In this paper, we will consider the problem of constructing stable maps between two closed orientable surfaces M and N with a given branch set of curves immersed on N. We will study, from a global point of view, the behavior of its families in different isotopies classes on the space of smooth maps. The main goal is to obtain different relationships between invariants. We will provide a new proof of Quine’s Theorem.

中文翻译：

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闭合可定向曲面之间的稳定映射的不变量

在本文中，我们将考虑在两个闭合的可定向曲面M和N之间构造给定分支曲线浸入N的稳定贴图的问题。我们将从全局的角度研究其在平滑地图空间上在不同同位素类别中的族的行为。主要目标是获得不变量之间的不同关系。我们将提供Quine定理的新证明。