量化同源类
Quantifying Homology Classes
摘要：
我们开发了一种测量同源类的方法。这涉及三个问题。首先，我们使用相对同调的思想定义同调类的大小。其次，我们定义了一个同调群的最优基为元素大小和最小的基。给出了一个计算最优基和最优基中度量类的贪心算法，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.第三，我们讨论了同调类的不同局部化方法，并证明了一些困难的结果。
英文摘要：
We develop a method for measuring homology classes. This involves three problems. First, we define the size of a homology class, using ideas from relative homology. Second, we define an optimal basis of a homology group to be the basis whose elements' size have the minimal sum. We provide a greedy algorithm to compute the optimal basis and measure classes in it. 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. Third, we discuss different ways of localizing homology classes and prove some hardness results.
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