• Corpus ID: 118417737

Persistent Homology of Filtered Covers

@article{Fraser2012PersistentHO,
  title={Persistent Homology of Filtered Covers},
  author={Maia Fraser},
  journal={arXiv: Algebraic Topology},
  year={2012}
}
  • M. Fraser
  • Published 28 February 2012
  • Mathematics
  • arXiv: Algebraic Topology
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… 
2 Citations

Local homology of abstract simplicial complexes

This survey describes some useful properties of the local homology of abstract simplicial complexes. Although the existing literature on local homology is somewhat dispersed, it is largely dedicated

Contact spectral invariants and persistence

This sketch shows that the usual generating function based capacities have an interpretation in the language of persistent homology as persistences of certain homology classes in the persistence

References

SHOWING 1-10 OF 20 REFERENCES

Computing Persistent Homology

The analysis establishes the existence of a simple description of persistent homology groups over arbitrary fields and derives an algorithm for computing individual persistent homological groups over an arbitrary principal ideal domain in any dimension.

Stability of Persistence Diagrams

The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane. We prove that under mild assumptions on the function, the persistence diagram

Weak feature size and persistent homology: computing homology of solids in Rn from noisy data samples

It is proved that under quite general assumptions one can deduce the topology of a bounded open set in Rn from a Hausdorff distance approximation of it and the weak feature size (wfs) is introduced that generalizes the notion of local feature size.

Proximity of persistence modules and their diagrams

This paper presents new stability results that do not suffer from the restrictions of existing stability results, and makes it possible to compare the persistence diagrams of functions defined over different spaces, thus enabling a variety of new applications of the concept of persistence.

Algebraic Topology

The focus of this paper is a proof of the Nielsen-Schreier Theorem, stating that every subgroup of a free group is free, using tools from algebraic topology.

The union of balls and its dual shape

Efficient algorithms are described for computing topological, combinatorial, and metric properties of the union of finitely many spherical balls in ℝd. These algorithms are based on a simplicial

Persistent Homology — a Survey

Persistent homology is an algebraic tool for measuring topological features of shapes and functions. It casts the multi-scale organization we frequently observe in nature into a mathematical

Topological Persistence and Simplification

Fast algorithms for computing persistence and experimental evidence for their speed and utility are given for topological simplification within the framework of a filtration, which is the history of a growing complex.

Computing and comprehending topology: persistence and hierarchical morse complexes

The authors begin by extending the concept of addition to binary operations, which is based on abstracting from algebra its core properties, and studying algebra in terms of those properties.

Topological estimation using witness complexes

This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object, and produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology.