A new approach to light bulb tricks: Disks in 4-manifolds

Abstract

For a 4-manifold $M$ and a knot $k:{\mathbb{S}}^1\hookrightarrow \partial M$ with dual sphere $G:{\mathbb{S}}^2\hookrightarrow \partial M$, we compute the set $\mathbb{D}(M;k)$ of smooth isotopy classes of neat embeddings ${\mathbb{D}}^2\hookrightarrow M$ with boundary $k$, using an invariant going back to Dax. Moreover, we construct a group structure on $\mathbb{D}(M;k)$ and show that it is usually neither abelian nor finitely generated. We recover all previous results for isotopy classes of spheres with framed duals and relate the group $\mathbb{D}(M;k)$ to the mapping class group of $M$.

41 pages, 10 figures. The initial version of arXiv:2105.13032 has been split into two parts, and this is the part about surfaces in 4-manifolds. v2: Version accepted for publication in Duke Math. J

References in corpus (1)

• On homotopy groups of spaces of embeddings of an arc or a circle: the Dax invariant

Cited by in corpus (2)

• Diffeomorphisms of 4-manifolds from graspers
• Homotopy versus isotopy: 2-spheres in 5-manifolds