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8
votes
0answers
143 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 ...
6
votes
1answer
108 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
votes
1answer
173 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 ...
7
votes
0answers
106 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, ...
42
votes
2answers
1k views

How many unit cylinders can touch a unit ball?

What is the maximum number $k$ of unit-radius cylinders with mutually disjoint interiors that can touch a unit ball? By a cylinder I mean a set congruent to the Cartesian product of a line and a ...
1
vote
0answers
92 views

On 'Very Movable' Geometric Configurations (Configurations with a large degree of freedom)

Let $C$ be an $(n_r, b_k)$ combinatorial configuration that admits a geometric realization in the plane. I'm interested in the maximum number of points/lines $M$ of $C$ we can place in general ...
4
votes
2answers
406 views

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?
16
votes
2answers
482 views

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 ...
5
votes
4answers
437 views

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 ...
8
votes
3answers
381 views

Configuration spaces of the torus

I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
2
votes
1answer
378 views

Contractibility of a configuration space

For a topological space $X$ and a positive integer $k\in \mathbb{N}_{>0}$ let $F_k(X):= \{ (x_1,\ldots,x_k)\in X^k |x_i\neq x_j \text{ for } i\neq j \}$ be its $k$-configuration space. Let ...
2
votes
1answer
166 views

holomorphic automorphisms of universal cover of configuration spaces

Hello everyone, I have been trying (without success) to determine the following. Let $P$ denote the space of monic polynomials of degree $n$ with complex coefficients, which have distinct roots. It ...
5
votes
1answer
249 views

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. $\Sigma^n\setminus \Delta$ ...
10
votes
4answers
659 views

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 ...
9
votes
2answers
574 views

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. ...
8
votes
1answer
567 views

Orbifold fundamental group and configuration space

Hi, 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 ...
3
votes
1answer
532 views

Almost-direct product and 1-formality

Hi everyone, 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 ...
6
votes
3answers
513 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 ...
20
votes
3answers
920 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 ...
7
votes
1answer
303 views

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 ...
10
votes
2answers
963 views

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 ...
16
votes
1answer
799 views

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 ...
3
votes
1answer
192 views

A k-component link defines a map T^k --> Conf_k S^3. Does the homotopy type capture Milnor's invariants?

A k-component link defines a map T^k --> Conf_k S^3. Does the homotopy type of this map capture the Milnor invariants? Some ...