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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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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Change of coordinates and effects on the tangent bundle of a Manifold

I am doing ex. 2.3(2) from Frankel's book "The geometry of Physics". He says to consider the tangent bundle to a manifold M and show: i) that under a change of coordinated in M, $\partial / \partial ...
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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,...