Organized collapse : an introduction to discrete Morse theory / Dmitry N. Kozlov.

Format:

Book

Author/Creator:

Kozlov, D. N. (Dmitriĭ Nikolaevich), author.

Series:

Graduate studies in mathematics ; Volume 207.

Graduate Studies in Mathematics ; Volume 207

Language:

English

Subjects (All):

Combinatorial topology.

Morse theory.

Homology theory.

Homotopy theory.

Physical Description:

1 online resource (339 pages).

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2020.

Language Note:

English

Summary:

Applied topology is a modern subject which emerged in recent years at a crossroads of many methods, all of them topological in nature, which were used in a wide variety of applications in classical mathematics and beyond. Within applied topology, discrete Morse theory came into light as one of the main tools to understand cell complexes arising in different contexts, as well as to reduce the complexity of homology calculations.The present book provides a gentle introduction into this beautiful theory. Using a combinatorial approach-the author emphasizes acyclic matchings as the central object of study. The first two parts of the book can be used as a stand-alone introduction to homology, the last two parts delve into the core of discrete Morse theory. The presentation is broad, ranging from abstract topics, such as formulation of the entire theory using poset maps with small fibers, to heavily computational aspects, providing, for example, a specific algorithm of finding an explicit homology basis starting from an acyclic matching.The book will be appreciated by graduate students in applied topology, students and specialists in computer science and engineering, as well as research mathematicians interested in learning about the subject and applying it in context of their fields.

Contents:

The first steps

Simplicial homology

Beyond the simplicial setting

Category of chain complexes

Chain homotopy

Connecting homomorphism

Singular homology

Cellular homology

Simplicial collapses

Organizing collapsing sequences

Internal collapses and discrete Morse theory

Explicit homology classes associated to critical cells

The critical Morse complex

Implications and variations

Algebraic Morse theory

Discrete Morse theory for posets

Discrete Morse theory for CW complexes

Discrete Morse theory and persistence.

Notes:

Description based on print version record.

Includes bibliographical references and index.

ISBN:

9781470460082

1470460084