Compactification of the space of branched coverings of the two-dimensional sphere

Abstract

For a closed oriented surface $\Sigma$ we define its degenerations into singular surfaces that are locally homeomorphic to wedges of disks. Let $X_{\Sigma ,n}$ be the set of isomorphism classes of orientation preserving $n$-fold branched coverings $\Sigma \to S^2$ of the two-dimensional sphere. We complete $X_{\Sigma ,n}$ with the isomorphism classes of mappings that cover the sphere by the degenerations of $\Sigma$. In case $\Sigma =S^2$, the topology that we define on the obtained completion ${\bar{X}}_{\Sigma ,n}$ coincides on $X_{S^2,n}$ with the topology induced by the space of coefficients of rational functions $P/Q$, where $P,Q$ are homogeneous polynomials of degree $n$ on $\mathbb{C}{\mathrm{P}}^1\cong S^2$. We prove that ${\bar{X}}_{\Sigma ,n}$ coincides with the Diaz-Edidin-Natanzon-Turaev compactification of the Hurwitz space $H(\Sigma ,n)\subset X_{\Sigma ,n}$ consisting of isomorphism classes of branched coverings with all critical values being simple.

17 pages. arXiv admin note: text overlap with arXiv:2002.12402

References in corpus (1)

• Covering semigroups

Cited by in corpus (1)

• A cell structure of the space of branched coverings of the two-dimensional sphere