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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1 vote
0 answers
92 views

Homeomorphism between interiors of simplex and permutohedron

The $n$-dimensional permutohedron $P_n$ is a polytope whose facets (i.e.\ codimension $1$ faces) are in 1-to-1 correspondence with all faces (of codimension${}\geq 1$) of the $n$-simplex $\Delta_n$, ...
2 votes
1 answer
80 views

Is there a way to parametrize the configuration space of all convex polyhedra of a given combinatorial type as a convex set?

I'm sure this is easy/known, but I'm not hitting an appropriate search term for finding the answer and the coffee hasn't kicked in enough to come up with it myself: Let $T$ be a simplicial 2-complex ...
1 vote
0 answers
78 views

What is the meaning of universal family of Fulton Macpherson configuration space?

Fulton and Macpherson suggests the way to compactify the set of $n$-labelled distinct point on variety in their paper, "A Compactification of Configuration Spaces" In this paper, the process ...
3 votes
0 answers
111 views

Universal cover of the configuration space of points on surface

Let $S$ be a closed oriented surface and $C(S, n)$ be the configuration space of $n$ points on $S$, i.e., the space of $n$-tuples of distinct points of $S$ with the topology induced from $S^n$. Let $V ...
1 vote
1 answer
144 views

Compact locus in (ordered) configuration spaces

Let $\mathit{Conf}_n(\mathbb{R}^2)$ be the configuration space of $n$ ordered distinct points in the plane. I'd like to know if the topological subspace $C_n$ consisting of points $(p_1,...,p_n)$ with ...
3 votes
1 answer
294 views

How many configurations of tubes are there?

Can $n$ disjoint lines in $\boldsymbol R^3$ be knotted? No... Let $X_n$ be the configuration space of $n$ disjoint lines in $\boldsymbol R^3$. It is not hard to see that $X_n$ is path connected: Let $...
0 votes
1 answer
138 views

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 ...
12 votes
1 answer
1k views

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 $C_2$ ...
3 votes
0 answers
49 views

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 ...
4 votes
0 answers
96 views

"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/...
2 votes
1 answer
128 views

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-...
1 vote
0 answers
30 views

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 ...
8 votes
1 answer
315 views

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, ...
6 votes
0 answers
177 views

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)\...
4 votes
0 answers
113 views

Going between the abstract and the concrete notions of chiral homology

Let $X$ be a smooth algebraic curve over $\mathbf{C}$, and let $\mathcal{V}$ be a factorisation algebra over $X$, whose fibre above $x\in X$ is the vertex algebra $V$. Note that $\mathcal{V}\in\...
0 votes
0 answers
267 views

Finite morphisms between two varieties

Let $X$ and $Y$ be varieties in some projective space. Furthermore let's assume these two varieties are intersecting at the subvariety $Z$. For this problem we are assuming that $\text{dim}(X)\ll\text{...
27 votes
3 answers
2k views

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 ...
0 votes
0 answers
69 views

packing numbers of the unit balls in Euclidean spaces and the dimensions

Let $k$, $m$ and $n$ be positive integers. Let $r$ be a positive real number. The $n$-th ordered $r$-disk configuration space on the Euclidean space $\mathbb{R}^{mk}$ is $$ F_r(\mathbb{R}^{mk},...
5 votes
2 answers
305 views

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 $8$ ...
2 votes
0 answers
40 views

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 $C_n(\mathbb{R}^d)$ ...
0 votes
1 answer
103 views

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) ...
6 votes
1 answer
666 views

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 ...
6 votes
1 answer
126 views

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. (...
5 votes
1 answer
249 views

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 ...
8 votes
2 answers
402 views

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 \}...
10 votes
1 answer
357 views

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. ...
15 votes
2 answers
938 views

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 ...
3 votes
1 answer
156 views

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 $$ p\colon \pi_1(\mathcal{M}_{0,n})\longrightarrow H_1(\mathcal{M}_{0,n},\mathbb{Z}/2\mathbb{Z}) $$ ...
8 votes
2 answers
381 views

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
1 answer
220 views

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 ...
2 votes
0 answers
58 views

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?
2 votes
1 answer
305 views

Is it true, the space of embeddings segments is homotopy equivalent to the subspace of all line segments?

Consider the space of smooth embeddings of the segment in the plane with the compact-open topology. Denote by X the quotient space obtained from the equivalence relation $a \sim b$ if and only if $\...
6 votes
1 answer
500 views

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 ...
65 votes
3 answers
3k views

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 ...
5 votes
0 answers
154 views

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,.....
5 votes
1 answer
256 views

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 ...
6 votes
1 answer
153 views

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 ...
7 votes
0 answers
216 views

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 ...
9 votes
1 answer
422 views

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 ...
2 votes
2 answers
636 views

Cohomology of configuration space of a compact manifold

There is a reference or a methode which by it we can calculate the cohomology of a configuration space of a compact manifold simply connected? It is possible to find a spectral sequence converging to ...
8 votes
0 answers
293 views

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 ...
2 votes
1 answer
151 views

Very symmetric quadrangle in $\Bbb CP^2$

Is there a quadrangle $Q \subset \Bbb CP^2$, namely $Q$ is a set of four points, such that every permutation of $Q$ can be realizad by an isometric projectivity of $\Bbb CP^2$? Clearly the analogous ...
12 votes
1 answer
538 views

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 $\{...
12 votes
3 answers
935 views

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 ...
3 votes
2 answers
238 views

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 ...
10 votes
2 answers
412 views

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 \...
6 votes
0 answers
379 views

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 ...
4 votes
1 answer
300 views

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 $r$ ...
1 vote
0 answers
196 views

(Ordered) Configuration space in algebraic geometry

Let $X$ be a topological space and denote by $F_n(X)$ the following subspace: $$F_n(X):=\{(x_1,\cdots ,x_n)\in X^n: x_i\neq x_j \forall i\neq j\}.$$ Note that, we are not considering the quotient of $...
4 votes
2 answers
366 views

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 ...