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Elements of Algebraic Topology
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Elements of Algebraic Topology provides the most concrete approach to the subject with coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorem of point-set topology.Abstract:
Elements of Algebraic Topology provides the most concrete approach to the subject. With coverage of homology and cohomology theory, universal coefficient theorems, Kunneth theorem, duality in manifolds, and applications to classical theorems of point-set topology, this book is perfect for comunicating complex topics and the fun nature of algebraic topology for beginners.read more
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Introduction to Smooth Manifolds
TL;DR: In this paper, a review of topology, linear algebra, algebraic geometry, and differential equations is presented, along with an overview of the de Rham Theorem and its application in calculus.
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Topological Persistence and Simplification
Edelsbrunner,Letscher,Zomorodian +2 more
TL;DR: 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.
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Computing Persistent Homology
TL;DR: In this article, it was shown that the persistent homology of a filtered d-dimensional simplicial complex is simply the standard homology over a polynomial ring of a particular graded module.
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Stability of Persistence Diagrams
TL;DR: The persistence diagram of a real-valued function on a topological space is a multiset of points in the extended plane and it is proved that under mild assumptions on the function, the persistence diagram is stable.
Proceedings ArticleDOI
Topological persistence and simplification
TL;DR: A notion of topological simplification is formalized within the framework of a filtration, which is the history of a growing complex, and a topological change that happens during growth is classified as either a feature or noise, depending on its life-time or persistence within the filTration.