Homotopy Theory of C*-Algebras

Homotopy Theory of C*-Algebras

Author: Paul Arne Østvær

Publisher: Springer Science & Business Media

Published: 2010-09-08

Total Pages: 142

ISBN-13: 303460565X

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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 suitable fodder for standard homotopy theoretic moves, leading to unstable and stable model structures. With the foundations in place one is led to natural definitions of invariants for C*-spaces such as homology and cohomology theories, K-theory and zeta-functions. The text is largely self-contained. It serves a wide audience of graduate students and researchers interested in C*-algebras, homotopy theory and applications.


Rational Homotopy Theory

Rational Homotopy Theory

Author: Yves Felix

Publisher: Springer Science & Business Media

Published: 2001

Total Pages: 589

ISBN-13: 0387950680

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This is a long awaited book on rational homotopy theory which contains all the main theorems with complete proofs, and more elementary proofs for many results that were proved ten or fifteen years ago. The authors added a frist section on classical algebraic topology to make the book accessible to students with only little background in algebraic topology.


Modern Classical Homotopy Theory

Modern Classical Homotopy Theory

Author: Jeffrey Strom

Publisher: American Mathematical Soc.

Published: 2011-10-19

Total Pages: 862

ISBN-13: 0821852868

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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.


Rings, Modules, and Algebras in Stable Homotopy Theory

Rings, Modules, and Algebras in Stable Homotopy Theory

Author: Anthony D. Elmendorf

Publisher: American Mathematical Soc.

Published: 1997

Total Pages: 265

ISBN-13: 0821843036

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This book introduces a new point-set level approach to stable homotopy theory that has already had many applications and promises to have a lasting impact on the subject. Given the sphere spectrum $S$, the authors construct an associative, commutative, and unital smash product in a complete and cocomplete category of ``$S$-modules'' whose derived category is equivalent to the classical stable homotopy category. This construction allows for a simple and algebraically manageable definition of ``$S$-algebras'' and ``commutative $S$-algebras'' in terms of associative, or associative and commutative, products $R\wedge SR \longrightarrow R$. These notions are essentially equivalent to the earlier notions of $A {\infty $ and $E {\infty $ ring spectra, and the older notions feed naturally into the new framework to provide plentiful examples. There is an equally simple definition of $R$-modules in terms of maps $R\wedge SM\longrightarrow M$. When $R$ is commutative, the category of $R$-modules also has a


Nilpotence and Periodicity in Stable Homotopy Theory

Nilpotence and Periodicity in Stable Homotopy Theory

Author: Douglas C. Ravenel

Publisher: Princeton University Press

Published: 1992-11-08

Total Pages: 228

ISBN-13: 9780691025728

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Nilpotence and Periodicity in Stable Homotopy Theory describes some major advances made in algebraic topology in recent years, centering on the nilpotence and periodicity theorems, which were conjectured by the author in 1977 and proved by Devinatz, Hopkins, and Smith in 1985. During the last ten years a number of significant advances have been made in homotopy theory, and this book fills a real need for an up-to-date text on that topic. Ravenel's first few chapters are written with a general mathematical audience in mind. They survey both the ideas that lead up to the theorems and their applications to homotopy theory. The book begins with some elementary concepts of homotopy theory that are needed to state the problem. This includes such notions as homotopy, homotopy equivalence, CW-complex, and suspension. Next the machinery of complex cobordism, Morava K-theory, and formal group laws in characteristic p are introduced. The latter portion of the book provides specialists with a coherent and rigorous account of the proofs. It includes hitherto unpublished material on the smash product and chromatic convergence theorems and on modular representations of the symmetric group.


Categorical Homotopy Theory

Categorical Homotopy Theory

Author: Emily Riehl

Publisher: Cambridge University Press

Published: 2014-05-26

Total Pages: 371

ISBN-13: 1139952633

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This book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Emily Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence.


Cohomology Operations and Applications in Homotopy Theory

Cohomology Operations and Applications in Homotopy Theory

Author: Robert E. Mosher

Publisher: Courier Corporation

Published: 2008-01-01

Total Pages: 226

ISBN-13: 0486466647

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Cohomology operations are at the center of a major area of activity in algebraic topology. This treatment explores the single most important variety of operations, the Steenrod squares. It constructs these operations, proves their major properties, and provides numerous applications, including several different techniques of homotopy theory useful for computation. 1968 edition.


C*-Algebras and Operator Theory

C*-Algebras and Operator Theory

Author: Gerald J. Murphy

Publisher: Academic Press

Published: 2014-06-28

Total Pages: 297

ISBN-13: 0080924964

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This book constitutes a first- or second-year graduate course in operator theory. It is a field that has great importance for other areas of mathematics and physics, such as algebraic topology, differential geometry, and quantum mechanics. It assumes a basic knowledge in functional analysis but no prior acquaintance with operator theory is required.


Cubical Homotopy Theory

Cubical Homotopy Theory

Author: Brian A. Munson

Publisher: Cambridge University Press

Published: 2015-10-06

Total Pages: 649

ISBN-13: 1107030250

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A modern, example-driven introduction to cubical diagrams and related topics such as homotopy limits and cosimplicial spaces.