AI 中文总结
本文将多重截面推广到带边界的高维流形，定义相关图，研究相对纤维化流形的性质，证明边界相对纤维化的5维流形存在相对多重截面，并提出胶合定理。
AI 中文摘要
多重截面（由Ben Aribi、Courte、Golla和Moussard定义）是将闭定向流形分解为模型块的结构，这些模型块是最高维的1-柄体，其子集合沿低维1-柄体相交，整体交集为闭曲面，该概念将Heegaard分裂和三分法推广到更高维。本文将多重截面适配到带边界的紧致流形，把缝合Heegaard分裂和相对三分法推广到所有维度，定义了包含缝合Heegaard图和相对三分图的相关图。相对多重截面会诱导流形边界的特定分解，称为相对纤维化。我们证明，在n>3维时，相对纤维化的连通流形必为球面或圆周与n-1维球面乘积的连通和；证明了边界相对纤维化的紧致5维流形存在相对多重截面；还提出了胶合定理，可将合适的带边界相对多重截面流形组合为多重截面闭流形。
英文摘要
A multisection (as defined by Ben Aribi, Courte, Golla and Moussard) is a decomposition of a closed orientable manifold into model pieces. These model pieces are top-dimensional 1-handlebodies, whose subcollections intersect along 1-handlebodies of lower dimensions, and whose global intersection is a closed surface. This concept extends the notions of Heegaard splittings and trisections to higher dimensions. In this article, we adapt multisections to compact manifolds with boundary, generalizing sutured Heegaard splittings and relative trisections to every dimension. We define the associated diagrams, which encompass sutured Heegaard diagrams and relative trisection diagrams. A relative multisection induces a particular decomposition of the boundary of the manifold, which we call a relative fibration. We show that, in dimension n greater than 3, a connected manifold which relatively fibers is necessarily either a sphere, or a connected sum of copies of the product of the circle with the sphere of dimension n-1. We prove that a compact 5-manifold whose boundary relatively fibers admits a relative multisection. We also state a gluing theorem that allows to combine suitable relatively multisected manifolds with boundary into multisected closed manifolds.
Comments34 pages, 26 figures
