Trace Embeddings from Zero Surgery Homeomorphisms

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.