University of Minnesota Combinatorics Seminar: 2020-2021

Date: 09/18/2020

Speaker: Yifeng Huang

Betti Numbers of Unordered Configuration Spaces of a Punctured Torus
Let $X$ be a elliptic curve over C with one point removed, and consider the unordered configuration spaces $Conf^n(X)=\{(x_1,...,x_n): x_i\neq x_j \text{ for } i\neq j\} / S_n$. We present a rational function in two variables from whose coefficients we can read off the i-th Betti numbers of $Conf^n(X)$ for all $i$ and $n$. The key of the proof is a property called "purity", which was known to Kim for (ordered or unordered) configuration spaces of the complex plane with $r \geq 0$ points removed. We show that the unordered configuration spaces of $X$ also have purity (but with different weights). This is a joint work with G. Cheong.

Betti Numbers of Unordered Configuration Spaces of a Punctured Torus

Let $X$ be a elliptic curve over C with one point removed, and consider the unordered configuration spaces $Conf^n(X)=\{(x_1,...,x_n): x_i\neq x_j \text{ for } i\neq j\} / S_n$. We present a rational function in two variables from whose coefficients we can read off the i-th Betti numbers of $Conf^n(X)$ for all $i$ and $n$. The key of the proof is a property called "purity", which was known to Kim for (ordered or unordered) configuration spaces of the complex plane with $r \geq 0$ points removed. We show that the unordered configuration spaces of $X$ also have purity (but with different weights). This is a joint work with G. Cheong.