AI 中文总结
本文研究固定有限单纯复形上n-滤子的多参数持续同调逆问题，通过赋予相关空间分层结构证明MPH映射的性质，给出纤维维数上界并推广了已知单参数结果。
AI 中文摘要
有限单纯复形上的向量值函数会产生多参数下的子水平集滤过及对应的持续同调。我们研究固定有限单纯复形上n-滤子的多参数持续同调（MPH）的相关逆问题。我们为滤子空间和本质有限持续模的模空间都赋予了分层结构，其中分层由单位n-立方体的保序同构的自然作用的轨道给出。随后证明MPH映射是等变且强分层的。在像空间的每个分层上，MPH映射限制为一个平凡纤维丛，其纤维是一个多面体复形。我们根据多重分次Betti数给出了纤维维数的上界，作为特例得到了Leygonie和Tillmann（2022）得到的已知单参数界。
英文摘要
Vector-valued functions on finite simplicial complexes give rise to multiparameter sublevel set filtrations and corresponding persistent homology. We study the associated inverse problem for multiparameter persistent homology (MPH) of $n$-filters on a fixed finite simplicial complex. We endow both the space of filters and the moduli space of essentially finite persistence modules with stratifications, where strata are given by orbits of natural actions of order-isomorphisms of the unit $n$-cube. The MPH map is then shown to be equivariant and strongly stratified. Over each stratum in the image, the MPH map restricts to a trivial fiber bundle whose fiber is a polyhedral complex. We provide an upper bound on the dimension of the fibers in terms of multigraded Betti numbers, recovering as a special case the known one-parameter bound obtained by Leygonie and Tillmann (2022).
Comments51 pages, 5 figures
