Liftable homeomorphisms of rank two finite abelian branched covers

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 $\Sigma \rightarrow S^2$ is a regular A-covering branched over n points such that every homeomorphism $f:S^2 \rightarrow S^2$ 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 $\pi :\Sigma \rightarrow S^2$, where $A={{\mathbb {Z}}}_{p^r}\times {{\mathbb {Z}}}_{p^t}, \ 1\le r\le t$, all homeomorphisms $f:S^2 \rightarrow S^2$ lift to those of $\Sigma$ if and only if $t=r$ or $t=r+1$ and $p=3$.