Liftable homeomorphisms of rank two finite abelian branched covers

Atalan F., Medetoğulları E., OZAN Y.

ARCHIV DER MATHEMATIK, vol.116, no.1, pp.37-48, 2021 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 116 Issue: 1
  • Publication Date: 2021
  • Doi Number: 10.1007/s00013-020-01501-z
  • Journal Name: ARCHIV DER MATHEMATIK
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.37-48
  • Keywords: Branched covers, Mapping class group, Automorphisms of groups
  • TED University Affiliated: No

Abstract

We investigate branched regular finite abelian A-covers of the 2-sphere, where every homeomorphism of the base (preserving the branch locus) lifts to a homeomorphism of the covering surface. In this study, we prove that if A is a finite abelian p-group of rank k and Sigma -> S-2 is a regular A-covering branched over n points such that every homeomorphism f:S-2 -> S-2 lifts to Sigma, then n = k + 1. We will also give a partial classification of such covers for rank two finite p-groups. In particular, we prove that for a regular branched A-covering pi : Sigma -> S-2, where A = ZprxZpt, 1 <= r <= t , all homeomorphisms f:S-2 -> S-2 lift to those of Sigma if and only if t = r or t = r + 1 and p = 3.