度量空间的离散同调论
Discrete Homology Theory for Metric Spaces
摘要：
本文定义并研究了度量空间的离散同调论。而不是单纯的同源性工作，我们的链复形的Lipschitz映射从一个$n$维立方体到一个固定的度量空间。我们证明，由此产生的同调论验证了一个离散的模拟Eilenberg-Steenrod公理，并证明了离散的模拟迈耶-Vietoris的确切序列。此外，这个离散的同调论是相关的离散同伦理论的度量空间通过离散模拟的Hurewicz定理。我们研究了可以作为离散同调群出现的群类，并且在这种情况下，我们证明了光滑、连通、可度量化、紧致流形的基本群同构于流形的“足够精细”矩形化的离散基本群。最后，我们表明，这种离散的同调论可以粗化，导致一个新的非平凡的粗不变的度量空间。
英文摘要：
In this paper we define and study a notion of discrete homology theory for metric spaces. Instead of working with simplicial homology, our chain complexes are given by Lipschitz maps from an $n$-dimensional cube to a fixed metric space. We prove that the resulting homology theory verifies a discrete analogue of the Eilenberg-Steenrod axioms, and prove a discrete analogue of the Mayer-Vietoris exact sequence. Moreover, this discrete homology theory is related to the discrete homotopy theory of a metric space through a discrete analogue of the Hurewicz theorem. We study the class of groups that can arise as discrete homology groups and, in this setting, we prove that the fundamental group of a smooth, connected, metrizable, compact manifold is isomorphic to the discrete fundamental group of a `fine enough' rectangulation of the manifold. Finally, we show that this discrete homology theory can be coarsened, leading to a new non-trivial coarse invariant of a metric space.
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