# 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.

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### 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 ...
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### "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
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### 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 ...
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### 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-...
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### 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 ...
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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 $8$ ...
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### 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 $C_n(\mathbb{R}^d)$ ...
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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) ...
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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 ...
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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. (...
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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 ...
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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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### 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 $$p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z})$$ ...
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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 ...
1 vote
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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 ...
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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?
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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 ...
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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 ...
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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 $F(\mathbb{R}P^... 7 votes 0 answers 179 views ### mod$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$be ... 5 votes 0 answers 204 views ### cohomology ring of configuration spaces on$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, $$B(M,n)=\{(... 1 vote 0 answers 100 views ### coefficient of homology of configuration spaces over real projective spaces In the slides Characteristic Classes of Surface Bundles and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology? Could the coefficient be an ... 6 votes 1 answer 243 views ### factorization of the cohomology of configuration space This question is a follow-up to my previous question factorization of the regular representation of the symmetric group, which was answered in a very satisfactory way. Let \operatorname{Conf}(n,\... 4 votes 2 answers 344 views ### homotopy equivalence between configuration spaces Let M be a m-dimensional compact manifold without boundary and W(M) the non-compact CW-complex obtained by glueing [0,1)\times (0,1)^{m-1} to M, identifying the boundary \{0\}\times(0,1)^{... 1 vote 1 answer 274 views ### torsion part of the cohomology module of configuration spaces of manifolds Let M be a manifold and C_n(M) the n-th unordered configuration space consisting of unordered n-tuples of distinct points in M. The mod p homology module, p prime, and the rational ... 3 votes 2 answers 176 views ### equivariant embeddings from the k-th configuration space to the k+1-th configuration space Let S be a closed, orientable surface in \mathbb{R}^3 and S' be the manifold S\setminus\text{one point}. Let F(S',k) be the k-th (ordered) configuration space on S'. It is claimed in ... 5 votes 0 answers 140 views ### configuration space of Riemannian manifolds with a parameter on the distance of distinct points Let M be a Riemannian manifold. For any \epsilon\geq 0, we define the k-th (k=1,2,\cdots) "\epsilon-configuration space" as$$ F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\... 5 votes 0 answers 140 views ### 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 ... 2 votes 0 answers 79 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,\...
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 ...
Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be $$B(X,k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j\}$$...