Computational Homological Algebra
- 2026
- Book
- Author
- Michael Robinson
- Book Series
- Mathematical Engineering
- Publisher
- Springer Nature Switzerland
About this book
This book is an attempt to reduce the barrier to entry for the key tools of homological algebra and develops the basic notions of homological algebra by emphasizing concrete, elementary, and computational examples in finite dimensional vector spaces. Linear algebra is the study of linear maps between vector spaces.
The broad success of linear algebra in applications is due to the dimension theorem and the algorithms that exploit it, like Gaussian elimination and QR factorizations.
Homological algebra is the study of what happens when linear maps are chained together, one after the next.
Unlike linear algebra, homological algebra is little known outside of mathematics, but is poised to become useful in engineering and data science.
The material covered in this book can be used for a one semester elementary course in computational homological algebra, but could also comfortably occupy a two-semester sequence.
This book is written for mid-division undergraduate students who have a solid background in linear algebra, but no background in abstract algebra, topology, or category theory.
Instead readers build insight by computation.
By working the examples and exercises, the requisite background material is covered as needed, and the powerful tools of homological algebra are unlocked.
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Table of Contents
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Frontmatter
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Chapter 1. Quotients of Vector Spaces
Michael RobinsonThis chapter delves into the fundamentals of vector space quotients, their properties, and the algorithms for computing them. It begins by defining vector space quotients and exploring their properties through calculations, culminating in the dimension theorem. The chapter then examines how quotients interact with linear maps and how maps can be induced. Additionally, it introduces the dual space and recasts the dimension theorem using it to obtain a more algorithmically useful version. The chapter concludes with an introduction to categories and their role in linear algebra. Throughout, the chapter emphasizes the practical applications of these concepts, providing a comprehensive overview of the use of vector space quotients in linear algebra.AI Generated
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AbstractThis first chapter introduces vector space quotients, and develops both theoretical properties and computational algorithms for working with them. Special emphasis will be placed on effective algorithms for computing the quotient of two vector spaces, because this task will be done many times in the book. In the latter chapters, vector space quotients allow the interactions between linear maps in a sequence to be disentangled. While the first few sections are likely to be review material, they are essential for setting the stage since they establish basic notation and computational conventions that will be used throughout the book. The centerpiece of this chapter is the dimension theorem, which characterizes a single linear map and is proven with a constructive, algorithmic proof. -
Chapter 2. Sequences and Chain Complexes
Michael RobinsonThis chapter delves into the study of sequences of linear maps, focusing on their structure and invariants. It begins by defining sequences and chain complexes, and then explores the barcode decomposition, which is a method for characterizing sequences by their barcode diagrams. The chapter also introduces the concept of homology, which measures how far a chain complex is from being exact. Practical algorithms for computing homology and barcode decomposition are provided, along with examples and exercises to illustrate the concepts. The chapter concludes with a discussion of the Euler characteristic and its relation to homology. Readers will gain a deep understanding of these topics and learn how to apply them in their own research.AI Generated
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AbstractOur study of homological algebra begins in earnest in this chapter, by studying sequences of linear maps. In this chapter, we prove the barcode decomposition theorem, which is the first of two theorems for sequences that are analogous to the dimension theorem. In the latter half of the chapter, we turn our attention to chain complexes, which are sequences that play a role in many homological applications. Our study of chain complexes is preparation for the Escolar-Hiraoka decomposition theorem, proven in the next chapter. Along the way, we develop algorithms for computing the barcode decomposition and its properties. -
Chapter 3. Chain Maps
Michael RobinsonThis chapter delves into the Escolar-Hiraoka decomposition, a powerful tool for understanding chain maps in linear algebra. The decomposition simplifies proofs and leads to Algorithm 1, which completely characterizes chain maps. The chapter explores two equivalences between chain complexes—quasi-isomorphism and chain homotopy equivalence—and their implications in homology. It also discusses the Snake Lemma, which relates exact sequences of sequences to homology. The chapter provides detailed examples and exercises to illustrate these concepts, making it a comprehensive guide for professionals in mathematical research and data analysis.AI Generated
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AbstractThis chapter is the fulcrum against which the force of linear algebra is magnified into a powerful tool for understanding chain maps. The Escolar-Hiraoka decomposition theorem enables us to reason about chain maps in terms of their canonical decomposition into 10 different kinds of simpler chain maps. Not only does this dramatically simplify proofs of useful properties, it completely characterizes chain maps. With the Escolar-Hiraoka decomposition in hand, we learn a few surprising facts about chain complexes of vector spaces. There are two—apparently different—equivalences between chain complexes, quasi-isomorphism and chain homotopy equivalence. While these two equivalences are quite different in general for chain complexes of vector spaces, they agree. In a roundabout way, these two equivalences are also related to exact sequences of sequences. If one starts with a three-term exact sequence of chain complexes (instead of vector spaces), one can derive a much longer exact sequence involving the homology of these three chain complexes, a fact called the Snake Lemma. -
Chapter 4. Abstract Simplicial Complexes
Michael RobinsonThis chapter delves into the concept of abstract simplicial complexes, which are used to model abstract spaces with rich structural properties. It begins by defining abstract simplicial complexes and exploring their key combinatorial properties, such as the star, closure, and link of a simplex. The chapter then discusses how to construct abstract simplicial complexes from other combinatorial objects and defines the category of abstract simplicial complexes. A significant focus is on simplicial chain complexes, which generalize the Euler characteristic and provide algebraic summaries of the structure of simplicial complexes. The chapter also covers simplicial maps, which are transformations that preserve the structure of simplicial complexes, and operations on simplicial complexes such as coning and suspending. The text includes numerous examples and exercises to illustrate these concepts, making it a comprehensive guide for professionals interested in the intersection of algebra and topology. The chapter concludes with a discussion on the simplicial chain complex as a functor, demonstrating how it preserves the structure of simplicial complexes through induced chain maps. This detailed exploration provides a solid foundation for understanding the role of abstract simplicial complexes in modeling and analyzing complex spatial structures.AI Generated
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AbstractChain complexes were not conceived on their own, but were discovered in the course of exploring various abstract models of space. An abstract simplicial complex is a representation of a space that is both structurally rich yet requires few axioms to describe. Many of these properties are elementary to state and lead to useful computational algorithms.This chapter forges the connection between abstract simplicial complexes and the algebraic constructions of the previous chapter. Given an abstract simplicial complex, one can construct its simplicial chain complex and compute its simplicial homology as algebraic summaries of its structure. -
Chapter 5. Simplicial Homology and Homotopy
Michael RobinsonThis chapter delves into the fascinating world of simplicial homology and homotopy, exploring how these concepts capture the topological features of simplicial complexes. The text begins by defining simplicial homology as a natural consequence of the previous chapters, exploring its properties and how it measures the presence of certain structures in a simplicial complex. It then introduces the concept of relative simplicial homology and elementary simplicial homotopies, which provide concrete examples of chain homotopies induced by simplicial maps. The chapter also discusses the impact of subdividing a simplicial complex on its homology and how to compose subdivision with simplicial maps. Furthermore, it explores the duality of simplicial homology to obtain simplicial cohomology, which leads to additional algebraic structure. The text concludes with a discussion on persistent homology and its applications in topological data analysis. Throughout the chapter, the theory of chain complexes developed in the previous chapters is utilized to provide a comprehensive understanding of these concepts. The chapter is filled with examples, exercises, and propositions that illustrate the theoretical concepts and their practical applications. It also includes diagrams and figures that visually represent the geometric and algebraic structures discussed. The chapter is a valuable resource for professionals seeking to deepen their understanding of simplicial homology and homotopy and their role in topological data analysis.AI Generated
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AbstractGiven that every simplicial complex can be transformed into its simplicial chain complex in a structure-preserving way, it is natural to explore the meaning of the homology of the simplicial chain complex in terms of the original simplicial complex. It happens that the simplicial homology captures certain topological features, which are those properties that are invariant under simplicial maps. Moreover, there is a notion of simplicial homotopy that is related to the chain homotopies in Chap. 3, which ultimately provides a concrete interpretation of the meaning of simplicial homology. -
Chapter 6. Sequences and Chain Complexes of Sequences
Michael RobinsonThis chapter delves into the fascinating realm of sequences and chain complexes within homological algebra. It begins by explaining how homology theories can be applied to various algebraic objects, including modules, which are generalizations of vector spaces. The text introduces the concept of sequences of sequences, which serve as concrete examples of modules, and explores their properties and structures. A significant portion of the chapter is dedicated to the study of chain complexes of sequences, including their homology, quasi-isomorphisms, and the construction of projective and injective resolutions. The chapter also covers double complexes, which are chain complexes of chain complexes, and discusses their unique properties and applications. Throughout the text, numerous examples and exercises are provided to illustrate key concepts and reinforce understanding. The chapter concludes with a brief treatment of derived functors, which generalize the idea of an induced map on homology. This comprehensive exploration of sequences and chain complexes offers valuable insights into the broader field of homological algebra and its practical applications.AI Generated
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AbstractThe business of homological algebra is to ascribe a homology theory to algebraic objects, including those that may not initially be amenable to such a theory. For instance, one may ascribe a homology theory to a module, which is a generalization of a vector space. This book does not assume familiarity with module theory, but one can still explore the beginnings of the theory by instead studying sequences of sequences. Sequences of sequences are strictly richer than sequences of vector spaces but provide enough concrete structure to reveal the outlines of the general theory. To study a given module homologically, one starts by constructing resolutions, which are exact sequences that either start or end at the given module of interest. The book ends by using these resolutions to understand homology in this generalized setting, exploring the notion of the derived category and derived functors. -
Backmatter
- Title
- Computational Homological Algebra
- Author
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Michael Robinson
- Copyright Year
- 2026
- Publisher
- Springer Nature Switzerland
- Electronic ISBN
- 978-3-032-08634-1
- Print ISBN
- 978-3-032-08633-4
- DOI
- https://doi.org/10.1007/978-3-032-08634-1
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