Summary: 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 \(O(\beta^4 n^3 \log^2 n)\) time, where \(n\) is the size of the simplicial complex and \(\beta\) is the Betti number of the homology group. Third, we discuss different ways of localizing homology classes and prove some hardness results.
For the entire collection see [Zbl 1213.68020].