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URN: urn:nbn:de:0030-drops-13434
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### Quantifying Homology Classes

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### Abstract

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.

### BibTeX - Entry

@InProceedings{chen_et_al:LIPIcs:2008:1343,
author =	{Chao Chen and Daniel Freedman},
title =	{{Quantifying Homology Classes}},
booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
pages =	{169--180},
series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN =	{978-3-939897-06-4},
ISSN =	{1868-8969},
year =	{2008},
volume =	{1},
editor =	{Susanne Albers and Pascal Weil},
publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},