*As the title suggests, this book is concerned with the elementary portion of the subject of homotopy theory.*

**Author**: George W. Whitehead

**Publisher:** Springer Science & Business Media

**ISBN:** 9781461263180

**Category:**

**Page:** 746

**View:** 144

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.

*2 Elements of homotopy theory In this section we give a rather rough account of homotopy groups . The first homotopy group ( it is called fundamental ) is closely related with the first homology group , since the latter is obtained from ...*

**Author**: Sergeĭ Vladimirovich Matveev

**Publisher:** European Mathematical Society

**ISBN:** 303719023X

**Category:**

**Page:** 112

**View:** 182

Algebraic topology is the study of the global properties of spaces by means of algebra. It is an important branch of modern mathematics with a wide degree of applicability to other fields, including geometric topology, differential geometry, functional analysis, differential equations, algebraic geometry, number theory, and theoretical physics. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications. The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. The numerous illustrations in the text also serve this purpose. Two features make the text different from the standard literature: first, special attention is given to providing explicit algorithms for calculating the homology groups and for manipulating the fundamental groups. Second, the book contains many exercises, all of which are supplied with hints or solutions. This makes the book suitable for both classroom use and for independent study.

*TAKEUTI/ZARING Introduction to Axiomatic Set Theory, 2nd ed. OxTOBY. Measure and Category, 2nd ed. ... COHEN A Course in Simple Homotopy Theory. CONWAY. ... Introduction to Operator Theory I: Elements of Functional Analysis. MASSEY.*

**Author**: Yves Felix

**Publisher:** Springer Science & Business Media

**ISBN:** 9781461301059

**Category:**

**Page:** 539

**View:** 460

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.

*f-admissible elements of Qom(K, L; own(Y)) form a set Af", which will be called the f-admissible set in Q,”(K, L; own (Y)). The following proposition is obvious. Proposition 12.1. For any two maps f,g: K -> Y in the (n − 1)-homotopy ...*

**Author**:

**Publisher:** Academic Press

**ISBN:** 0080873162

**Category:**

**Page:** 346

**View:** 842

Homotopy Theory

**elements**. of. n-categories. The observation that the **theory** of strict n-groupoids is not enough to give a good model for **homotopy** n-types, led Grothendieck to ask for a **theory** of n-categories with weakly associative composition.

**Author**: Carlos Simpson

**Publisher:** Cambridge University Press

**ISBN:** 9781139502191

**Category:**

**Page:**

**View:** 405

The study of higher categories is attracting growing interest for its many applications in topology, algebraic geometry, mathematical physics and category theory. In this highly readable book, Carlos Simpson develops a full set of homotopical algebra techniques and proposes a working theory of higher categories. Starting with a cohesive overview of the many different approaches currently used by researchers, the author proceeds with a detailed exposition of one of the most widely used techniques: the construction of a Cartesian Quillen model structure for higher categories. The fully iterative construction applies to enrichment over any Cartesian model category, and yields model categories for weakly associative n-categories and Segal n-categories. A corollary is the construction of higher functor categories which fit together to form the (n+1)-category of n-categories. The approach uses Tamsamani's definition based on Segal's ideas, iterated as in Pelissier's thesis using modern techniques due to Barwick, Bergner, Lurie and others.

*We content ourselves with simply stating the resulting homotopy groups of TMF2). These are displayed in Figure 6.4.2. Our choice of names for elements in the descent spectral sequence (and our abusive practice of giving the elements of ...*

**Author**: Haynes Miller

**Publisher:** CRC Press

**ISBN:** 9781351251600

**Category:**

**Page:** 990

**View:** 274

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. The origins of the subject date back to work of Henri Poincaré and Heinz Hopf in the early 20th century, but it has seen enormous progress in the 21st century. A highlight of this volume is an introduction to and diverse applications of the newly established foundational theory of ¥ -categories. The coverage is vast, ranging from axiomatic to applied, from foundational to computational, and includes surveys of applications both geometric and algebraic. The contributors are among the most active and creative researchers in the field. The 22 chapters by 31 contributors are designed to address novices, as well as established mathematicians, interested in learning the state of the art in this field, whose methods are of increasing importance in many other areas.

**Elements of Homotopy Theory** This chapter and the next are adapted from a lecture published in the Proceedings of the Third Physics Workshop, held at the Institute of Theoretical and Experimental Physics, Moscow, in 1975.

**Author**: Albert S. Schwarz

**Publisher:** Springer Science & Business Media

**ISBN:** 9783662029435

**Category:**

**Page:** 276

**View:** 518

In recent years topology has firmly established itself as an important part of the physicist's mathematical arsenal. It has many applications, first of all in quantum field theory, but increasingly also in other areas of physics. The main focus of this book is on the results of quantum field theory that are obtained by topological methods. Some aspects of the theory of condensed matter are also discussed. Part I is an introduction to quantum field theory: it discusses the basic Lagrangians used in the theory of elementary particles. Part II is devoted to the applications of topology to quantum field theory. Part III covers the necessary mathematical background in summary form. The book is aimed at physicists interested in applications of topology to physics and at mathematicians wishing to familiarize themselves with quantum field theory and the mathematical methods used in this field. It is accessible to graduate students in physics and mathematics.