We compare two naturally arising notions of ‘unknotting number’ for 2-spheres in the 4-sphere: namely, the minimal number of 1-handle stabilizations needed to obtain an unknotted surface, and the minimal number of Whitney moves required in a regular homotopy to the unknotted 2-sphere. We refer to these invariants as the stabilization number and the Casson–Whitney number of the sphere, respectively. Using both algebraic and geometric techniques, we show that the stabilization number is bounded above by one more than the Casson–Whitney number. We also provide explicit families of spheres for which these invariants are equal, as well as families for which they are distinct. Furthermore, we give additional bounds for both invariants, concrete examples of their non-additivity, and applications to classical unknotting number of 1-knots.

中文翻译：

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解开 4 球体中的 2 球体数

我们比较了 4 球体中 2 球体的两个自然产生的“解结数”概念：即，获得解结表面所需的 1 手柄稳定化的最小数量，以及常规同伦所需的惠特尼移动的最小数量到未打结的 2 球体。我们将这些不变量称为稳定数和Casson-Whitney 数分别为球体。使用代数和几何技术，我们表明稳定数比 Casson-Whitney 数多一。我们还提供了这些不变量相等的领域的明确族，以及它们不同的族。此外，我们为这两个不变量、它们的非可加性的具体示例以及经典解开 1 结数的应用提供了额外的界限。