Unknottedness of symplectic submanifold fillings

TL;DR

The paper proves that symplectic fillings of the standard contact sphere are smoothly unknotted for dimensions n≥2, and provides a new proof of an intersection formula for punctured holomorphic curves.

Contribution

It establishes unknottedness of symplectic fillings in higher dimensions and offers a self-contained proof of a key intersection formula in symplectic geometry.

Findings

Symplectic fillings of standard contact spheres are smoothly unknotted for n≥2.

A self-contained proof of the Siefring intersection formula is provided.

The results extend understanding of symplectic topology in higher dimensions.

Abstract

We show that any symplectic filling of the standard contact submanifold $(\mathbb{S}^{2n-1},\xi_{\mathrm{std}})$ of $(\mathbb{S}^{2n+1},\xi_{\mathrm{std}})$ in $(\mathbb{D}^{n+1},\omega_{\mathrm{std}})$ is smoothly unknotted if $n\ge 2$. We also give a self-contained proof of the Siefring intersection formula between punctured holomorphic curves and holomorphic hypersurfaces used in the proof using the $L$-simple setup of Bao-Honda.