15
votes
2answers
436 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
423 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
341 views

Configuration spaces of the torus

I would like a reference that calculates the rational homology of the unordered configuration spaces of the torus.
5
votes
1answer
242 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
595 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
523 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
1answer
505 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
490 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
879 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
297 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
1answer
766 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 ...