Questions tagged [configuration-space]

Configuration spaces refer to topological spaces that consist of ordered or unordered subsets of a topological space, of a given (finite) cardinality.

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The configuration space of a rolling ball.

This Wikipedia article mentions that the configuration space of a rolling ball is $\Bbb{C}^5$. I don't understand why that is. The position of the center of mass, that's a point in $\Bbb{R}^3$. The ...
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Configuration space of a circle

I was unable to find a direct description of $\operatorname{Conf}^{\,n}(S^1)$ (configuration space of $n$ distinct points on a circle). It is pretty clear that $\operatorname{Conf}^{\,2}(S^1)\simeq S^...
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Isomorphism between braid groups

Let M be a connected topological manifold of dimension $\geq2$ and let $M^n=M\times\dots\times M$ be the product on $n\geq1$ copies of M with the product topology. Set $\mathcal{F}_n(M)=\{(u_1,u_2,\...
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Is the set of collections of directions satisfying the following convexity conditions “geometric”?

Let $X_n$ denote the set of all collections $(v_{ij})$ of points on the sphere $S^2$, for $1 \leq i,j \leq n$, $i \neq j$, such that: $v_{ji} = -v_{ij}$, the origin $O$ in $\mathbb{R}^3$ is in the ...
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Fundamental groups of the configuration spaces of all triangles and right triangles

This is a question from a past comprehensive exam: Consider triangles in the plane, with vertices given by non-colinear points as usual. The space $T$ of all plane triangles can be given a natural ...
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Generalising dual triangulation of manifolds

We know the following “geometric” version of Poincaré duality: Let $M$ be a closed $m$-dimensional manifold and let $\mathfrak{X}_*$ be a finite simplicial complex with $|\mathfrak{X}_*|=M$. We can ...
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Path-connected components of the configuration space $F_n(R^d)$ [duplicate]

Let $X$ be a topological space, for $n \geq 1$ we define $F_n(X) = \lbrace{ (x_1,...,x_n) \in X^n | x_i \neq x_j for i \neq j \rbrace} $ the configuration space of n points of X. My question is ...
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Configurations whose convex hull contains the origin

Let $x_1,\ldots,x_n$ be $n$ points in $\mathbb{R}^3$. Are there known necessary and sufficient conditions on the $x_i$'s so that the origin belongs to the convex hull of the $x_i$'s? I did a (not so ...
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simple double pendulum and four equilibrium configurations

I have difficulty in understanding the notion of "configuration space" and "toroidal spaces" in the following explanation: The configuration space of any double pendulum can be represented as the ...
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Given three distinct points on a sphere, find the unique round circle they live in

Say you have three (distinct) points on the unit sphere in Euclidean space $$p_1, p_2, p_3 \in S^n = \{ x \in \mathbb R^{n+1} : |x| = 1 \}$$ I'd like to find, as efficiently and robustly as possible,...
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Cohomology ring of a configuration space

Consider the following configuration space of triples of points. $$\begin{align}C &= \left\lbrace (z_1,z_2,z_3) \in (\mathbb C^*)^3, z_1 \ne z_2, z_1 \ne z_3, z_2 \ne z_3\right\rbrace \\&\...