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

中文翻译：

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二阶有限阿贝尔分支覆盖的可提升同胚

我们研究了 2 球体的分支规则有限阿贝尔 A 覆盖，其中基的每个同胚（保留分支轨迹）提升到覆盖表面的同胚。在这项研究中，我们证明如果 A 是秩为 k 的有限阿贝尔 p 群，并且 $$\Sigma \rightarrow S^2$$ Σ → S 2 是在 n 个点上分支的正则 A -覆盖，使得每个同胚 $ $f:S^2 \rightarrow S^2$$ f : S 2 → S 2 提升到 $$\Sigma $$ Σ ，​​然后 $$n=k+1$$ n = k + 1 。我们还将对二阶有限 p 群的此类覆盖进行部分分类。特别地，我们证明了对于规则分支 A -covering $$\pi :\Sigma \rightarrow S^2$$ π : Σ → S 2 ，其中 $$A={{\mathbb {Z}}}_{ p^r}\times {{\mathbb {Z}}}_{p^t}, \ 1\le r\le t$$ A = Z pr × Z pt , 1 ≤ r ≤ t , 所有同胚 $$ f:S^2 \rightarrow S^2$$ f :