2022/03/27 by Kai Nakamura, Nakamura, Kai
Mathematics · #Advanced Combinatorial Mathematics #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.GT
paper · pdf · doi:10.48550/arxiv.2203.14270
18 pages, 8 figures. Comments welcome!
arxiv created 2022/03/27 · openalex publication_date 2022/03/27 · arxiv updated 2022/03/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Manolescu and Piccirillo recently initiated a program to construct an exotic S4 or # n \mathbbCP2 by using zero surgery homeomorphisms and Rasmussen's s-invariant. They find five knots that if any were slice, one could construct an exotic S4 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 S4 or # n \mathbbCP2 as Manolescu and Piccirillo proposed. We explain the limits of these methods hoping this will inform and invite new attempts to construct an exotic S4 or # n \mathbbCP2. We also show a family of homotopy spheres constructed by Manolescu and Piccirillo using annulus twists of a ribbon knot are all standard.