论文标题
零手术同构的痕量嵌入
Trace Embeddings from Zero Surgery Homeomorphisms
论文作者
论文摘要
Manolescu和Piccirillo最近启动了一个程序，用于构建一个异国情调的$ S^4 $或$ \＃n \ Mathbb {Cp}^2 $，通过使用零手术同构和Rasmussen的$ S $ INVARIANT。他们发现五个结，如果有的话，一个人可以构建一个异国情调的$ s^4 $，并证明了光滑的$ 4 $维度的庞加莱猜想。我们排除了这种令人兴奋的可能性，并表明这些结不是切片。为此，我们使用零手术同构形态形态，在连接的总和（约4美元的manifold）连接的总和之后，将两个节属性\ textit {specy}联系起来。此外，我们表明我们的技术将扩展到由Manolescu和Piccirillo建立的整个零手术同构的无限家族。但是，我们的方法并不完全排除构建异国情调的$ s^4 $或$ \＃n \ Mathbb {cp}^2 $作为Manolescu和Piccirillo提议的可能性。我们解释了这些方法的局限性，希望这将为您提供信息并邀请新的尝试构建异国情调的$ S^4 $或$ \＃n \ MATHBB {CP}^2 $。我们还显示了由Manolescu和Piccirillo使用丝带结的Annulus Twists构建的一系列同型球体。
Manolescu and Piccirillo 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 \textit{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 a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.
