Homotopy equivalence for proper holomorphic mappings

TL;DR

This paper explores homotopy equivalence relations for proper holomorphic mappings between balls, showing finiteness of homotopy classes in certain dimensions and introducing Whitney sequences as higher-dimensional analogues of finite Blaschke products.

Contribution

It introduces new homotopy equivalence relations, proves finiteness of homotopy classes in specific dimensions, and defines Whitney sequences as higher-dimensional analogues of Blaschke products.

Findings

The degree of rational proper mappings is not a homotopy invariant.

Finitely many homotopy classes exist for rational proper mappings between balls in certain dimensions.

Uncountably many spherical inequivalence classes can be connected by homotopies.

Abstract

We introduce several homotopy equivalence relations for proper holomorphic mappings between balls. We provide examples showing that the degree of a rational proper mapping between balls (in positive codimension) is not a homotopy invariant. In domain dimension at least 2, we prove that the set of homotopy classes of rational proper mappings from a ball to a higher dimensional ball is finite. By contrast, when the target dimension is at least twice the domain dimension, it is well known that there are uncountably many spherical equivalence classes. We generalize this result by proving that an arbitrary homotopy of rational maps whose endpoints are spherically inequivalent must contain uncountably many spherically inequivalent maps. We introduce Whitney sequences, a precise analogue (in higher dimensions) of the notion of finite Blaschke product (in one dimension). We show that terms in a Whitney sequence are homotopic to monomial mappings, and we establish an additional result about the target dimensions of such homotopies.