m-连续性距离
m-Contiguity Distance
AI总结：
本文系统发展了单纯映射间的m-连续性距离，作为simplicial复形中同调复杂度的离散近似框架，构建了逼近连续性距离的递增不变量序列，并建立了其在巴列中心细分、组合和积不等式下的基本性质，定义了m-单纯 Lusternik-Schnirelmann范畴和m-离散拓扑复杂度。
摘要：
在本文中，我们系统地发展了单纯映射之间的m-连续性距离，作为simplicial复形中同调复杂度的离散近似框架。我们构建了一个递增的不变量序列，该序列从下界逼近连续性距离。建立了m-连续性距离的基本性质，包括其在巴列中心细分、组合以及积不等式下的行为。作为该理论的应用，我们定义了m-单纯Lusternik-Schnirelmann范畴和m-离散拓扑复杂度，并证明每个都自然地作为m-连续性距离的特殊情形出现。
英文摘要：
In this paper, we systematically develop the $m$-contiguity distance between simplicial maps as a discrete approximation framework for homotopical complexity in the category of simplicial complexes. We construct an increasing sequence of invariants that approximate the contiguity distance from below. We prove that $m$-contiguity distance is invariant under strong homotopy equivalence and that $m$-contiguity distance coincides with the usual contiguity distance provided that the dimension of the domain simplicial complex is $m$. The fundamental properties of $m$-contiguity distance are established, including its behaviour under barycentric subdivision, under compositions, and a categorical poduct inequality. As applications of this theory, we define the $m$-simplicial Lusternik-Schnirelmann category and the $m$-discrete topological complexity, proving that each arises naturally as a special case of $m$-contiguity distance. We also showed that $SD_1(φ,ψ)=SD(φ,ψ)$ under some conditions related to aspherical spaces.
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