We introduce a collection of $\frac{1}{2}$‐ ${\pi }_1$‐null four‐dimensional surgery problems. This is an intermediate notion between the classically studied universal surgery models and the ${\pi }_1$‐null kernels which are known to admit a solution in the topological category. Using geometric applications of the group‐theoretic 2‐Engel relation, we show that the $\frac{1}{2}$‐ ${\pi }_1$‐null surgery problems are universal, in the sense that solving them is equivalent to establishing four‐dimensional topological surgery for all fundamental groups. As another application of these methods, we formulate a weaker version of the ${\pi }_1$‐null disk lemma and show that it is sufficient for proofs of topological surgery and s‐cobordism theorems for good groups.

中文翻译：

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恩格尔团体和通用手术模型

我们介绍了 $\frac{\mathrm{1个}}{2}$‐ ${\pi }_{\mathrm{1个}}$无效的四维手术问题。这是经典研究的全能外科手术模型与常规手术模型之间的中间概念。${\pi }_{\mathrm{1个}}$-已知可以接受拓扑类别中的解决方案的空内核。使用群理论2恩格尔关系的几何应用，我们证明了$\frac{\mathrm{1个}}{2}$‐ ${\pi }_{\mathrm{1个}}$空手术问题是普遍的，从某种意义上讲，解决这些问题等同于为所有基本组建立三维拓扑手术。作为这些方法的另一个应用，我们制定了较弱的${\pi }_{\mathrm{1个}}$空磁盘引理，证明它足以证明拓扑手术和s-cobordism定理对于好的群体。