Here, we introduce a new definition of regular point for piecewise-linear (PL) functions on combinatorial (PL triangulated) manifolds. This definition is given in terms of the restriction of the function to the link of the point. We show that our definition of regularity is distinct from other definitions that exist in the combinatorial topology literature. Next, we stratify the Jacobi set/critical locus of such a map as a poset stratified space. As an application, we consider the Reeb space of a PL function, stratify the Reeb space as well as the target of the function, and show that the Stein factorization is a map of stratified spaces.

翻译：本文针对组合（PL三角剖分）流形上的分段线性（PL）函数，引入了一种新的正则点定义。该定义基于函数在点链环上的限制。我们证明，本文提出的正则性定义与组合拓扑学文献中已有的其他定义存在区别。进一步地，我们将此类映射的雅可比集/临界轨迹分化为偏序集分层空间。作为应用，我们考察PL函数的瑞布空间，对该瑞布空间及函数目标空间进行分层，并证明斯坦因分解是分层空间之间的映射。