Questions tagged [configuration-spaces]
for questions on configuration spaces, both in the sense of spaces that parameterizes collections of points in a manifold, and in the sense of the space of possible states of a classical mechanical physical system.
91
questions
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Going between the abstract and the concrete notions of chiral homology
Let $X$ be a smooth algebraic curve over $\mathbf{C}$, and let $\mathcal{V}$ be a factorisation algebra over $X$, whose fibre above $x\in X$ is the vertex algebra $V$.
Note that $\mathcal{V}\in\...
- 4,004
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0
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218
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Finite morphisms between two varieties
Let $X$ and $Y$ be varieties in some projective space. Furthermore let's assume these two varieties are intersecting at the subvariety $Z$. For this problem we are assuming that $\text{dim}(X)\ll\text{...
- 4,926
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42
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packing numbers of the unit balls in Euclidean spaces and the dimensions
Let $k$, $m$ and $n$ be positive integers. Let $r$ be a positive real number.
The $n$-th ordered $r$-disk configuration space on the Euclidean space $\mathbb{R}^{mk}$ is
$$
F_r(\mathbb{R}^{mk},...
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0
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81
views
packing numbers and configuration spaces of the torus
Let $S^1$ be the unit circle of radius $1$.
For any $k\geq 1$, let the $k$-dimensional torus $T^k= \underbrace{S^1\times S^1\times\cdots\times S^1}_k$ be the $k$-fold self-Cartesian ...
- 1,940
5
votes
2
answers
277
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Dimension of configuration space of triangulated convex polyhedron
The configuration space of all tetrahedra is $5$-dimensional, perhaps a non-obvious fact.
There are $12$ face angles, but the sum of each of the four faces angles is $\pi$,
reducing $12$ to $8$ ...
- 144k
1
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0
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26
views
On the higher-dimensional Berry-Robbins problem
Let $C_n(\mathbb{R}^d)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^d$, say $\mathbf{x}_1, \ldots, \mathbf{x}_n$. The symmetric group $\Sigma_n$ acts on $C_n(\mathbb{R}^d)$ ...
- 3,594
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1
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78
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Fundamental group to groupoid : bijection between homotopy classes?
I'm looking at the fundamental group $\pi_{1}(M)$ of the $n^{th}$ unordered configuration space $M$ of $\mathbb{R}^{d}$. In particular, it's well-known that $\pi_{1}(M)\cong S_{n}$ (symmetric group) ...
- 205
6
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1
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457
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The symmetric square of a sphere
$\DeclareMathOperator\Sym{Sym}$Recently, I have been thinking about the space $\Sym^2(S^n)$ of pairs of points in the sphere (including the repeated pairs $(p,p)$ for each $p\in S^n$) and possible ...
- 945
6
votes
1
answer
120
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Is there a filtered splitting of product labelling spaces?
For a well-based space $X$ denote by $C(\mathbb{R};X)$ the unordered configuration space of points on the real line with labels in $X$, and a point can vanish if its label reaches the basepoint. (...
- 1,563
5
votes
1
answer
212
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Integral homology of braid groups as a ring
Let $Br_k$ denote the braid group on $k$ strands. In Corollary A.4 of "Homology of Iterated Loop Spaces" (Page 348), Cohen-Lada-May compute $H_i(Br_k;\mathbb Z)$ as an abelian group for each ...
- 833
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1
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289
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A piecewise-linear or topological Fulton-MacPherson compactification
The Fulton-MacPherson compactifications of configuration spaces are smooth manifolds with corners which have the ordered configuration spaces of distinct points in a smooth manifold as their interior. ...
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3
votes
1
answer
119
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Cohomology of the moduli space of rational curves with $n$ marked points with spin structure
Consider $\mathcal{M}_{0,n}$- the moduli space of rational curves with $n$ marked points. A map
$$
p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z})
$$
...
- 4,044
8
votes
2
answers
299
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Is there a Riemannian metric on the configuration space of $n$ distinct points with "nice" geodesics?
Let $C_n = C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in $\mathbb{R}^3$. Is there a Riemannian metric $g$ on $C_n$ such that given any two configurations in $C_n$, there ...
- 3,594
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1
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132
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Visualising the moduli space of stable disks with interior marked point and 4 marked point on the boundary
Is there any nice description/picture of the moduli space of stable disks with 1 interior marked point and 4 marked points on the boundary?
I'm expecting it to be a 3-dimensional polytope, because ...
- 1,838
2
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0
answers
52
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Homology of configuration space of punctured projective spaces?
Let $M=\mathbb{C}P^n$ or $\mathbb{R}P^n$ with $m$ punctures, is it known what the homology of the configuration space, $H_*(C_k(M))$ is? How are cases $\mathbb{C}P^n$ and $\mathbb{R}P^n$ different?
- 1,365
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1
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Is it true, the space of embeddings segments is homotopy equivalent to the subspace of all line segments?
Consider the space of smooth embeddings of the segment in the plane with the compact-open topology. Denote by X the quotient space obtained from the equivalence relation $a \sim b$ if and only if $\...
- 2,279
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votes
1
answer
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Zero differential in Serre spectral sequence for configuration spaces
I moved this question from Math StackExchange.
I am trying to compute homology of $Conf(n, \mathbb{R}^2)$ - ordered configurations of $n$ points on the plane - using Serre spectral sequence. I know ...
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Where have you encountered "arrangement spaces"?
I am compiling a paper in which I advertise (and use) the following notion of arrangement spaces (I made up the name, as I found no standard name in the literature).
Let $v_i\in\Bbb R^d,i\in N:=\{1,.....
- 9,471
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votes
1
answer
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Graded commutativity of the $n$th Browder bracket
Let $\mathcal{O}$ be a topological operad and $X$ an algebra over $\mathcal{O}$. Then $H_*(X)$ is an algebra (in the category of $\mathbb{Z}$-graded $R$-modules) over $H_*(\mathcal{O})$. Each $e\in ...
- 1,563
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votes
1
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134
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Configurations of $n$ points modulo isometries of the ambient space
Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate ...
- 63
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Cohomology of little disks and dg algebras over $\mathbb{F}_p$
This a alternative form of the question I posted some time ago.
We, the people who don't know topology, are told that in characteristic p the formalizm of DG algebras is not quite adequate for ...
9
votes
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Does the Serre spectral sequence of the Fadell-Neuwirth fibration collapse if there is a cross-section?
I had asked a vague question in MSE where a useful pointer to the Leray-Hirsch theorem was mentioned by Mike Miller in the comments, but received no answers. Here I will specialize to an interesting ...
- 1,266
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2
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Is there a proof of the formality of configuration spaces of Euclidean spaces that do not involve operads?
By "configuration spaces of $\mathbb{R}^n$" I mean ordered configuration spaces:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}...
- 4,167
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0
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271
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Genus=2 theta functions, Arnold's relation, and KZ connection
Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...
2
votes
1
answer
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Very symmetric quadrangle in $\Bbb CP^2$
Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$?
Clearly the analogous ...
- 2,233
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1
answer
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Configuration spaces, Ran spaces, free semilattices, Vietoris spaces and power objects
These are five important constructions and I would like to know how they are related.
The $n$th unordered configuration space of a space $X$ is
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
- 16.2k
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votes
2
answers
232
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Topological Complexity $TC$ of two robots moving on number $8$
I have been working on my research as a student at Wilbur Wright College on Topological Complexity. We solved the problem of two robots moving on a circle and letter $T$ using Farber's theorem but ...
- 33
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325
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What's the meaning of the Johnson filtration in terms of configuration spaces?
This question is inspired of course by the remarkable paper of Tetsuhiro Moriyama from 2008.
Let $\Sigma$ be a genus $g \geq 3$ closed surface.
Let $\phi : \Sigma \to \Sigma$ be an orientation ...
- 1,885
10
votes
2
answers
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Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?
If $X$ is a topological space, write $C_n(X)$ for the configuration space of distinct ordered tuples of points in $X$:
$$C_n(X) = \{(x_1, \ldots, x_n) \in X^n \mbox{ so that $i \neq j \implies x_i \...
- 5,642
4
votes
1
answer
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fundamental group of configuration spaces of ordered points on open Riemann surfaces
Let $\bar{X}$ be a compact Riemann surface of genus $g>0$. Let $X$ be $\bar{X}$ minus a finite set of points $\{a_1,\ldots,a_n\}$ ($n\geq 1$). Let $X^{(r)}$ be the configuration space of $r$ ...
- 299
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0
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(Ordered) Configuration space in algebraic geometry
Let $X$ be a topological space and denote by $F_n(X)$ the following subspace:
$$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$
Note that, we are not considering the quotient of $...
- 73
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votes
2
answers
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cohomology of configuration space of punctured variety
Given a smooth projective variety $X$ of dimension $l$, we denote with $F(X,n)$ the configuration space of points
$$
F(X,n):=\{(x_{1}, \dots, x_{n})\in X^{n}\: : \: x_{i}\neq x_{j}\text{ for each }i,j ...
- 1,394
3
votes
0
answers
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group actions of $S^3$ on the configuration space of projective plane
Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...
- 2,203
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votes
0
answers
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views
mod $p$ homology module of unordered configuration spaces of the projective plane
Let $M$ be a manifold and $k$ be a positive integer. Let $F(M,k)$ be the $k$-th ordered configuration space over $M$, consisting of all ordered $k$-tuples of distinct points in $M$. Let $\Sigma_k$ be ...
- 2,203
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votes
0
answers
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views
cohomology ring of configuration spaces on $S^2$ and the projective plane
For a manifold $M$ and a positive integer $n$, the unordered configuration space $B(M,n)$ is the space consisting of all unordered collections of $n$ distinct points on $M$. Precisely,
$$
B(M,n)=\{(...
- 519
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0
answers
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views
coefficient of homology of configuration spaces over real projective spaces
In the slides Characteristic Classes of Surface Bundles
and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology?
Could the coefficient be an ...
- 1,940
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votes
1
answer
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factorization of the cohomology of configuration space
This question is a follow-up to my previous question factorization of the regular representation of the symmetric group, which was answered in a very satisfactory way.
Let $\operatorname{Conf}(n,\...
- 2,288
4
votes
2
answers
333
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homotopy equivalence between configuration spaces
Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary $\{0\}\times(0,1)^{...
- 1,940
1
vote
1
answer
260
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torsion part of the cohomology module of configuration spaces of manifolds
Let $M$ be a manifold and $C_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational ...
- 1,940
3
votes
2
answers
172
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equivariant embeddings from the k-th configuration space to the k+1-th configuration space
Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
- 1,940
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votes
0
answers
134
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configuration space of Riemannian manifolds with a parameter on the distance of distinct points
Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\...
- 2,203
5
votes
0
answers
139
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cohomology ring of stable configuration spaces
Let $M$ be a compact Riemannian manifold without boundary. Distinct $k$-points $x_1,\cdots,x_k\in M$ are called stable if the potential energy given by coulomb forces among $k$ electrons reaches ...
- 543
2
votes
0
answers
76
views
symmetric points on symmetric spaces
Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\...
- 543
6
votes
1
answer
348
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electron configuration on manifolds
Let $M$ be a Riemannian manifold. For $k\geq 2$, suppose there are $k$ particles whose mass and volume can be regarded as zero and negatively charged with electricity equally. These $k$ particles move ...
- 543
2
votes
1
answer
87
views
distinct multiple points in a space with at least one point lying in a subspace
Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be
$$
B(X,k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j\}
$$...
- 2,203
15
votes
1
answer
542
views
Cohomology of configuration space as a representation of the symmetric group
Let $X_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S_n$. It is well known that the total cohomology of $X_n$ is ...
- 2,288
0
votes
1
answer
167
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cohomology ring of the fundamental group of unordered configuration space
From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR
APPLICATIONS, p. 18, I find:
Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...
- 2,203
1
vote
1
answer
138
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dimension of generators of cohomology ring of iterated loop-suspension
In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32
Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...
- 2,203
6
votes
1
answer
272
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Stiefel-Whitney class of unordered configuration space
Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-...
- 2,203
4
votes
1
answer
267
views
unordered configuration space over spheres and Euclidean spaces
For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then
$$
B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1},
$$
$$
B(S^n,2)\simeq \mathbb{R}P^n.
$$
Hence
$
(*)
$
$$
B(\...
- 2,203