On holomorphic contractibility of Teichmuller spaces

TL;DR

This paper discusses the holomorphic contractibility of Teichmuller spaces of punctured spheres, providing improved results and an alternative proof for low-dimensional cases, building on previous work from the 1970s and recent advances.

Contribution

The paper refines a key lemma and offers an alternative proof for the holomorphic contractibility of low-dimensional Teichmuller spaces, enhancing understanding of their complex structure.

Findings

Improved the statement of Lemma 3 from previous work.

Provided an alternative proof for low-dimensional Teichmuller spaces.

Confirmed holomorphic contractibility in specific cases.

Abstract

The problem of holomorphic contractibilty of the Teichmuller spaces $T(0, n)$ of punctured spheres ($n > 4$) arose in the 1970s in connection with solving algebraic equations in Banach algebras. Recently it was solved by the author in \cite{Kr2}. In the present paper we improve the statement of Lemma 3 in \cite{Kr2} and provide an alternate proof of holomorphic contractibility of low dimensional Teichmuller spaces.