Univalent Foundations Program, 2013. — 589 p.
The present work has its origins in our collective attempts to develop a new style of informal type theory that can be read and understood by a human being, as a complement to a formal proof that can be checked by a machine. Univalent foundations is closely tied to the idea of a foundation of mathematics that can be implemented in a...
New York: Springer, 2002. — 510 p. The purpose of this book is to introduce algebraic topology using the novel approach of homotopy theory, an approach with clear applications in algebraic geometry as understood by Lawson and Voevodsky. This method allows the authors to cover the material more efficiently than the more common method using homological algebra. The basic concepts...
Springer, 2011. — 359 p. — (Universitext). — ISBN: 1441973281. This is a book in pure mathematics dealing with homotopy theory, one of the main branches of algebraic topology. The principal topics are as follows: Basic homotopy; H-spaces and co-H-spaces; Fibrations and cofibrations; Exact sequences of homotopy sets, actions, and coactions; Homotopy pushouts and pullbacks;...
The Univalent Foundations Program, 2013. — xii, 475 p. Each of the above-named individuals contributed something to the Special Year — and so to this book — in the form of ideas, words, or deeds. The spirit of collaboration that prevailed throughout the year was truly extraordinary. Homotopy type theory is a new branch of mathematics that combines aspects of several different...
Cambridge University Press, 2020. — 431 p. — ISBN: 978-1-108-48278-3. The beginning graduate student in homotopy theory is confronted with a vast literature on spectra that is scattered across books, articles and decades. There is much folklore but very few easy entry points. This comprehensive introduction to stable homotopy theory changes that. It presents the foundations of...
Cambridge University Press, 2008. — 486 p. — (Cambridge Studies in Advanced Mathematics 15). — ISBN-13 9780521055314. This book gives a general outlook on homotopy theory; fundamental concepts, such as homotopy groups and spectral sequences, are developed from a few axioms and are thus available in a broad variety of contexts. Many examples and applications in topology and...
Springer, 2020. — 248 p. — (Lecture Notes in Mathematics 2269). — ISBN: 978-3-030-51334-4. Providing a new approach to assembly maps, this book develops the foundations of coarse homotopy using the language of infinity categories. It introduces the category of bornological coarse spaces and the notion of a coarse homology theory, and further constructs the universal coarse...
New York : Springer, 1973. — 114 p. — ISBN: 0387900551. This book contains a semi-historical and geometrically motivated exposition of J. H. C. Witehead'sl theory of simple-homotopy types. Homotopy equivalence Whitehead's combinatorial approach to homotopy theory CW complexes A Geometric Approach to Homotopy Theory Formal deformations Mappings Cylinders and deformations The...
World Scientific Publishing, 2018. — 446 p. — (Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore: Volume 35). — ISBN: 978-981-3226-56-2. This volume consists of introductory lectures on the topics in the new and rapidly developing area of toric homotopy theory, and its applications to the current research in configuration spaces and...
Springer-Verlag, 1976. - 302 p. ISBN: 0387078630.
The Cech construction associates to a topological space an inverse system of CW-complexes. A similar construction in algebraic geometry leads to tale homotopy theory. The Postnikov and pro finite completion constructions also associate inverse systems of complexes to complexes.
Berlin, Germany: Walter de Gruyter, 2007. — 315 p. — (de Gruyter Series in Nonlinear Analysis and Applications 11). — ISBN: 3110189429. This monograph provides a thorough treatment of parameter-dependent extremal problems with local minimum values that remain unchanged under changes of the parameter. The authors consider the theory as well the practical treatment of those...
2nd ed. — Springer, 2016. — xii + 627 p. This textbook on algebraic topology updates a popular textbook from the golden era of the Moscow school of I. M. Gelfand. The first English translation, done many decades ago, remains very much in demand, although it has been long out-of-print and is difficult to obtain. Therefore, this updated English edition will be much welcomed by the...
Springer, 1967. — 168 p. Introduction. Categories of Fractions. Simplicial Sets. Geometric Realization of Simplicial Sets. The Homotopic Category. Exact Sequences of Algebraic Topology. Exact Sequences of the Homotopic Category. Combinatorial Description of Topological Spaces. Appendix I: Coverings. Appendix II: The Homology Groups of a Simplicial Set.
Oxford University Press, 2019. — 546 p. The book is an introduction to elementary algebraic topology for those with an interest in computers and computer programming. It expertly illustrates how the basics of the subject can be implemented on a computer through its focus on fully-worked examples designed to develop problem solving techniques. The transition from basic theory to...
2nd Edition. — Springer, 2013. — 228 p. — (Progress in Mathematics 16). — ISBN: 978-1-4614-8467-7. This completely revised and corrected version of the well-known Florence notes circulated by the authors together with E. Friedlander examines basic topology, emphasizing homotopy theory. Included is a discussion of Postnikov towers and rational homotopy theory. This is then...
Cambridge: Cambridge University Press, 2021. — 880 p. The long-standing Kervaire invariant problem in homotopy theory arose from geometric and differential topology in the 1960s and was quickly recognised as one of the most important problems in the field. In 2009 the authors of this book announced a solution to the problem, which was published to wide acclaim in a landmark...
Springer, 2015. — 508 p. — ISBN10: 1493922998 This monograph on the homotopy theory of topologized diagrams of spaces and spectra gives an expert account of a subject at the foundation of motivic homotopy theory and the theory of topological modular forms in stable homotopy theory. Beginning with an introduction to the homotopy theory of simplicial sets and topos theory, the...
Dordrecht: Springer. – 2006. – 325 p. The notion of a fixed point plays a crucial role in numerous branches of mathematics and its applications. Information about the existence of such points is often the crucial argument in solving a problem. In particular, topological methods of fixed point theory have been an increasing focus of interest over the last century. This book is an...
The University of Chicago Press, 161 p. — ISBN 0-226-51180-4. Simplicial sets are discrete analogs of topological spaces. They have played a central role in algebraic topology ever since their introduction in the late 1940s, and they also play an important role in other areas such as geometric topology and algebraic geometry. On a formal level, the homotopy theory of simplicial...
Boca Raton: CRC Press, 2019. — 990 p. — (CRC Press/Chapman and Hall Handbooks in Mathematics Series). — ISBN: 0815369700. The Handbook of Homotopy Theory provides a panoramic view of an active area in mathematics that is currently seeing dramatic solutions to long-standing open problems , and is proving itself of increasing importance across many other mathematical disciplines....
Harper and Row Publishers, 1968. — 214 p. Introduction to cohomology operations. Construction of the steenrod squares. Properties of the squares. Application: the hopf invariant. Application: vector fields on spheres. The steenrod algebra. Exact couples and spectral sequences. Fibre spaces. Cohomology of k(\pi,n). Classes of abellAN Groups. More about fiber spaces....
Springer-Verlag, 1977. — 268 p. — (Lecture Notes in Mathematics). Introduction l functors Coordinate-free spectra Orientation Theory E_infinity ring spectra On kO-oriented bundle theories E_infinity ring spaces and bipermutative categories The recognition principle for E_infinity ring spaces Algebraic and topological K-Theory Pairings in infinite loop space theory
Graylock Press, 1952. — 99 p. — ISBN: 0486406857. Hailed by The Mathematical Gazette as "an extremely valuable addition to the literature of algebraic topology," this concise but rigorous introductory treatment focuses on applications to dimension theory and fixed-point theorems. The lucid text examines complexes and their Betti groups, including Euclidean space, application to...
American Mathematical Society, 2011. — 835 p. The Language of Categories Categories and Functors Limits and Colimits Semi-Formal Homotopy Theory Categories of Spaces Homotopy Cofibrations and Fibrations Homotopy Limits and Colimits Homotopy Pushout and Pullback Squares Tools and Techniques Topics and Examples Model Categories Four Topological Inputs The Concept of Dimension in...
Singapore: World Scientific, 2021. — 337 p. This book is an introduction to fiber bundles and fibrations. But the ultimate goal is to make the reader feel comfortable with basic ideas in homotopy theory. The author found that the classification of principal fiber bundles is an ideal motivation for this purpose. The notion of homotopy appears naturally in the classification....
Springer, 1978. — 764 p. As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory. It is assumed that the reader is familiar with the fundamental group and with singular homology theory, including the Universal Coefficient and Kiinneth Theorems. Some acquaintance with manifolds and Poincare duality is desirable, but not essential.
Springer, 2010. — 142 p. — (Frontiers in Mathematics). — ISBN: 978-3-0346-0564-9. Homotopy theory and C*-algebras are central topics in contemporary mathematics. This book introduces a modern homotopy theory for C*-algebras. One basic idea of the setup is to merge C*-algebras and spaces studied in algebraic topology into one category comprising C*-spaces. These objects are...
Пер. с англ. под ред. Д. Каледина, с добавлениями А. Хаттори и В. М. Бухштабера. — М.: Московский центр непрерывного математического образования (МЦНМО), 2014. — 432 с. — ISBN: 978-5-4439-2058-0. Эта книга –– классический, стандартный в англоязычном мире учебник алгебраической топологии, наконец-то переведенный на русский язык. Тема книги –– стабильная гомотопическая теория,...
Перевод и редактирование Г. Чернышёва. 2013. — 502 с. Изложены основы нового, унивалентного, направления в основаниях математики. Часть I содержит относительно неформальное описание гомотопической теории типов. В части II и приложении, содержащих в основном, не связанные между собой темы, описывается то, что унивалентные основания предлагают по конкретной теме.
М.: Наука, 1976. — 176 с. В книге выдающегося советского математика Л. С. Понтрягина изложена теория гладких многообразий в связи с гомотопической классификацией непрерывных отображений. Значение этой монографии как классического труда, лежащего в основе целого направления, очень велико.
Пер. с англ. — М.: Наука. Главная редакция физико-математической литературы, 1985.— 608 с. Книга содержит описание, по существу, всех основных задач, методов и результатов современной алгебраической топологии. Принятый автором способ изложения позволил ему при сравнительно небольшом объеме познакомить читателя с обширным материалом, до сих пор не включавшимся в учебную...
М.: Наука, 1982. — 224 с.
Книга принадлежит перу знаменитого японского тополога Хироси Тода. Она содержит вычисление групп с явным построением образующих и с описанием соответствующей части композиционного умножения. Книга может служить пособием для изучения классических методов гомотопической топологии и является незаменимым справочником по гомотопическим группам сфер....
М.: Наука, Главная редакция физико-математической литературы, 1989. — 528 с. — ISBN 978-5-02-013929-7. Первый в отечественной литературе учебный курс гомотопической топологии и ее многочисленных приложений. Среди основных тем, затронутых в книге: теория клеточных комплексов, гомотопические группы, гомологии и когомологии, метод спектральных последовательностей, гомотопические...
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