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 ...
8
votes
1answer
532 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 ...
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 ...