Braid groups, elliptic curves, and resolving the quartic

TL;DR

This paper classifies all non-trivial holomorphic maps between certain configuration spaces of complex points, revealing their connection to elliptic curves and braid groups, and resolves related conjectures in complex analysis and algebraic geometry.

Contribution

It completes the classification of holomorphic maps between configuration spaces for small cases, linking them to elliptic curves and braid group homomorphisms, and resolves conjectures of Farb and Castel.

Findings

Classified holomorphic maps between configuration spaces for m ≤ n.

Connected these maps to elliptic curves via inflection points.

Proved a conjecture on braid group endomorphisms.

Abstract

We show that, up to a natural equivalence relation, the only non-trivial, non-identity holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ between unordered configuration spaces, where $m\in\{3,4\}$, are the resolving quartic map $R\colon\mathrm{Conf}_4\mathbb{C}\to\mathrm{Conf}_3\mathbb{C}$, a map $\Psi_3\colon\mathrm{Conf}_3\mathbb{C}\to\mathrm{Conf}_4\mathbb{C}$ constructed from the inflection points of elliptic curves in a family, and $\Psi_3\circ R$. This completes the classification of holomorphic maps $\mathrm{Conf}_n\mathbb{C}\to\mathrm{Conf}_m\mathbb{C}$ for $m\leq n$, extending results of Lin, Chen and Salter, and partially resolves a conjecture of Farb. We also classify the holomorphic families of elliptic curves over $\mathrm{Conf}_n\mathbb{C}$. To do this we classify homomorphisms between braid groups with few strands and $\mathrm{PSL}_2\mathbb{Z}$, then apply powerful results from complex analysis and Teichm\"uller theory. Furthermore, we prove a conjecture of Castel about the equivalence classes of endomorphisms of the braid group with three strands.