Stable equivalence of handlebody decompositions whose partitions are multibranched surfaces

In this paper, we consider decompositions of closed orientable 3-manifolds with more than 3 handlebodies, where the union of intersections of handlebodies is a multibranched surface. We define stabilization operations for such decompositions and show the stable equivalence. & COPY; 2023 Elsevier B.V. All rights reserved.