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  "body": "\n## Statement\n\nLet $X$ be a compact connected [[Riemann surface|5K0N8QMV6Q]] and let $B \\subset X$ be a finite set.\n\n1. Let $p^\\circ \\colon Y^\\circ \\to X \\setminus B$ be a finite topological covering map with $Y^\\circ$ connected.\nThen there is a compact connected Riemann surface $Y$ that contains $Y^\\circ$ as the complement of a finite set.\nThere is a holomorphic [[branched covering|YARENZRXTD]] $p \\colon Y \\to X$ that extends $p^\\circ$.\nThe pair $(Y, p)$ is unique up to a unique isomorphism over $X$ that is the identity on $Y^\\circ$.\n2. As a consequence, take $X = \\hat{\\C}$ and $B = \\set{b_1, \\ldots, b_n}$, and let $d \\ge 1$.\nConsider the connected holomorphic branched coverings $Y \\to \\hat{\\C}$ of degree $d$ that are unbranched outside $B$, up to isomorphism over $\\hat{\\C}$.\nConsider the tuples $(\\sigma_1, \\ldots, \\sigma_n)$ in the [[symmetric group|0515A6D2Y1]] $S_d$ with $\\sigma_1 \\cdots \\sigma_n = 1$ that generate a [[transitive|02GK6HZYBT]] subgroup.\nCall two such tuples equivalent when they are conjugate by one element of $S_d$.\nThe coverings correspond one to one to the equivalence classes of such tuples.\n\n## Remarks\n\nFor part 1, the complex structure of $X$ pulls back to $Y^\\circ$ through the local homeomorphism $p^\\circ$.\nNear a point $b \\in B$, each component of the preimage of a small punctured disc is a punctured disc that covers it like $z \\mapsto z^e$, and one fills in the puncture.\nPart 2 follows by applying part 1 to the coverings of $\\hat{\\C} \\setminus B$, which correspond to transitive actions of the [[fundamental group|S1XK3HXHFX]].\nThat group is generated by loops $\\gamma_1, \\ldots, \\gamma_n$ around the points $b_i$ with the single relation $\\gamma_1 \\cdots \\gamma_n = 1$, and $\\sigma_i$ is the permutation of the $d$ sheets given by $\\gamma_i$.\n\nExample: for $n = 2$, $B = \\set{0, \\infty}$ and $\\sigma_1 = (1\\,2\\,\\cdots\\,d)$, $\\sigma_2 = \\sigma_1^{-1}$, the covering is $z \\mapsto z^d$.\n\nSince every compact Riemann surface carries enough meromorphic functions to be an algebraic curve, every such datum is realized by a polynomial equation $F(z, w) = 0$.\n\n## Explanations\n\n### L1 human\n\nThe theorem says that one can build Riemann surfaces from purely combinatorial data.\nTo describe a surface lying over the sphere with $d$ sheets, one only needs to say how the sheets are permuted when one walks around each branch point.\nAny choice of permutations that multiply to the identity and connect all sheets gives a surface, and only one.\n\n### L2 human\n\nThis is the link between the topology of coverings and the algebra of function fields.\nCompact Riemann surfaces are the same as smooth projective complex curves.\nSo the theorem says that the finite extensions of $\\C(z)$ unramified outside $B$ correspond to conjugacy classes of finite-index subgroups of the free group $\\pi_1(\\hat{\\C} \\setminus B)$.\nIt is used to count coverings, as in Hurwitz numbers, and in inverse Galois theory to show that every finite group is a Galois group over $\\C(z)$.\nTogether with the [[uniformization theorem|5EJ7SW1RCV]] it is one of the main existence results for Riemann surfaces.\n",
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