The Gauss--skizze decomposition is a Goresky-MacPherson stratification

TL;DR

This paper introduces a new stratification of the space of degree n complex polynomials with distinct roots, demonstrating it forms a Goresky--MacPherson stratification and enabling explicit cohomology computations.

Contribution

It establishes that the proposed stratification is a Goresky--MacPherson stratification and provides a method for explicit cohomology calculations.

Findings

The stratification is a Goresky--MacPherson stratification.

A good cover can be constructed from thickening strata.

Explicit cohomology computations are possible using this approach.

Abstract

We consider a new stratification of the space of configurations of $n$ marked points on the complex plane. Recall that this space can be differently interpreted as the space $^{\rm D}{\rm Pol}_{n}$ of degree $n>1$ complex, monic polynomials with distinct roots, the sum of which is 0. A stratum $A_{\sigma}$ is the set of polynomials having $P^{-1}(\mathbb{R}\cup\imath\mathbb{R})$ in the same isotopy class, relative to their asymptotic directions. We show that this stratification is a Goresky--MacPherson stratification and that from thickening strata a good cover in the sense of \v{C}ech can be constructed, allowing an explicit computation of the cohomology groups of this space.