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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How many unit cylinders can touch a unit ball?
What is the maximum number $k$ of unit radius, infinitely long cylinders with mutually disjoint interiors that can touch a unit ball?
By a cylinder I mean a set congruent to the Cartesian product of ...
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Configuration space of little disks inside a big disk
The space of configurations of $k$ distinct points in the plane
$$F(\mathbb{R}^2,k)=\lbrace(z_1,\ldots , z_k)\mid z_i\in \mathbb{R}^2, i\neq j\implies z_i\neq z_j\rbrace$$
is a well-studied object ...
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Distinct manifolds with the same configuration spaces?
For a space $X$, let $C_k X$ denote the space of configurations of $k$ distinct unordered points in $X$.
What is an example of a pair of smooth manifolds $M$ and $N$ that are not homeomorphic but ...
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Fundamental groups of the spaces of rational functions
Here is a question which I asked myself (and couldn't answer) while reading "The topology of spaces of rational functions" by G. Segal.
Let $X$ be a smooth complete complex curve (=a compact Riemann ...
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Cohomology of configuration space as a representation of the symmetric group
Let $X_n$ be the space of $n$ distinct labeled points in $\mathbb{R}^3$, which is equipped with an action of the symmetric group $S_n$. It is well known that the total cohomology of $X_n$ is ...
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Mixed Hodge structure on configuration spaces
Let $X$ be a smooth complex projective variety. Let $F(X,n)$ be the configuration space parametrizing $n$ distinct ordered points in $X$. The cohomology groups $H^k(F(X,n),\mathbf Q)$ carry a mixed ...
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The fibers of M_{g,n} \to M_g and the Fulton-MacPherson compactification
Let $g \geq 2$, and consider the moduli space $\bar M_{g,n}$ of stable n-pointed curves of genus g. There is a natural forgetful map to $\bar M_g$, which forgets the markings and contracts any ...
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Relation between cohomology of ordered and unordered configuration spaces?
For any manifold $M$, the unordered configuration space of $k$ points is obtained as a quotient of ordered configuration space of $k$ points by the group action of symmetric group on $k$ letters. Does ...
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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
$$
\operatorname{UConf}_n(X):=\{\text{embeddings of $\{...
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Orbifold fundamental group and configuration space
I'm not very familiar with (even simple examples of) orbifolds, so my first question is:
Let $C_2$ be $\mathbb{C}$ with one cone singularity at 0 of index 2. What is the fundamental group of $C_2$ ...
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Beginning reference for configuration spaces
In my mathematical reading and thoughts, I keep running across the notion of configuration spaces, and while I essentially understand the idea behind them, I don't have much intuition for them (not ...
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Configuration spaces and non homeomorphic vector bundles
Given two complex line bundles over the complex projective line ${\mathbb CP}^1$, prove or disprove that their total spaces are homeomorphic if and only if their Chern numbers are equal up to sign.
...
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Is the configuration space of ordered triples of distinct points in the four-edge banana graph homotopy equivalent to a surface of genus 13?
If $X$ is a topological space, write $C_n(X)$ for the configuration space of distinct ordered tuples of points in $X$:
$$C_n(X) = \{(x_1, \ldots, x_n) \in X^n \mbox{ so that $i \neq j \implies x_i \...
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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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The example of mechanical system that has a Mobius strip as their configuration space
Can you give examples for mechanical system that has a Mobius strip as their configuration space?
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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 ...
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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 ...
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Configuration spaces of the torus
I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
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Almost-direct product and 1-formality
Let $G$ be a finitely presented group. To $G$ is associated in a functorial way a Malcev Lie algebra which can be constructed in several equivalent ways. Roughly speaking, it is the quotient of the ...
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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:$$\operatorname{Conf}_k(\mathbb{R}^n) = \{ (x_1,\dots,x_k) \in (\mathbb{R}^n)^k \mid x_i \neq x_j, \, \forall i \neq j \}...
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A $k$-component link defines a map $T^k\rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type capture Milnor's invariants?
A $k$-component link defines a map $T^k \rightarrow \operatorname{Conf}_k S^3$. Does the homotopy type of this map capture the Milnor invariants?
Some special cases:
$k=2$, no, it's null homologous, ...
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Configuration space like subspace of sphere product
For $k \geq 2, n \geq 1$ let $$M^{n,k} = \{(x_1,\dots,x_k) \in S^n \times \dots \times S^n \ | \ x_1 + \dots + x_k = 0\}$$ This is a compact CW-complex and almost, but not quite, a manifold. I ...
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The homology of the braid group with coefficients in the Burau representation
Let $B_n$ denote the braid group with $n$ braids. The Burau representation $B_n\to GL_n(\mathbb{Z}[t^{\pm1}])$ makes $(\mathbb{Q}[t^{\pm1}])^n$ a $B_n$-module. I am curious in knowing what $H_i(B_n, (\...
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Genus=2 theta functions, Arnold's relation, and KZ connection
Let $C_5:=\{{(z_1 \dots, z_5) \in (\mathbb{C})^5 | z_i \neq z_j \forall i\neq j }\}$ be the configuration space of five distinct ordered points in $\mathbb{C}$. Arnold showed that the holomorphic one ...
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contractible configuration spaces
Let $F(M,k)=\{(x_1,\cdots,x_k)\mid x_1\cdots,x_k\in M,x_i\neq x_j, \text{ for } i\neq j \}$. It is known that $F(\mathbb{R}^\infty,k)$ is contractible for each $k$.
My question: is $F(S^\infty,k)$ ...
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Aspherical homotopy orbit space of configurations on the 2-sphere
The group SO(3) acts naturally on $S^2$ and thus on $Conf(S^2, q)$, the configuration space of $q$ distinct points on the 2-sphere, via the diagonal action on $S^2 \times...\times S^2$. This is a ...
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An analogue of cabling for configuration spaces
There is a well-known operation known as cabling for knots, and also for braid groups, where it is a homomorphism
$$\beta_k \times \beta_\ell \longrightarrow \beta_{k\ell}$$
given by thickening up the ...
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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 ...
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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 ...
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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 ...
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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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electron configuration on manifolds
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 ...
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Stiefel-Whitney class of unordered configuration space
Let $S^m$ be the $m$-sphere and $$F(S^m,2)/\mathbb{Z}_2=\{(a,b)\mid a,b\in S^m, a\neq b\}/(a,b)\sim (b,a)$$ be the $2$-nd unordered configuration space on $S^m$. How to compute the total Stiefel-...
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Configurations of $n$ points modulo isometries of the ambient space
Let $M$ be a Riemannian manifold and let $n$ a positive integer. Denote by $F_n(M) \subset M^n$ the space of all $n$-tuples of pairwise distinct points from $M$. The isometries of $M$ act co-ordinate ...
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factorization of the cohomology of configuration space
This question is a follow-up to my previous question factorization of the regular representation of the symmetric group, which was answered in a very satisfactory way.
Let $\operatorname{Conf}(n,\...
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Does combinatorial deleted product become equivalent to the topological deleted product after enough subdivision?
Suppose $X$ is a topological space. Define the (topological) $n$-fold deleted product of $X$ to be the space or ordered $n$-tuples of pairwise distinct points in $X$.
$$F(X, n):= \{(x_1, \ldots, x_n)\...
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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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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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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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Configuration topos?
Let ${\bf Fin}$ denote the category of finite sets. If $X$ is a topological space, then for any natural number $k\in{\mathbb N}$, the slice category ${\bf Fin}/X$ contains the configuration space $C_k(...
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Spaces parametrizing ramified covers of surfaces
Let $\Sigma$ be a surface (let's say oriented and of finite type). We can consider the configuration space $F(\Sigma,n)$ of $n$ ordered distinct points on $\Sigma$, i.e. $\Sigma^n\setminus \Delta$ ...
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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 ...
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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,.....
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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,
$$
B(M,n)=\{(...
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configuration space of Riemannian manifolds with a parameter on the distance of distinct points
Let $M$ be a Riemannian manifold. For any $\epsilon\geq 0$, we define the $k$-th ($k=1,2,\cdots$) "$\epsilon$-configuration space" as
$$
F(M,k,\epsilon)=\{(x_1,\cdots,x_k)\in M^k\mid d(x_i, x_j)>\...
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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 ...
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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 ...
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1
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Relation between cohomology of ordered and unordered configuration spaces
Let $M$ be a manifold. Then $F(M,k)/\Sigma_k$, the unordered configuration space of $k$ points, is obtained as a quotient of $F(M,k)$, the ordered configuration space of $k$ points, by the group ...
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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 $r$ ...
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