# Holomorphic contractibility of Teichm\"uller spaces

## TL;DR

This paper proves that Teichmüller spaces of punctured spheres with more than four punctures are holomorphically contractible, resolving a longstanding problem from the 1970s related to Banach algebra equations.

## Contribution

It provides the first positive proof of holomorphic contractibility for these Teichmüller spaces, a problem posed since the 1970s.

## Key findings

- Teichmüller spaces T(0, n) are holomorphically contractible for n > 4
- Resolution of a problem from the 1970s
- Connections to algebraic equations in Banach algebras

## Abstract

The problem of holomorphic contractibilty of Teichm\"uller spaces T(0, n) of the punctures spheres (n > 4) arose in the 1970s in connection with solving the algebraic equations in Banach algebras. We provide a positive solution of this problem.