# 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.

101
questions

1
vote

0
answers

75
views

### What is the meaning of universal family of Fulton Macpherson configuration space?

Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces"
In this paper, the process ...

3
votes

0
answers

107
views

### Universal cover of the configuration space of points on surface

Let $S$ be a closed oriented surface and $C(S, n)$ be the configuration space of $n$ points on $S$, i.e., the space of $n$-tuples of distinct points of $S$ with the topology induced from $S^n$. Let $V ...

1
vote

1
answer

143
views

### Compact locus in (ordered) configuration spaces

Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with ...

3
votes

1
answer

292
views

### How many configurations of tubes are there?

Can $n$ disjoint lines in $\boldsymbol R^3$ be knotted? No... Let $X_n$ be the configuration space of $n$ disjoint lines in $\boldsymbol R^3$. It is not hard to see that $X_n$ is path connected: Let $...

2
votes

1
answer

79
views

### Is there a way to parametrize the configuration space of all convex polyhedra of a given combinatorial type as a convex set?

I'm sure this is easy/known, but I'm not hitting an appropriate search term for finding the answer and the coffee hasn't kicked in enough to come up with it myself:
Let $T$ be a simplicial 2-complex ...

3
votes

0
answers

49
views

### Are there invariants of configurations of points in space obtainable via the moduli space of solutions of the Berry-Robbins problem?

Let $C_n(\mathbb{R}^3)$ denote the configuration space of $n$ distinct points in Euclidean $3$-space and let $U(n)/T^n$ denote the flag manifold associated to the unitary group $U(n)$, i.e. the ...

4
votes

0
answers

96
views

### "Standard computations" with stable Hopf invariants

I am struggling in understanding the proof of Lemma 10.6 of the paper "Mapping class groups and function spaces" by Bodigheimer, Cohen and Peim http://www.math.uni-bonn.de/people/cfb/...

1
vote

0
answers

30
views

### How does configuration or phase space change in pseudo-Hermitian (or just non-Hermtiian) QM vs Hermitian QM?

I was wondering if there is some relaxation of the configuration (or phase) space when considering pseudo-Hermitian physical situations vs Hermitian? For instance in "$C^*$-Algebras of Energy ...

2
votes

1
answer

127
views

### On the zero-dimensional strata of the Fulton-MacPherson conpactification

Let $\operatorname{Conf}_n(\mathbb{R})$ be the configuration space of $n$ marked points on the real line. What is the difference between $\operatorname{Conf}_n(\mathbb{R})$ and the locus of zero-...

6
votes

0
answers

170
views

### Does combinatorial deleted product become equivalent to the topological deleted product after enough subdivision?

Suppose $X$ is a topological space. Define the (topological) $n$-fold deleted product of $X$ to be the space or ordered $n$-tuples of pairwise distinct points in $X$.
$$F(X, n):= \{(x_1, \ldots, x_n)\...

4
votes

0
answers

113
views

### 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\...

0
votes

0
answers

264
views

### 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{...

0
votes

0
answers

66
views

### 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},...

0
votes

1
answer

137
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 ...

5
votes

2
answers

305
views

### 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 $8$ ...

2
votes

0
answers

40
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 $C_n(\mathbb{R}^d)$ ...

0
votes

1
answer

102
views

### 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) ...

6
votes

1
answer

655
views

### 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 ...

6
votes

1
answer

126
views

### 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. (...

5
votes

1
answer

246
views

### 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 ...

10
votes

1
answer

350
views

### 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. ...

3
votes

1
answer

154
views

### 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
$$
p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z})
$$
...

8
votes

2
answers

380
views

### 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 ...

1
vote

1
answer

216
views

### 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 ...

2
votes

0
answers

58
views

### 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?

2
votes

1
answer

304
views

### 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 $\...

6
votes

1
answer

497
views

### 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 ...

5
votes

0
answers

154
views

### 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,.....

5
votes

1
answer

254
views

### 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 ...

6
votes

1
answer

153
views

### 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 ...

7
votes

0
answers

216
views

### 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

1
answer

420
views

### 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 ...

8
votes

2
answers

398
views

### 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 \}...

8
votes

0
answers

293
views

### 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

151
views

### 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 ...

12
votes

1
answer

519
views

### 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 $\{...

3
votes

2
answers

238
views

### 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 ...

6
votes

0
answers

375
views

### 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 ...

10
votes

2
answers

409
views

### 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 \...

4
votes

1
answer

299
views

### 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 $r$ ...

1
vote

0
answers

195
views

### (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 $...

4
votes

2
answers

366
views

### 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 ...

3
votes

0
answers

123
views

### 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^...

7
votes

0
answers

189
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 ...

5
votes

0
answers

213
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)=\{(...

1
vote

0
answers

101
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 ...

6
votes

1
answer

253
views

### 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,\...

4
votes

2
answers

360
views

### 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
vote

1
answer

288
views

### 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 ...

3
votes

2
answers

182
views

### 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 ...