Manolescu and Piccirillo (2023) recently initiated a program to construct an exotic $S^4$ or $\#n{\mathbb{CP}}^2$ by using zero surgery homeomorphisms and Rasmussen's $s$-invariant. They find five knots that if any were slice, one could construct an exotic $S^4$ and disprove the Smooth 4-dimensional Poincaré conjecture. We rule out this exciting possibility and show that these knots are not slice. To do this, we use a zero surgery homeomorphism to relate slice properties of two knots stably after a connected sum with some 4-manifold. Furthermore, we show that our techniques will extend to the entire infinite family of zero surgery homeomorphisms constructed by Manolescu and Piccirillo. However, our methods do not completely rule out the possibility of constructing an exotic $S^4$ or $\#n{\mathbb{CP}}^2$ as Manolescu and Piccirillo proposed. We explain the limits of these methods hoping this will inform and invite new attempts to construct an exotic $S^4$ or $\#n{\mathbb{CP}}^2$. We also show that a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.

中文翻译：

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从零手术同胚追踪嵌入

Manolescu 和 Piccirillo (2023) 最近启动了一项计划，旨在建造一个异国情调的$S^4$或者$\#n{\mathbb{CP}}^2$通过使用零手术同胚和拉斯穆森$s$-不变的。他们发现了五个结，如果将其中任何一个切开，就可以建造一个异国情调的结$S^4$并反驳平滑 4 维庞加莱猜想。我们排除了这种令人兴奋的可能性，并证明这些结不是切片。为此，我们使用零手术同胚在与某个 4 流形进行连通和后稳定地关联两个结的切片属性。此外，我们表明我们的技术将扩展到由 Manolescu 和 Piccirillo 构建的整个无限家族的零手术同胚。然而，我们的方法并没有完全排除构建奇异的可能性$S^4$或者$\#n{\mathbb{CP}}^2$正如马诺莱斯库和皮奇里洛所提议的那样。我们解释了这些方法的局限性，希望这能为构建一个异国情调的新尝试提供信息并引发新的尝试。$S^4$或者$\#n{\mathbb{CP}}^2$。我们还表明，Manolescu 和 Piccirillo 使用丝带结的环形扭曲构造的同伦球族都是标准的。