Quantifying Homology Classes

Chao Chen (1) , Daniel Freedman (1)

1 Department of Computer Science

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.

Keywords: Computational Topology; Computational Geometry; Homology; Persistent Homology; Localization; Optimization

Domains: Computational Geometry [cs.CG]; Discrete Mathematics [cs.DM]

Chao Chen, Daniel Freedman. Quantifying Homology Classes. STACS 2008, Feb 2008, Bordeaux, France. pp.169-180. ⟨hal-00256542⟩