4,658 research outputs found
Quantifying Homology Classes
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 time, where is the size of the simplicial
complex and is the Betti number of the homology group. Third, we
discuss different ways of localizing homology classes and prove some hardness
results
Algebraic Topology
The chapter provides an introduction to the basic concepts of Algebraic
Topology with an emphasis on motivation from applications in the physical
sciences. It finishes with a brief review of computational work in algebraic
topology, including persistent homology.Comment: This manuscript will be published as Chapter 5 in Wiley's textbook
\emph{Mathematical Tools for Physicists}, 2nd edition, edited by Michael
Grinfeld from the University of Strathclyd
Persistent Homology of Filtered Covers
We prove an extension to the simplicial Nerve Lemma which establishes
isomorphism of persistent homology groups, in the case where the covering
spaces are filtered. While persistent homology is now widely used in
topological data analysis, the usual Nerve Lemma does not provide isomorphism
of persistent homology groups. Our argument involves some homological algebra:
the key point being that although the maps produced in the standard proof of
the Nerve Lemma do not commute as maps of chain complexes, the maps they induce
on homology do.Comment: 12 pages, no figure
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