topology (point-set topology, point-free topology)
see also differential topology, algebraic topology, functional analysis and topological homotopy theory
Basic concepts
fiber space, space attachment
Extra stuff, structure, properties
Kolmogorov space, Hausdorff space, regular space, normal space
sequentially compact, countably compact, locally compact, sigma-compact, paracompact, countably paracompact, strongly compact
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 $X$.
(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.
On Euclidean spaces (and maybe more generally onframed manifolds) any configuration space of points gets the mathematical 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).
Write
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.
(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 electric field map/scanning 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 electric field map/scanning 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 spheres 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 scanning 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 scanning 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 $S^k$.
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
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.
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.
General accounts:
Edward Fadell, Lee Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111-118, MR141126, pdf
Edward Fadell, Sufian Husseini, Geometry and topology of configuration spaces, Springer Monographs in Mathematics (2001), MR2002k:55038, xvi+313
Craig Westerland, Configuration spaces in geometry and topology, 2011 (pdf)
Ben Knudsen, Configuration spaces in algebraic topology (arXiv:1803.11165)
(in algebraic topology)
The 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. MR 0331377 (pdf)
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)
and generalization to equivariant homotopy theory is discussed in
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)
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, Particle Spaces on Manifolds and Generalized Poincaré Dualities (arXiv:math/9810067)
Shingo Okuyama, Kazuhisa Shimakawa, Interactions of strings and equivariant homology theories, (arXiv:0903.4667)
For relation to instantons via topological Yang-Mills theory:
In speculation regarding Galois theory over the sphere spectrum:
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
…
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)
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)
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)
Thomas Church, Homological stability for configuration spaces of manifolds (arxiv:1602.04748)
Ben Knudsen, Betti numbers and stability for configuration spaces via factorization homology, Algebr. Geom. Topol. 17 (2017) 3137-3187 (arXiv:1405.6696)
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)
Roberto Pagaria, The cohomology rings of the unordered configuration spaces of the torus (arxiv:1901.01171)
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)
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)
On loop spaces of configuration spaces of points:
Edward Fadell, Sufian Husseini, The space of loops on configuration spaces and the Majer-Terracini index, Topol. Methods Nonlinear Anal. Volume 11, Number 2 (1998), 249-271 (euclid:tmna/1476842829)
Fred Cohen, Samuel Gitler, Loop spaces of configuration spaces, braid-like groups, and knots, In: Aguadé J., Broto C., Carles Casacuberta (eds.) Cohomological Methods in Homotopy Theory. Progress in Mathematics, vol 196. Birkhäuser, Basel (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)
(on ordinary homology of loop spaces of configuration spaces)
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
Last revised on October 16, 2019 at 09:29:36. See the history of this page for a list of all contributions to it.