Gromov's Oka principle, fiber bundles and the conformal module

TL;DR

This paper explores the conformal module of braid conjugacy classes as an invariant linked to entropy, providing a conceptual proof of its properties and examining its role as an obstruction to holomorphic deformations within the framework of Gromov's Oka principle.

Contribution

It offers a new conceptual proof relating the conformal module to braid entropy and investigates its role as an obstruction in the context of Gromov's Oka principle.

Findings

Conformal module is inversely proportional to braid entropy.

It acts as an obstruction to homotopies to holomorphic objects.

The paper delineates the limited applicability of Gromov's Oka principle in certain braid contexts.

Abstract

The conformal module of conjugacy classes of braids is an invariant that appeared earlier than the entropy of conjugacy classes of braids, and is inverse proportional to the entropy. Using the relation between the two invariants we give a short conceptional proof of an earlier result on the conformal module. Mainly, we consider situations, when the conformal module of conjugacy classes of braids serves as obstruction for the existence of homotopies (or isotopies) of smooth objects involving braids to the respective holomorphic objects, and present theorems on the restricted validity of Gromov's Oka principle in these situations.