Mathematics Stack Exchange
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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.

Configuration Spaces typically refer to $C_n X = X^n \setminus \Delta$ where $\Delta$ refers to the subset of $X^n$ where any two coordinates agree,

$$\Delta^n = \{ (x_1, \cdots, x_n) \in X^n : x_i = x_j \text{ for some } i \neq j\}$$

$X$ is a topological space.

There is a natural right action of the symmetric group $\Sigma_n$ on $C_n X$. For example, $\pi_1 C_n D^2$ is the pure braid group on $n$ strands, and $\pi_1 (C_n D^2 / \Sigma_n)$ is the braid group on $n$ strands.

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Various proofs: pairs of points in a circle = Möbius strip

It is well-known that the space of "pairs of points in a circle" (also called the symmetric square of $S^1$) can be identified with a Möbius strip. I don't know where the idea originates, but it is a neat elementary result in topology, which has a…
Nikhil Sahoo
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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 \\&\phantom{abcde}\setminus \left\lbrace (z_1,z_2,z_3) \in…
user79456
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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 convex hull of $v_{ab}$, $v_{bc}$ and $v_{ca}$…
Malkoun
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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 build a dual cell complex $\mathfrak{X}^*$: For…
FKranhold
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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 $\operatorname{Conf}^{\,2}(S^1)\simeq S^1,$ however I don't think that the same method…
Grisha Taroyan
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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, a description of the unique round circle in $S^n$…
Ryan Budney
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Similarity between Configuration Spaces and Monad of little cube Operad.

I am trying to understand May's Recognition Principle, specifically its proof. I will now recall some definition, which can be found in [Geometry of Iterated Loop Space] (or, a concise survey of Maru Sarazola) to make the question a bit more…
Alessandro Fenu
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What is the rational homology of the unordered configuration space of points in the plane?

What is the rational homology of the unordered configuration space of $n$ points in the Euclidean plane? In the unordered configuration space, I know there is torsion in homology, but I just want to know the Betti numbers, i.e. what is $$\beta_j =…
Matthew Kahle
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What is the geometric realization of the configuration simplicial space?

Take a path-connected space $X$ and consider the family of non-empty, ordered configuration spaces $(\text{Conf}_{\bullet+1}(X))$. This collection of spaces forms a semi-simplicial space with face maps $d_i =$ forget the $i$-th point. This can be…
Nicolas Guès
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Configuration spaces to moduli spaces

In Segal's paper on Mapping Configuration spaces to moduli spaces, I'm not understanding what the map $\Phi$ is, explicitly. Also in section 2, he goes on to say $M_{g,2} \simeq BHomeo^{+}(F_{g,2}; \partial$). (In the paper, $F_{g,2}$ is a surface…
May
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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,\dots,u_n)\in M^n\mid u_i\neq u_j, \forall i\neq…
バレリオ
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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 quotient topology: $T$ is the quotient of an open…
user392347
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The configuration space of 3 unordered points in $\mathbb{R}^2$ with distinct distances

Let $(X, d)$ be a metric space and let $n \in \mathbb{N}$. Define $A_n(X) = \{ (x_i)_{i=1}^n \in X^n \mid \forall i \neq j: x_i \neq x_j \}$ to be the space of $n$ ordered distinct points in $X$. Define $B_n(X) = \{ (x_i)_{i=1}^n \in A_n(X) \mid…
Smiley1000
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Is this quotient space a manifold?

Consider the configuration space $\text{Conf}_n(\mathbb R^k)$, and consider the subgroup $G=\mathbb R^k\rtimes \mathbb R^{\times}\leq \mathbb R^k\rtimes \text{GL}(k,\mathbb R)=\text{Aff}(\mathbb R^k)$ of dilations and translations in affine group,…
Eric
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