度量和定位同调类
Measuring and Localing Homology Classes
摘要：
我们开发了一种方法来测量和定位同源类。这涉及两个问题。首先，我们定义了相关的概念的大小为一个同源类和一个同源群的基础上，使用相对同源的想法。其次，我们提出了一个算法来计算最佳的同调基，使用持久同调和有限域代数的技术。类的计算的最佳基础是本地化的循环传达他们的大小。该算法在$O(β^4 n^3 \log^2 n)$时间内运行，其中$n$是单纯复形的大小，$β$是同调群的Betti数。
英文摘要：
We develop a method for measuring and localizing homology classes. This involves two problems. First, we define relevant notions of size for both a homology class and a homology group basis, using ideas from relative homology. Second, we propose an algorithm to compute the optimal homology basis, using techniques from persistent homology and finite field algebra. Classes of the computed optimal basis are localized with cycles conveying their sizes. The algorithm runs in $O(β^4 n^3 \log^2 n)$ time, where $n$ is the size of the simplicial complex and $β$ is the Betti number of the homology group.
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