Homotopy equivalence for proper holomorphic mappings

John P. D'Angelo, Jiří Lebl

Research output: Contribution to journalArticlepeer-review

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.

Original languageEnglish (US)
Pages (from-to)160-180
Number of pages21
JournalAdvances in Mathematics
Volume286
DOIs
StatePublished - Jan 2 2016

Keywords

  • Blaschke product
  • CR Geometry
  • Homotopy equivalence
  • Proper holomorphic mappings
  • Spherical equivalence
  • Unit ball
  • Unit sphere
  • Whitney mapping

ASJC Scopus subject areas

  • Mathematics(all)

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