**Author**: George W. Whitehead

**Publisher:** Springer Science & Business Media

**ISBN:** 1461263182

**Category:** Mathematics

**Page:** 746

**View:** 7557

Skip to content
# Free eBooks PDF

## Elements of Homotopy Theory

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. Anyone who has taught a course in algebraic topology is familiar with the fact that a formidable amount of technical machinery must be introduced and mastered before the simplest applications can be made. This phenomenon is also observable in the more advanced parts of the subject. I have attempted to short-circuit it by making maximal use of elementary methods. This approach entails a leisurely exposition in which brevity and perhaps elegance are sacrificed in favor of concreteness and ease of application. It is my hope that this approach will make homotopy theory accessible to workers in a wide range of other subjects-subjects in which its impact is beginning to be felt. It is a consequence of this approach that the order of development is to a certain extent historical. Indeed, if the order in which the results presented here does not strictly correspond to that in which they were discovered, it nevertheless does correspond to an order in which they might have been discovered had those of us who were working in the area been a little more perspicacious.
## Modern Classical Homotopy Theory

The core of classical homotopy theory is a body of ideas and theorems that emerged in the 1950s and was later largely codified in the notion of a model category. This core includes the notions of fibration and cofibration; CW complexes; long fiber and cofiber sequences; loop spaces and suspensions; and so on. Brown's representability theorems show that homology and cohomology are also contained in classical homotopy theory. This text develops classical homotopy theory from a modern point of view, meaning that the exposition is informed by the theory of model categories and that homotopy limits and colimits play central roles. The exposition is guided by the principle that it is generally preferable to prove topological results using topology (rather than algebra). The language and basic theory of homotopy limits and colimits make it possible to penetrate deep into the subject with just the rudiments of algebra. The text does reach advanced territory, including the Steenrod algebra, Bott periodicity, localization, the Exponent Theorem of Cohen, Moore, and Neisendorfer, and Miller's Theorem on the Sullivan Conjecture. Thus the reader is given the tools needed to understand and participate in research at (part of) the current frontier of homotopy theory. Proofs are not provided outright. Rather, they are presented in the form of directed problem sets. To the expert, these read as terse proofs; to novices they are challenges that draw them in and help them to thoroughly understand the arguments.
## Rational Homotopy Theory

Rational homotopy theory is a subfield of algebraic topology. Written by three authorities in the field, this book contains all the main theorems of the field with complete proofs. As both notation and techniques of rational homotopy theory have been considerably simplified, the book presents modern elementary proofs for many results that were proven ten or fifteen years ago.
## Introduction to Homotopy Theory

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; classical theorems, including those of Serre, Hurewicz, Blakers-Massey, and Whitehead; homotopy Sets; homotopy and homology decompositions of spaces and maps; and obstruction theory. The underlying theme of the entire book is the Eckmann-Hilton duality theory. The book can be used as a text for the second semester of an advanced ungraduate or graduate algebraic topology course.
## Algebraic Topology

To the Teacher. This book is designed to introduce a student to some of the important ideas of algebraic topology by emphasizing the re lations of these ideas with other areas of mathematics. Rather than choosing one point of view of modem topology (homotopy theory, simplicial complexes, singular theory, axiomatic homology, differ ential topology, etc.), we concentrate our attention on concrete prob lems in low dimensions, introducing only as much algebraic machin ery as necessary for the problems we meet. This makes it possible to see a wider variety of important features of the subject than is usual in a beginning text. The book is designed for students of mathematics or science who are not aiming to become practicing algebraic topol ogists-without, we hope, discouraging budding topologists. We also feel that this approach is in better harmony with the historical devel opment of the subject. What would we like a student to know after a first course in to pology (assuming we reject the answer: half of what one would like the student to know after a second course in topology)? Our answers to this have guided the choice of material, which includes: under standing the relation between homology and integration, first on plane domains, later on Riemann surfaces and in higher dimensions; wind ing numbers and degrees of mappings, fixed-point theorems; appli cations such as the Jordan curve theorem, invariance of domain; in dices of vector fields and Euler characteristics; fundamental groups
## Elements of Homology Theory

The book is a continuation of the previous book by the author (Elements of Combinatorial and Differential Topology, Graduate Studies in Mathematics, Volume 74, American Mathematical Society, 2006). It starts with the definition of simplicial homology and cohomology, with many examples and applications. Then the Kolmogorov-Alexander multiplication in cohomology is introduced. A significant part of the book is devoted to applications of simplicial homology and cohomology to obstruction theory, in particular, to characteristic classes of vector bundles. The later chapters are concerned with singular homology and cohomology, and Cech and de Rham cohomology. The book ends with various applications of homology to the topology of manifolds, some of which might be of interest to experts in the area. The book contains many problems; almost all of them are provided with hints or complete solutions.
## Classical Topology and Combinatorial Group Theory

In recent years, many students have been introduced to topology in high school mathematics. Having met the Mobius band, the seven bridges of Konigsberg, Euler's polyhedron formula, and knots, the student is led to expect that these picturesque ideas will come to full flower in university topology courses. What a disappointment "undergraduate topology" proves to be! In most institutions it is either a service course for analysts, on abstract spaces, or else an introduction to homological algebra in which the only geometric activity is the completion of commutative diagrams. Pictures are kept to a minimum, and at the end the student still does nr~ understand the simplest topological facts, such as the rcason why knots exist. In my opinion, a well-balanced introduction to topology should stress its intuitive geometric aspect, while admitting the legitimate interest that analysts and algebraists have in the subject. At any rate, this is the aim of the present book. In support of this view, I have followed the historical development where practicable, since it clearly shows the influence of geometric thought at all stages. This is not to claim that topology received its main impetus from geometric recreations like the seven bridges; rather, it resulted from the l'isualization of problems from other parts of mathematics-complex analysis (Riemann), mechanics (Poincare), and group theory (Dehn). It is these connec tions to other parts of mathematics which make topology an important as well as a beautiful subject.
## Differential Forms in Algebraic Topology

Developed from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes. By using the de Rham theory of differential forms as a prototype of cohomology, the machineries of algebraic topology are made easier to assimilate. With its stress on concreteness, motivation, and readability, this book is equally suitable for self-study and as a one-semester course in topology.
## Topology and Geometry

This book offers an introductory course in algebraic topology. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. From the reviews: "An interesting and original graduate text in topology and geometry...a good lecturer can use this text to create a fine course....A beginning graduate student can use this text to learn a great deal of mathematics."—-MATHEMATICAL REVIEWS
## The Homotopy Theory of (∞,1)-Categories

The notion of an (∞,1)-category has become widely used in homotopy theory, category theory, and in a number of applications. There are many different approaches to this structure, all of them equivalent, and each with its corresponding homotopy theory. This book provides a relatively self-contained source of the definitions of the different models, the model structure (homotopy theory) of each, and the equivalences between the models. While most of the current literature focusses on how to extend category theory in this context, and centers in particular on the quasi-category model, this book offers a balanced treatment of the appropriate model structures for simplicial categories, Segal categories, complete Segal spaces, quasi-categories, and relative categories, all from a homotopy-theoretic perspective. Introductory chapters provide background in both homotopy and category theory and contain many references to the literature, thus making the book accessible to graduates and to researchers in related areas.
## Homology Theory

This book is designed to be an introduction to some of the basic ideas in the field of algebraic topology. In particular, it is devoted to the foundations and applications of homology theory. The only prerequisite for the student is a basic knowledge of abelian groups and point set topology. The essentials of singular homology are given in the first chapter, along with some of the most important applications. In this way the student can quickly see the importance of the material. The successive topics include attaching spaces, finite CW complexes, the Eilenberg-Steenrod axioms, cohomology products, manifolds, Poincare duality, and fixed point theory. Throughout the book, the approach is as illustrative as possible, with numerous examples and diagrams. Extremes of generality are sacrificed when they are likely to obscure the essential concepts involved. The book is intended to be easily read by students as a textbook for a course or as a source for individual study. This second edition has been expanded to include a new chapter on covering spaces, as well as additional illuminating exercises. The conceptual approach is again used to show how lifting problems give rise to the fundamental group and its properties.
## Homotopy Theory: An Introduction to Algebraic Topology

Homotopy Theory: An Introduction to Algebraic Topology
## Cohomology of Groups

Aimed at second year graduate students, this text introduces them to cohomology theory (involving a rich interplay between algebra and topology) with a minimum of prerequisites. No homological algebra is assumed beyond what is normally learned in a first course in algebraic topology, and the basics of the subject, as well as exercises, are given prior to discussion of more specialized topics.
## Simplicial Homotopy Theory

## Fundamentals of Algebraic Topology

This rapid and concise presentation of the essential ideas and results of algebraic topology follows the axiomatic foundations pioneered by Eilenberg and Steenrod. The approach of the book is pragmatic: while most proofs are given, those that are particularly long or technical are omitted, and results are stated in a form that emphasizes practical use over maximal generality. Moreover, to better reveal the logical structure of the subject, the separate roles of algebra and topology are illuminated. Assuming a background in point-set topology, Fundamentals of Algebraic Topology covers the canon of a first-year graduate course in algebraic topology: the fundamental group and covering spaces, homology and cohomology, CW complexes and manifolds, and a short introduction to homotopy theory. Readers wishing to deepen their knowledge of algebraic topology beyond the fundamentals are guided by a short but carefully annotated bibliography.
## Homotopical Topology

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 mathematical community. Distinctive features of this book include: a concise but fully rigorous presentation, supplemented by a plethora of illustrations of a high technical and artistic caliber; a huge number of nontrivial examples and computations done in detail; a deeper and broader treatment of topics in comparison to most beginning books on algebraic topology; an extensive, and very concrete, treatment of the machinery of spectral sequences. The second edition contains an entirely new chapter on K-theory and the Riemann-Roch theorem (after Hirzebruch and Grothendieck).
## Metric Structures in Differential Geometry

This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.
## Differential Topology

"A very valuable book. In little over 200 pages, it presents a well-organized and surprisingly comprehensive treatment of most of the basic material in differential topology, as far as is accessible without the methods of algebraic topology....There is an abundance of exercises, which supply many beautiful examples and much interesting additional information, and help the reader to become thoroughly familiar with the material of the main text." —MATHEMATICAL REVIEWS
## Algebraic Topology

This book surveys the fundamental ideas of algebraic topology. The first part covers the fundamental group, its definition and application in the study of covering spaces. The second part turns to homology theory including cohomology, cup products, cohomology operations and topological manifolds. The final part is devoted to Homotropy theory, including basic facts about homotropy groups and applications to obstruction theory.
## Women in Topology: Collaborations in Homotopy Theory

This volume contains the proceedings of the WIT: Women in Topology workshop, held from August 18-23, 2013, at the Banff International Research Station, Banff, Alberta, Canada. The Women in Topology workshop was devoted primarily to active collaboration by teams of five to seven participants, each including senior and junior researchers, as well as graduate students. This volume contains papers based on the results obtained by team projects in homotopy theory, including -infinity structures, equivariant homotopy theory, functor calculus, model categories, orbispaces, and topological Hochschild homology.

Just another PDF Download site

Mathematics

**Author**: George W. Whitehead

**Publisher:** Springer Science & Business Media

**ISBN:** 1461263182

**Category:** Mathematics

**Page:** 746

**View:** 7557

Mathematics

**Author**: Jeffrey Strom

**Publisher:** American Mathematical Soc.

**ISBN:** 0821852868

**Category:** Mathematics

**Page:** 835

**View:** 1423

Mathematics

**Author**: Yves Felix,Stephen Halperin,J.-C. Thomas

**Publisher:** Springer Science & Business Media

**ISBN:** 146130105X

**Category:** Mathematics

**Page:** 539

**View:** 2093

Mathematics

**Author**: Martin Arkowitz

**Publisher:** Springer Science & Business Media

**ISBN:** 9781441973290

**Category:** Mathematics

**Page:** 344

**View:** 9660

Mathematics

*A First Course*

**Author**: William Fulton

**Publisher:** Springer Science & Business Media

**ISBN:** 1461241804

**Category:** Mathematics

**Page:** 430

**View:** 8010

Mathematics

**Author**: Viktor Vasilʹevich Prasolov

**Publisher:** American Mathematical Soc.

**ISBN:** 0821838121

**Category:** Mathematics

**Page:** 418

**View:** 5703

Mathematics

**Author**: John Stillwell

**Publisher:** Springer Science & Business Media

**ISBN:** 1461243726

**Category:** Mathematics

**Page:** 336

**View:** 887

Mathematics

**Author**: Raoul Bott,Loring W. Tu

**Publisher:** Springer Science & Business Media

**ISBN:** 1475739516

**Category:** Mathematics

**Page:** 338

**View:** 3137

Mathematics

**Author**: Glen E. Bredon

**Publisher:** Springer Science & Business Media

**ISBN:** 1475768486

**Category:** Mathematics

**Page:** 131

**View:** 3219

Mathematics

**Author**: Julia E. Bergner

**Publisher:** Cambridge University Press

**ISBN:** 1108565042

**Category:** Mathematics

**Page:** N.A

**View:** 1193

Mathematics

*An Introduction to Algebraic Topology*

**Author**: James W. Vick

**Publisher:** Springer Science & Business Media

**ISBN:** 9780387941264

**Category:** Mathematics

**Page:** 242

**View:** 9073

Mathematics

**Author**: N.A

**Publisher:** Academic Press

**ISBN:** 9780080873800

**Category:** Mathematics

**Page:** 367

**View:** 2640

Mathematics

**Author**: Kenneth S. Brown

**Publisher:** Springer Science & Business Media

**ISBN:** 1468493272

**Category:** Mathematics

**Page:** 306

**View:** 7370

Mathematics

**Author**: Paul G. Goerss,John F. Jardine

**Publisher:** Birkhäuser

**ISBN:** 3034887078

**Category:** Mathematics

**Page:** 510

**View:** 6673

Mathematics

**Author**: Steven Weintraub

**Publisher:** Springer

**ISBN:** 1493918443

**Category:** Mathematics

**Page:** 163

**View:** 2885

Mathematics

**Author**: Anatoly Fomenko,Dmitry Fuchs

**Publisher:** Springer

**ISBN:** 3319234889

**Category:** Mathematics

**Page:** 627

**View:** 3196

Mathematics

**Author**: Gerard Walschap

**Publisher:** Springer Science & Business Media

**ISBN:** 0387218262

**Category:** Mathematics

**Page:** 229

**View:** 1270

Mathematics

**Author**: Morris W. Hirsch

**Publisher:** Springer Science & Business Media

**ISBN:** 146849449X

**Category:** Mathematics

**Page:** 222

**View:** 2429

Mathematics

**Author**: Edwin H. Spanier

**Publisher:** Springer Science & Business Media

**ISBN:** 1468493221

**Category:** Mathematics

**Page:** 548

**View:** 8978

Algebraic topology -- Applied homological algebra and category theory -- Abstract and axiomatic homotopy theory

**Author**: Maria Basterra,Kristine Bauer,Kathryn Hess,Brenda Johnson

**Publisher:** American Mathematical Soc.

**ISBN:** 1470410133

**Category:** Algebraic topology -- Applied homological algebra and category theory -- Abstract and axiomatic homotopy theory

**Page:** 166

**View:** 8326