# Tagged Questions

**3**

votes

**1**answer

165 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 ...

**15**

votes

**2**answers

466 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

**4**answers

434 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

**3**answers

375 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

**1**answer

371 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 ...

**5**

votes

**1**answer

245 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

**4**answers

634 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

**2**answers

558 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.
...

**3**

votes

**1**answer

521 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

**3**answers

504 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

**3**answers

912 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

**1**answer

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 ...

**16**

votes

**1**answer

791 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 ...