mapping spaces, Algebraic topology. Springer 1987. 174-187 (pdf, pdf)
Richard Manthorpe, Ulrike Tillmann, Tubular configurations: equivariant scanning and splitting, Journal of the London Mathematical Society, Volume 90, Issue 3 (arxiv:1307.5669, doi:10.1112/jlms/jdu050)
Generalization to equivariant homotopy theory:
and strengthening in the special case $G = \mathbb{Z}/2$:
The relevant construction for the group completion of the configuration space
Paolo Salvatore, Configuration spaces with summable labels, In: Aguadé J., Broto C., Carles Casacuberta (eds.) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel, 2001 (arXiv:math/9907073)
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 $\mathbb{R}^2$):
See also
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:
An analogous statement for homotopy of rational maps related to Yang-Mills monopoles:
In the context of speculations regarding Galois theory over the sphere spectrum:
The appearance of configuration spaces as summands in stable splittings of mapping spaces is originally due to
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
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)
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.
General discussion of ordinary homology/ordinary cohomology of configuration spaces of points:
Vladimir Arnold, The cohomology ring of the colored braid group, Mat. Zametki, 1969, Volume 5, Issue 2, Pages 227–231 (mathnet:mz6827)
Fred Cohen, Cohomology of braid spaces, Bull. Amer. Math. Soc. Volume 79, Number 4 (1973), 763-766 (euclid:1183534761)
Fred Cohen, The homology of $C_{n+1}$-Spaces, $n \geq 0$, In: The Homology of Iterated Loop Spaces, Lecture Notes in Mathematics, vol 533. Springer 1976 (doi:10.1007/BFb0080467)
Carl-Friedrich Bödigheimer, Fred Cohen, L. Taylor, On the homology of configuration spaces, Topology Vol. 28 No. 1, p. 111-123 1989 (pdf)
E. Ossa, On the cohomology of configuration spaces, In: Broto C., Carles Casacuberta, Mislin G. (eds.), Algebraic Topology: New Trends in Localization and Periodicity, Progress in Mathematics, vol 136. Birkhäuser Basel (1996) (doi:10.1007/978-3-0348-9018-2_26)
Yves Félix, Jean-Claude Thomas, Rational Betti numbers of configuration spaces, Topology and its Applications, Volume 102, Issue 2, 8 April 2000, Pages 139-149 (doi:10.1016/S0166-8641(98)00148-5)
Oscar Randal-Williams, Homological stability for unordered configuration spaces, The Quarterly Journal of Mathematics, Volume 64, Issue 1, March 2013, Pages 303–326 (arXiv:1105.5257)
Yves Félix, Daniel Tanré, The cohomology algebra of unordered configuration spaces, Journal of the LMS, Vol 72, Issue 2 (arxiv:math/0311323, doi:10.1112/S0024610705006794)
Martin Palmer, Configuration spaces and homological stability, Oxford University Research Archive (2012) $[$pdf, web$]$
Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Algebr. Geom. Topol. 17 (2017) 3137-3187 (arXiv:1405.6696)
Thomas Church, Homological stability for configuration spaces of manifolds (arxiv:1602.04748)
Christoph Schiessl, Betti numbers of unordered configuration spaces of the torus (arxiv:1602.04748)
Christoph Schiessl, Integral cohomology of configuration spaces of the sphere (arxiv:1801.04273)
Weiyan Chen, Twisted cohomology of configuration spaces and spaces of maximal tori via point-counting (arXiv:1603.03931)
Dan Petersen, Cohomology of generalized configuration spaces (arXiv:1807.07293)
Victor Vassiliev, Twisted homology of configuration spaces, homology of spaces of equivariant maps, and stable homology of spaces of non-resultant systems of real homogeneous polynomials (arXiv:1809.05632)
Victor Vassiliev, Cohomology of spaces of Hopf equivariant maps of spheres (arXiv:2102.07157)
Roberto Pagaria, The cohomology rings of the unordered configuration spaces of the torus, Algebr. Geom. Topol. 20 (2020) 2995–3012 (doi:10.2140/agt.2020.20.2995)
Christian Blanchet, Martin Palmer, Awais Shaukat, Heisenberg homology on surface configurations $[$arXiv:2109.00515$]$
Review:
Expressing the rational cohomology of ordered configuration spaces of points via factorization homology and Ran spaces:
Discussion of the Morava E-theory of configuration spaces of points:
Discussion of homotopy groups of configuration spaces:
Pascal Lambrechts, Victor Tourtchine, Homotopy graph-complex for configuration and knot spaces, Transactions of the AMS, Volume 361, Number 1, January 2009, Pages 207–222 (arxiv:math/0611766)
Sadok Kallel, Ines Saihi, Homotopy Groups of Diagonal Complements, Algebr. Geom. Topol. 16 (2016) 2949-2980 (arXiv:1306.6272)
Discussion of the rational homotopy type:
Second Series, Vol. 139, No. 2 (Mar., 1994), pp. 227-237 (jstor:2946581)
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
Najib Idrissi, The Lambrechts-Stanley Model of Configuration Spaces, Invent. Math, 2018 (arXiv:1608.08054, doi:10.1007/s00222-018-0842-9)
Ricardo Campos, Thomas Willwacher, A model for configuration spaces of points (arXiv:1604.02043)
Ricardo Campos, Najib Idrissi, Pascal Lambrechts, Thomas Willwacher, Configuration Spaces of Manifolds with Boundary (arXiv:1802.00716)
Ricardo Campos, Julien Ducoulombier, Najib Idrissi, Thomas Willwacher, A model for framed configuration spaces of points (arXiv:1807.08319)
On 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:
Toshitake Kohno, Vassiliev invariants and de Rham complex on the space of knots,
In: Yoshiaki Maeda, Hideki Omori, Alan Weinstein (eds.), Symplectic Geometry and Quantization, Contemporary Mathematics 179 (1994): 123-123 (doi:10.1090/conm/179)
Fred Cohen, Samuel Gitler, 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)
Toshitake Kohno, Loop spaces of configuration spaces and finite type invariants, Geom. Topol. Monogr. 4 (2002) 143-160 (arXiv:math/0211056)
Fred Cohen, Samuel Gitler, On loop spaces of configuration spaces, Trans. Amer. Math. Soc. 354 (2002), no. 5, 1705–1748, (jstor:2693715, MR2002m:55020)
In physics (solid state physics/particle physics) the configuration space of points appears early on in the discussion of anyon statistics, originally in:
Michael G. G. Laidlaw, Cécile Morette DeWitt, Feynman Functional Integrals for Systems of Indistinguishable Particles, Phys. Rev. D 3 (1971) 1375 $[$doi:10.1103/PhysRevD.3.1375$]$
Jon Magne Leinaas, Jan Myrheim, pp. 22 of: On the theory of identical particles, К теории тождествениых частиц, Nuovo Cim B 37, 1–23 (1977) (doi:10.1007/BF02727953)
Frank Wilczek, p. 959 of: Quantum Mechanics of Fractional-Spin Particles, Phys. Rev. Lett. 49 (1982) 957 (reprinted in Wilczek 1990, p. 166-168) $[$doi:10.1103/PhysRevLett.49.957$]$
Concretely, anyon-wavefunctions are identified with multi-valued functions on a configuration space of points, see there:
Tom Imbo, Chandni Shah Imbo, E. C. G. Sudarshan, Identical particles, exotic statistics and braid groups, Physics Letters B 234 1–2, (1990) 103-107 $[$doi:10.1016/0370-2693(90)92010-G, pdf$]$
Jürg Fröhlich, Fabrizio Gabbiani, Pieralberto Marchetti, around (1.1) in: Braid statistics in three-dimensional local quantum field theory, in: H.C. Lee (ed.) Physics, Geometry and Topology NATO ASI Series, 238 Springer (1990) $[$doi:10.1007/978-1-4615-3802-8_2, pdf$]$
J. Mund, Robert Schrader, Hilbert Spaces for Nonrelativistic and Relativistic “Free” Plektons (Particles with Braid Group Statistics), in Advances in dynamical systems and quantum physics (Capri, 1993), World Sci. (1995) 235–259 $[$arXiv:hep-th/9310054v2$]$
Garth A. Baker, Geoff S. Canright, Shashikant B. Mulay, Carl Sundberg, On the spectral problem for anyons, Communications in Mathematical Physics 153 (1993) 277–295 $[$doi:10.1007/BF02096644$]$
Gianfausto Dell’Antonio, Rodolfo Figari & Alessandro Teta, Statistics in Space Dimension Two, Letters in Mathematical Physics 40 (1997) 235–256 $[$doi:10.1023/A:1007361832622$]$
M.V.N. Murthy, Ramamurti Shankar, p. 7, 50Exclusion Statistics: From Pauli to Haldane (1999, 2009) $[$dspace:123456789/334, pdf, pdf$]$
Moreover, in quantum field theory one may formalize correlators as differential forms on configuration spaces of points. This perspective was originally considered specifically for Chern-Simons theory in:
which was re-amplified in:
Raoul Bott, Alberto Cattaneo, Remark 3.6 in Integral invariants of 3-manifolds, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001)
Alberto Cattaneo, Pavel Mnev, Remark 11 in Remarks on Chern-Simons invariants, Commun.Math.Phys.293:803-836,2010 (arXiv:0811.2045)
Alberto Cattaneo, Pavel Mnev, Nicolai Reshetikhin, appendix B of Perturbative quantum gauge theories on manifolds with boundary, Communications in Mathematical Physics, January 2018, Volume 357, Issue 2, pp 631–730 (arXiv:1507.01221, doi:10.1007/s00220-017-3031-6)
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
see also
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
Christoph Bergbauer, Romeo Brunetti, Dirk Kreimer, Renormalization and resolution of singularities, ESI preprint 2010 (arXiv:0908.0633, ESI:2244)
Christoph Bergbauer, Renormalization and resolution of singularities, talks as IHES and Boston, 2009 (pdf)
Marko Berghoff, Wonderful renormalization, 2014 (pdf, doi:10.18452/17160)
Marko Berghoff, Wonderful compactifications in quantum field theory, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (arXiv:1411.5583)
Analogous discussion for Hilbert schemes of points:
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:
The “hypergeometric integral” construction of conformal blocks for affine Lie algebra/WZW model-2d CFTs and of more general solutions to the Knizhnik-Zamolodchikov equation, via twisted de Rham cohomology of configuration spaces of points, originates with:
Vadim Schechtman, Alexander Varchenko, Integral representations of N-point conformal correlators in the WZW model, Max-Planck-Institut für Mathematik, (1989) Preprint MPI/89- $[$cds:1044951$]$
Etsuro Date, Michio Jimbo, Atsushi Matsuo, Tetsuji Miwa, Hypergeometric-type integrals and the $\mathfrak{sl}(2,\mathbb{C})$-Knizhnik-Zamolodchikov equation, International Journal of Modern Physics B 04 05 (1990) 1049-1057 $[$doi:10.1142/S0217979290000528$]$
Atsushi Matsuo, An application of Aomoto-Gelfand hypergeometric functions to the $SU(n)$ Knizhnik-Zamolodchikov equation, Communications in Mathematical Physics 134 (1990) 65–77 $[$doi:10.1007/BF02102089$]$
Vadim Schechtman, Alexander Varchenko, Hypergeometric solutions of Knizhnik-Zamolodchikov equations, Lett. Math. Phys. 20 (1990) 279–283 $[$doi:10.1007/BF00626523$]$
Vadim Schechtman, Alexander Varchenko, Arrangements of hyperplanes and Lie algebra homology, Inventiones mathematicae 106 1 (1991) 139-194 $[$dml:143938, pdf$]$
following precursor observations due to:
Vladimir S. Dotsenko, Vladimir A. Fateev, Conformal algebra and multipoint correlation functions in 2D statistical models, Nuclear Physics B 240 3 (1984) 312-348 $[$doi:10.1016/0550-3213(84)90269-4$]$
Philippe Christe, Rainald Flume, The four-point correlations of all primary operators of the $d = 2$ conformally invariant $SU(2)$ $\sigma$-model with Wess-Zumino term, Nuclear Physics B
282 (1987) 466-494 $[$doi:10.1016/0550-3213(87)90693-6$]$
The proof that for rational levels this construction indeed yields conformal blocks is due to:
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by correlators in Wess-Zumino-Witten models, Lett Math Phys 20 (1990) 291–297 $[$doi:10.1007/BF00626525$]$
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. I, Commun. Math. Phys. 163 (1994) 173–184 $[$doi:10.1007/BF02101739$]$
Boris Feigin, Vadim Schechtman, Alexander Varchenko, On algebraic equations satisfied by hypergeometric correlators in WZW models. II, Comm. Math. Phys. 170 1 (1995) 219-247 $[$euclid:cmp/1104272957$]$
Review:
Alexander Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, Advanced Series in Mathematical Physics 21, World Scientific 1995 (doi:10.1142/2467)
Ivan Cherednik, Section 8.2 of: Lectures on Knizhnik-Zamolodchikov equations and Hecke algebras, Mathematical Society of Japan Memoirs 1998 (1998) 1-96 $[$doi:10.2969/msjmemoirs/00101C010$]$
Pavel Etingof, Igor Frenkel, Alexander Kirillov, Lecture 7 in: Lectures on Representation Theory and Knizhnik-Zamolodchikov Equations, Mathematical surveys and monographs 58, American Mathematical Society (1998) $[$ISBN:978-1-4704-1285-2, review pdf$]$
Toshitake Kohno, Local Systems on Configuration Spaces, KZ Connections and Conformal Blocks, Acta Math Vietnam 39 (2014) 575–598 $[$doi:10.1007%2Fs40306-014-0088-6, pdf$]$
Toshitake Kohno, Homological representations of braid groups and KZ connections, Journal of Singularities 5 (2012) 94-108 $[$doi:10.5427/jsing.2012.5g, pdf$]$
See also:
Alexander Varchenko, Asymptotic solutions to the Knizhnik-Zamolodchikov equation and crystal base, Comm. Math. Phys. 171 1 (1995) 99-137 $[$arXiv:hep-th/9403102, doi:10.1007/BF02103772$]$
Edward Frenkel, David Ben-Zvi, Section 14.3 in: Vertex Algebras and Algebraic Curves, Mathematical Surveys and Monographs 88, AMS 2004 $[$ISBN:978-1-4704-1315-6, web$]$
This “hypergeometric” construction uses results on the twisted de Rham cohomology of configuration spaces of points due to:
Peter Orlik, Louis Solomon, Combinatorics and topology of complements of hyperplanes, Invent Math 56 (1980) 167–189 $[$doi:10.1007/BF01392549$]$
Kazuhiko Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan 39 2 (1987) 191-208 $[$doi:10.2969/jmsj/03920191$]$
Hélène Esnault, Vadim Schechtman, Eckart Viehweg, Cohomology of local systems on the complement of hyperplanes, Inventiones mathematicae 109.1 (1992) 557-561 $[$pdf$]$
Vadim Schechtman, H. Terao, Alexander Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors, Journal of Pure and Applied Algebra 100 1–3 (1995) 93-102 $[$arXiv:hep-th/9411083, doi:10.1016/0022-4049(95)00014-N$]$
also:
reviewed in:
Discussion for the special case of level$=0$ (cf. at logarithmic CFT – Examples):
Fedor A. Smirnov, Remarks on deformed and undeformed Knizhnik-Zamolodchikov equations, $[$arXiv:hep-th/9210051$]$
Fedor A. Smirnov, Form factors, deformed Knizhnik-Zamolodchikov equations and finite-gap integration, Communications in Mathematical Physics 155 (1993) 459–487 $[$doi:10.1007/BF02096723, arXiv:hep-th/9210052$]$
S. Pakuliak, A. Perelomov, Relation Between Hyperelliptic Integrals, Mod. Phys. Lett. 9 19 (1994) 1791-1798 $[$doi:10.1142/S0217732394001647$]$
Interpretation of the hypergeometric construction as happening in twisted equivariant differential K-theory, showing that the K-theory classification of D-brane charge and the K-theory classification of topological phases of matter both reflect braid group representations as expected for defect branes and for anyons/topological order, respectively:
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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