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In classical mechanics, the parameters that define the configuration of a system are called generalized coordinates, and the vector space defined by these coordinates is called the configuration space of the physical system.

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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 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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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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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 ...
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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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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 ...
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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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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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226 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 $r$ ...
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(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 $...
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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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group actions of $S^3$ on the configuration space of projective plane

Let $\mathbb{R}P^2$ be the lines in $\mathbb{R}^3$ passing through the origin. Let $SO(3)$ act on $\mathbb{R}^3$ canonically. Then $SO(3)$ has an induced action on $\mathbb{R}P^2$. Let $F(\mathbb{R}P^...
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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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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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coefficient of homology of configuration spaces over real projective spaces

In the slides Characteristic Classes of Surface Bundles and Configuration Spaces, Miguel A. Xicot'encatl, page 38, what is the coefficient of the following homology? Could the coefficient be an ...
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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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296 views

homotopy equivalence between configuration spaces

Let $M$ be a $m$-dimensional compact manifold without boundary and $W(M)$ the non-compact $CW$-complex obtained by glueing $[0,1)\times (0,1)^{m-1}$ to $M$, identifying the boundary $\{0\}\times(0,1)^{...
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1answer
193 views

torsion part of the cohomology module of configuration spaces of manifolds

Let $M$ be a manifold and $C_n(M)$ the $n$-th unordered configuration space consisting of unordered $n$-tuples of distinct points in $M$. The mod $p$ homology module, $p$ prime, and the rational ...
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equivariant embeddings from the k-th configuration space to the k+1-th configuration space

Let $S$ be a closed, orientable surface in $\mathbb{R}^3$ and $S'$ be the manifold $S\setminus\text{one point}$. Let $F(S',k)$ be the $k$-th (ordered) configuration space on $S'$. It is claimed in ...
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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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symmetric points on symmetric spaces

Let $M$ by an $m$-dimensional symmetric space (or a general Riemannian manifold). The finite distinct points $p_1,p_2,\cdots,p_n\in M$ are said symmetric, if for any permutation $\sigma$ on $1,2,\...
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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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1answer
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distinct multiple points in a space with at least one point lying in a subspace

Let $X$ be a topological space and $A$ a subspace of $X$. Given $k\geq 2$, let the unordered configuration space be $$ B(X,k)=\{(x_1,x_2,\cdots,x_k)\in X^k\mid x_i\neq x_j \text{ for any } i\neq j\} $$...
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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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1answer
142 views

cohomology ring of the fundamental group of unordered configuration space

From the lecture notes INTRODUCTION TO CONFIGURATION SPACES AND THEIR APPLICATIONS, p. 18, I find: Os it possible to derive the cohomology ring $H^*(Conf(S,k)/\Sigma_k;\mathbb{Z}_2)$ from the above ...
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1answer
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dimension of generators of cohomology ring of iterated loop-suspension

In the book The unstable Adams spectral sequence for free iterated loop spaces, R.J. Wellington, Mem. Amer. Math. Soc. 258, 1982, p. 32 Question: When $p=2$, $k\geq 1$, $n=0$ to $\infty$, what kind ...
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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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unordered configuration space over spheres and Euclidean spaces

For a topological space $X$, let $B(X,k)$ be the $k$-th unordered configuration space. Then $$ B(\mathbb{R}^n,2)\simeq \mathbb{R}P^{n-1}, $$ $$ B(S^n,2)\simeq \mathbb{R}P^n. $$ Hence $ (*) $ $$ B(\...
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coproduct of the homology of iterated loop space on spheres

Let $\Omega^{n+1}S^{n+1}$ be the base-pointed $(n+1)$-iterated loop space on the $(n+1)$-sphere. In the paper The homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, Lecture notes in ...
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homotopy equivalence between configuration spaces on non-homeomorphic spaces

(1). Let $D^m$ be the closed $m$-disc in $\mathbb{R}^m$. For each $k$, does the $k$-th configuration space on $D^m$ homotopy equivalent to the $k$-th configuration space on $\mathbb{R}^m$ $$ F(D^m,k)\...
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1answer
191 views

Configuration spaces of positive and negative particles

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, page 3, line from bottom 1-3, it is given that for a $m$-manifold $M$, there is a map from the labelled ...
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maps from labelled configuration space to section space / iterated loop space

In the paper Mapping class group and function spaces: a survey, F. Cohen, M.A. Maldonado, 2014, page 3, Section 3: for a $m$-manifold $M$, consider the disc bundle $D(M)$ in the tangent bundle $T(M)$...
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Group completion of labelled configuration space on Euclidean spaces

In the lecture notes The Homology of $\mathcal{C}_{n+1}$-spaces, $n\geq 0$, F. Cohen, page 225 -226, it is obtained that there is a group completion on homology $$ \alpha_n: C(\mathbb{R}^n;X)\to \...
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configuration space and iterated loop space

Let the topological monoid $M$ be the configuration space $C(\mathbb{R}^n;X)=C_n(X)$ as in the book The geometry of iterated loop spaces, Theorem 5.2. I want to prove that the map $\alpha_n$ in ...
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iterated loop spaces and configuration spaces [closed]

In the lecture notes by J.P. May, The geometry of iterated loop spaces, Chapter 5, formula (1), (2) and (10), a map $$ \phi: Hom_T(X,\Omega Y)\to Hom_T(SX,Y) $$ is defined. And a map $$ \eta_n=\phi^{-...
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1answer
414 views

unordered configuration space of pointed space

Let $(X,*)$ be a pointed topological space. Let $F(X,k)=\{(x_1,\cdots,x_k)\in X^k\mid \forall i\neq j: x_i\neq x_j, \}$. Let $F(X,k)/S_k$ be the $k$-th unordered configuration space. Is there an ...
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Unordered configuration space of $\mathbb{R}P^1$

In the paper GEOMETRY OF TRUNCATED SYMMETRIC PRODUCTS AND REAL ROOTS OF REAL POLYNOMIALS, JACOB MOSTOVOY, Bull. London Math. Soc. (1998) 30 (2): 159-165, Theorem 2. (b): $TP^n(\mathbb{R}...
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How do we see the rank of the braid group?

The only presentation of the braid group that most people ever see is the standard Artin presentation $$B_n=\langle σ_1,\cdots,σ_{n−1}|\ σ_iσ_j=σ_jσ_i\ \ (|i−j|>1),\ σ_iσ_{i+1}σ_i=σ_{i+1}σ_i σ_{i+...
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1answer
288 views

cohomology ring of configuration spaces

In the paper configuration spaces: applications to Gelfand-Fuks cohomology, by F. Cohen and L. Taylor, Bull. Amer. Math. Soc., 1978, theorem 1, I did not find the proof. What method did the author use ...
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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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configuration spaces of real projective space

Let $F(\mathbb{R}P^n,k)$ be the $k$-th ordered configuration space on $\mathbb{R}P^n$. In http://arxiv.org/abs/1502.04258, the cohomology ring $$ H^*(F(\mathbb{R}P^n,k);R)$$ is obtained for any ...
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homology of configuration spaces of non-compact manifolds

Let $M$ be a manifold. Let $F(M,n)$ be the configuration space of $n$-tuples on $M$. Let $B(M,n)=F(M,n)/S_n$, where $S_n$ is the symmetric group of order $n$, be the corresponding unordered ...
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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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1answer
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cohomology algebra of unordered configuration space on Euclidean space

In Homology of $C_{n+1}$-spaces, $n\geq 0$, F.R. Cohen, Lecture Notes in Mathematics, Vol. 533, page 210 (the preface part before contents): Line 2: ... is used to compute the precise algebra ...
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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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519 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 ...
7
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1answer
136 views

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 ...
3
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1answer
239 views

Configuration spaces of trees are Eilenberg-MacLane spaces

I'm reading Ghrist's paper "Configuration spaces and braid groups on graphs in robotics". In this discussion, counterexamples are shown for both Theorem 2.3 and the implication "Theorem 2.3 $\...
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316 views

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, (\...