Amalgamations along surfaces with boundary in a handlebody
Siqi Ding, Fengchun Lei, Wei Lin, Andrei Vesnin
TL;DR
This work provides a complete characterization of when amalgamations and self-amalgamations of a handlebody along a surface yield a handlebody. Central to the result is a JD-/primitive-type criterion: under incompressibility of the amalgamating surface and when $F$ is not a disk, the manifold is a handlebody iff there exists a primitive curve set $\mathcal{J}$ on $F$ paired with disks $\mathcal{D}$ in the handlebody such that $|J_i\cap \partial D_j| = \delta_{ij}$ and $p = \operatorname{rank}(\pi_1(F))$. The paper develops a theory for incompressible and compressible surfaces in a handlebody, introduces boundary-compression hierarchies, and handles the annulus case separately. It also introduces the JD-pair framework to enable an induction-based proof of the main theorem and establishes the existence of maximal systems of compression disks. Together, these results advance understanding of when gluing along surfaces preserves the handlebody structure, with implications for Heegaard splittings and fundamental-group generation along the gluing surface.
Abstract
Let M be a connected orientable 3-manifold, and F a compact connected orientable surface properly embedded in M. If F cuts M into two connected 3-manifolds X and Y, that is, M=X \cup_F Y, we say that M is an amalgamation of X and Y along F; and if F cuts M into a connected 3-manifold X, we say that M is a self-amalgamation of X along F. A characterization of an amalgamation of two handlebodies along a surface, incompressible in both, to be a handlebody was obtained by Lei, Liu, Li, and Vesnin. The case of amalgamation of two handelbodies along a compressional surface was studdied by Xu, Fang, and Lei. In the present paper, a characterization of an amalgamation and self-amalgamation of a handlebody to be a handlebody is given.