The number of sheets of a connected cover is the index of its induced fundamental-group subgroup.Suggest a slogan
Let \(p\colon(\tilde X, \tilde x_0)\to(X, x_0)\) be a covering space with \(X\) and \(\tilde X\) path-connected. The number of sheets of \(p\), that is, the cardinality of \(p^{-1}(x_0)\), equals the index \[ [\pi_1(X, x_0) : p_*(\pi_1(\tilde X, \tilde x_0))] \] [@Hat02].