Download or read book Syzygies and Homotopy Theory written by F.E.A. Johnson and published by Springer Science & Business Media. This book was released on 2011-11-17 with total page 307 pages. Available in PDF, EPUB and Kindle. Book excerpt: The most important invariant of a topological space is its fundamental group. When this is trivial, the resulting homotopy theory is well researched and familiar. In the general case, however, homotopy theory over nontrivial fundamental groups is much more problematic and far less well understood. Syzygies and Homotopy Theory explores the problem of nonsimply connected homotopy in the first nontrivial cases and presents, for the first time, a systematic rehabilitation of Hilbert's method of syzygies in the context of non-simply connected homotopy theory. The first part of the book is theoretical, formulated to allow a general finitely presented group as a fundamental group. The innovation here is to regard syzygies as stable modules rather than minimal modules. Inevitably this forces a reconsideration of the problems of noncancellation; these are confronted in the second, practical, part of the book. In particular, the second part of the book considers how the theory works out in detail for the specific examples Fn ́F where Fn is a free group of rank n and F is finite. Another innovation is to parametrize the first syzygy in terms of the more familiar class of stably free modules. Furthermore, detailed description of these stably free modules is effected by a suitable modification of the method of Milnor squares. The theory developed within this book has potential applications in various branches of algebra, including homological algebra, ring theory and K-theory. Syzygies and Homotopy Theory will be of interest to researchers and also to graduate students with a background in algebra and algebraic topology.
Download or read book Une Degustation Topologique Homotopy Theory in the Swiss Alps written by Dominique Arlettaz and published by American Mathematical Soc.. This book was released on 2000 with total page 274 pages. Available in PDF, EPUB and Kindle. Book excerpt: The talks given at the Arolla Conference on Algebraic Topology covered a broad spectrum of current research in homotopy theory, offering participants the possibility to sample and relish selected morsels of homotopy theory, much as a participant in a wine tasting partakes of a variety of fine wines. True to the spirit of the conference, the proceedings included in this volume present a savory sampler of homotopical delicacies. Readers will find within these pages a compilation of articles describing current research in the area, including classical stable and unstable homotopy theory, configuration spaces, group cohomology, K-theory, localization, p-compact groups, and simplicial theory.
Download or read book Elements of Homotopy Theory written by George W. Whitehead and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 764 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book From Categories to Homotopy Theory written by Birgit Richter and published by Cambridge University Press. This book was released on 2020-04-16 with total page 401 pages. Available in PDF, EPUB and Kindle. Book excerpt: Bridge the gap between category theory and its applications in homotopy theory with this guide for graduate students and researchers.
Download or read book Homotopy Theory An Introduction to Algebraic Topology written by and published by Academic Press. This book was released on 1975-11-12 with total page 383 pages. Available in PDF, EPUB and Kindle. Book excerpt: Homotopy Theory: An Introduction to Algebraic Topology
Download or read book Introduction to Homotopy Theory written by Paul Selick and published by American Mathematical Soc.. This book was released on 2008 with total page 220 pages. Available in PDF, EPUB and Kindle. Book excerpt: Offers a summary for students and non-specialists who are interested in learning the basics of algebraic topology. This book covers fibrations and cofibrations, Hurewicz and cellular approximation theorems, topics in classical homotopy theory, simplicial sets, fiber bundles, Hopf algebras, and generalized homology and cohomology operations.
Download or read book Handbook of Homotopy Theory written by Haynes Miller and published by CRC Press. This book was released on 2020-01-23 with total page 1043 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Rational Homotopy Theory and Differential Forms written by Phillip Griffiths and published by Springer Science & Business Media. This book was released on 2013-10-02 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 followed by an in-depth look at differential forms and de Tham’s theorem on simplicial complexes. In addition, Sullivan’s results on computing the rational homotopy type from forms is presented. New to the Second Edition: *Fully-revised appendices including an expanded discussion of the Hirsch lemma *Presentation of a natural proof of a Serre spectral sequence result *Updated content throughout the book, reflecting advances in the area of homotopy theory With its modern approach and timely revisions, this second edition of Rational Homotopy Theory and Differential Forms will be a valuable resource for graduate students and researchers in algebraic topology, differential forms, and homotopy theory.
Download or read book Local Homotopy Theory written by John F. Jardine and published by Springer. This book was released on 2015-05-27 with total page 508 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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 book covers core topics such as the unstable homotopy theory of simplicial presheaves and sheaves, localized theories, cocycles, descent theory, non-abelian cohomology, stacks, and local stable homotopy theory. A detailed treatment of the formalism of the subject is interwoven with explanations of the motivation, development, and nuances of ideas and results. The coherence of the abstract theory is elucidated through the use of widely applicable tools, such as Barr's theorem on Boolean localization, model structures on the category of simplicial presheaves on a site, and cocycle categories. A wealth of concrete examples convey the vitality and importance of the subject in topology, number theory, algebraic geometry, and algebraic K-theory. Assuming basic knowledge of algebraic geometry and homotopy theory, Local Homotopy Theory will appeal to researchers and advanced graduate students seeking to understand and advance the applications of homotopy theory in multiple areas of mathematics and the mathematical sciences.
Download or read book Homotopy Theory of Schemes written by Fabien Morel and published by American Mathematical Soc.. This book was released on 2006 with total page 116 pages. Available in PDF, EPUB and Kindle. Book excerpt: In this text, the author presents a general framework for applying the standard methods from homotopy theory to the category of smooth schemes over a reasonable base scheme $k$. He defines the homotopy category $h(\mathcal{E} k)$ of smooth $k$-schemes and shows that it plays the same role for smooth $k$-schemes as the classical homotopy category plays for differentiable varieties. It is shown that certain expected properties are satisfied, for example, concerning the algebraic$K$-theory of those schemes. In this way, advanced methods of algebraic topology become available in modern algebraic geometry.
Download or read book Rational Homotopy Theory written by Yves Felix and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 574 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.
Download or read book Axiomatic Enriched and Motivic Homotopy Theory written by John Greenlees and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 396 pages. Available in PDF, EPUB and Kindle. Book excerpt: The NATO Advanced Study Institute "Axiomatic, enriched and rna tivic homotopy theory" took place at the Isaac Newton Institute of Mathematical Sciences, Cambridge, England during 9-20 September 2002. The Directors were J.P.C.Greenlees and I.Zhukov; the other or ganizers were P.G.Goerss, F.Morel, J.F.Jardine and V.P.Snaith. The title describes the content well, and both the event and the contents of the present volume reflect recent remarkable successes in model categor ies, structured ring spectra and homotopy theory of algebraic geometry. The ASI took the form of a series of 15 minicourses and a few extra lectures, and was designed to provide background, and to bring the par ticipants up to date with developments. The present volume is based on a number of the lectures given during the workshop. The ASI was the opening workshop of the four month programme "New Contexts for Stable Homotopy Theory" which explored several themes in greater depth. I am grateful to the Isaac Newton Institute for providing such an ideal venue, the NATO Science Committee for their funding, and to all the speakers at the conference, whether or not they were able to contribute to the present volume. All contributions were refereed, and I thank the authors and referees for their efforts to fit in with the tight schedule. Finally, I would like to thank my coorganizers and all the staff at the Institute for making the ASI run so smoothly. J.P.C.GREENLEES.
Download or read book Homotopy Theory written by and published by Academic Press. This book was released on 1959-01-01 with total page 363 pages. Available in PDF, EPUB and Kindle. Book excerpt: Homotopy Theory
Download or read book Primary Homotopy Theory written by Joseph Neisendorfer and published by American Mathematical Soc.. This book was released on 1980 with total page 73 pages. Available in PDF, EPUB and Kindle. Book excerpt: The author gives a systematic exposition of homotopy groups with coefficients in a cyclic group [italic]Z or [italic]Z[subscript italic]k. The text pays particular attention to low-dimensional cases and trouble with the small primes. The book gives a complete treatment of some topics--such as Samelson products--with a view toward applications.
Download or read book Homotopy of Operads and Grothendieck Teichmuller Groups written by Benoit Fresse and published by American Mathematical Soc.. This book was released on 2017-05-22 with total page 743 pages. Available in PDF, EPUB and Kindle. Book excerpt: The ultimate goal of this book is to explain that the Grothendieck–Teichmüller group, as defined by Drinfeld in quantum group theory, has a topological interpretation as a group of homotopy automorphisms associated to the little 2-disc operad. To establish this result, the applications of methods of algebraic topology to operads must be developed. This volume is devoted primarily to this subject, with the main objective of developing a rational homotopy theory for operads. The book starts with a comprehensive review of the general theory of model categories and of general methods of homotopy theory. The definition of the Sullivan model for the rational homotopy of spaces is revisited, and the definition of models for the rational homotopy of operads is then explained. The applications of spectral sequence methods to compute homotopy automorphism spaces associated to operads are also explained. This approach is used to get a topological interpretation of the Grothendieck–Teichmüller group in the case of the little 2-disc operad. This volume is intended for graduate students and researchers interested in the applications of homotopy theory methods in operad theory. It is accessible to readers with a minimal background in classical algebraic topology and operad theory.
Download or read book Homotopy Theory and Its Applications written by Alejandro Adem and published by American Mathematical Soc.. This book was released on 1995 with total page 250 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is the result of a conference held to examine developments in homotopy theory in honor of Samuel Gitler in July 1993 (Cocoyoc, Mexico). It includes several research papers and three expository papers on various topics in homotopy theory. The research papers discuss the following: BL application of homotopy theory to group theory BL fiber bundle theory BL homotopy theory The expository papers consider the following topics: BL the Atiyah-Jones conjecture (by C. Boyer) BL classifying spaces of finite groups (by J. Martino) BL instanton moduli spaces (by J. Milgram) Homotopy Theory and Its Applications offers a distinctive account of how homotopy theoretic methods can be applied to a variety of interesting problems.
Download or read book A Course in Simple Homotopy Theory written by M.M. Cohen and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 124 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book grew out of courses which I taught at Cornell University and the University of Warwick during 1969 and 1970. I wrote it because of a strong belief that there should be readily available a semi-historical and geo metrically motivated exposition of J. H. C. Whitehead's beautiful theory of simple-homotopy types; that the best way to understand this theory is to know how and why it was built. This belief is buttressed by the fact that the major uses of, and advances in, the theory in recent times-for example, the s-cobordism theorem (discussed in §25), the use of the theory in surgery, its extension to non-compact complexes (discussed at the end of §6) and the proof of topological invariance (given in the Appendix)-have come from just such an understanding. A second reason for writing the book is pedagogical. This is an excellent subject for a topology student to "grow up" on. The interplay between geometry and algebra in topology, each enriching the other, is beautifully illustrated in simple-homotopy theory. The subject is accessible (as in the courses mentioned at the outset) to students who have had a good one semester course in algebraic topology. I have tried to write proofs which meet the needs of such students. (When a proof was omitted and left as an exercise, it was done with the welfare of the student in mind. He should do such exercises zealously.
