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Author: William M. Singer
Publisher: American Mathematical Soc.
ISBN: 0821841416
Category : Mathematics
Languages : en
Pages : 170
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Book Description
This book develops a general theory of Steenrod operations in spectral sequences. It gives special attention to the change-of-rings spectral sequence for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces and homotopy orbit spaces. In treating the change-of-rings spectral sequence, the book develops from scratch the necessary properties of extensions of Hopf algebras and constructs the spectral sequence in a form particularly suited to the introduction of Steenrod squares. The resulting theory can be used effectively for the computation of the cohomology rings of groups and Hopf algebras, and of the Steenrod algebra in particular, and so should play a useful role in stable homotopy theory. Similarly the book offers a self-contained construction of the Eilenberg-Moore spectral sequence, in a form suitable for the introduction of Steenrod operations. The corresponding theory is an effective tool for the computation of t
Author: William M. Singer
Publisher: American Mathematical Soc.
ISBN: 0821841416
Category : Mathematics
Languages : en
Pages : 170
Get Book
Book Description
This book develops a general theory of Steenrod operations in spectral sequences. It gives special attention to the change-of-rings spectral sequence for the cohomology of an extension of Hopf algebras and to the Eilenberg-Moore spectral sequence for the cohomology of classifying spaces and homotopy orbit spaces. In treating the change-of-rings spectral sequence, the book develops from scratch the necessary properties of extensions of Hopf algebras and constructs the spectral sequence in a form particularly suited to the introduction of Steenrod squares. The resulting theory can be used effectively for the computation of the cohomology rings of groups and Hopf algebras, and of the Steenrod algebra in particular, and so should play a useful role in stable homotopy theory. Similarly the book offers a self-contained construction of the Eilenberg-Moore spectral sequence, in a form suitable for the introduction of Steenrod operations. The corresponding theory is an effective tool for the computation of t
Author: S P Novikov
Publisher: World Scientific
ISBN: 9814401323
Category : Mathematics
Languages : en
Pages : 592
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Book Description
The final volume of the three-volume edition, this book features classical papers on algebraic and differential topology published in the 1950s–1960s. The partition of these papers among the volumes is rather conditional. The original methods and constructions from these works are properly documented for the first time in this book. No existing book covers the beautiful ensemble of methods created in topology starting from approximately 1950. That is, from Serre's celebrated “singular homologies of fiber spaces.” Contents:Singular Homology of Fiber Spaces (J-P Serre)Homotopy Groups and Classes of Abelian Groups (J-P Serre)Cohomology Modulo 2 of Eilenberg–MacLane Complexes (J-P Serre)On Cohomology of Principal Fiber Bundles and Homogeneous Spaces of Compact Lie Groups (A Borel)Cohomology Mod 2 of Some Homogeneous Spaces (A Borel)The Steenrod Algebra and Its Dual (J Milnor)On the Structure and Applications of the Steenrod Algebra (J F Adams)Vector Bundles and Homogeneous Spaces (M F Atiyah and F Hirzebruch)The Methods of Algebraic Topology from Viewpoint of Cobordism Theory (S P Novikov) Readership: Researchers in algebraic topology, its applications, and history of topology. Keywords:Topology;Homeomorphism;Fundamental Group;Smooth Manifold;Homology;Homotopy;Fiber Spaces;Vector Bundles;Characteristic Classes;Homogeneous Spaces;Cobordism;Steenrod AlgebraKey Features:Serves as a tool in learning classical algebraic topologyAn essential book for topologistsReviews: "It is utmost useful to have these (interrelated) classics gathered together in one volume. This facilitates the study of the originals considerably, all the more as numerous editorial hints provide additional guidance. In this regard, the entire edition represents an invaluable source book for both students and researchers in the field." Zentralblatt MATH "This is a nice volume that should not be missing in any Mathematics Library." European Mathematical Society
Author: Robert E. Mosher
Publisher: Courier Corporation
ISBN: 0486466647
Category : Mathematics
Languages : en
Pages : 226
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Book Description
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.
Author: John McCleary
Publisher: Cambridge University Press
ISBN: 0521567599
Category : Mathematics
Languages : en
Pages : 579
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Book Description
Spectral sequences are among the most elegant and powerful methods of computation in mathematics. This book describes some of the most important examples of spectral sequences and some of their most spectacular applications. The first part treats the algebraic foundations for this sort of homological algebra, starting from informal calculations. The heart of the text is an exposition of the classical examples from homotopy theory, with chapters on the Leray-Serre spectral sequence, the Eilenberg-Moore spectral sequence, the Adams spectral sequence, and, in this new edition, the Bockstein spectral sequence. The last part of the book treats applications throughout mathematics, including the theory of knots and links, algebraic geometry, differential geometry and algebra. This is an excellent reference for students and researchers in geometry, topology, and algebra.
Author: John McCleary
Publisher:
ISBN:
Category : Mathematics
Languages : en
Pages : 448
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Book Description
Author: Stanley O. Kochman
Publisher: American Mathematical Soc.
ISBN: 0821871900
Category : Mathematics
Languages : en
Pages : 288
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Book Description
This book is a compilation of lecture notes that were prepared for the graduate course "Adams Spectral Sequences and Stable Homotopy Theory" given at The Fields Institute during the fall of 1995. The aim of this volume is to prepare students with a knowledge of elementary algebraic topology to study recent developments in stable homotopy theory, such as the nilpotence and periodicity theorems. Suitable as a text for an intermediate course in algebraic topology, this book provides a direct exposition of the basic concepts of bordism, characteristic classes, Adams spectral sequences, Brown-Peterson spectra and the computation of stable stems. The key ideas are presented in complete detail without becoming encyclopedic. The approach to characteristic classes and some of the methods for computing stable stems have not been published previously.
Author: Larry Smith
Publisher: Springer
ISBN: 354036286X
Category : Mathematics
Languages : en
Pages : 149
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Book Description
Author: Robert R. Bruner
Publisher: American Mathematical Society
ISBN: 1470469588
Category : Mathematics
Languages : en
Pages : 690
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Book Description
The connective topological modular forms spectrum, $tmf$, is in a sense initial among elliptic spectra, and as such is an important link between the homotopy groups of spheres and modular forms. A primary goal of this volume is to give a complete account, with full proofs, of the homotopy of $tmf$ and several $tmf$-module spectra by means of the classical Adams spectral sequence, thus verifying, correcting, and extending existing approaches. In the process, folklore results are made precise and generalized. Anderson and Brown-Comenetz duality, and the corresponding dualities in homotopy groups, are carefully proved. The volume also includes an account of the homotopy groups of spheres through degree 44, with complete proofs, except that the Adams conjecture is used without proof. Also presented are modern stable proofs of classical results which are hard to extract from the literature. Tools used in this book include a multiplicative spectral sequence generalizing a construction of Davis and Mahowald, and computer software which computes the cohomology of modules over the Steenrod algebra and products therein. Techniques from commutative algebra are used to make the calculation precise and finite. The $H$-infinity ring structure of the sphere and of $tmf$ are used to determine many differentials and relations.
Author: John Harold Palmieri
Publisher: American Mathematical Soc.
ISBN: 0821826689
Category : Mathematics
Languages : en
Pages : 193
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Book Description
This title applys the tools of stable homotopy theory to the study of modules over the mod $p$ Steenrod algebra $A DEGREES{*}$. More precisely, let $A$ be the dual of $A DEGREES{*}$; then we study the category $\mathsf{stable}(A)$ of unbounded cochain complexes of injective comodules over $A$, in which the morphisms are cochain homotopy classes of maps. This category is triangulated. Indeed, it is a stable homotopy category, so we can use Brown representability, Bousfield localization, Brown-Comenetz duality, and other homotopy-theoretic tools to study it. One focus of attention is the analogue of the stable homotopy groups of spheres, which in this setting is the cohomology of $A$, $\mathrm{Ext}_A DEGREES{**}(\mathbf{F}_p, \mathbf{F}_p)$. This title also has nilpotence theorems, periodicity theorems, a convergent chromatic tower, and a nu
Author: Douglas C. Ravenel
Publisher: American Mathematical Society
ISBN: 1470472937
Category : Mathematics
Languages : en
Pages : 417
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Book Description
Since the publication of its first edition, this book has served as one of the few available on the classical Adams spectral sequence, and is the best account on the Adams-Novikov spectral sequence. This new edition has been updated in many places, especially the final chapter, which has been completely rewritten with an eye toward future research in the field. It remains the definitive reference on the stable homotopy groups of spheres. The first three chapters introduce the homotopy groups of spheres and take the reader from the classical results in the field though the computational aspects of the classical Adams spectral sequence and its modifications, which are the main tools topologists have to investigate the homotopy groups of spheres. Nowadays, the most efficient tools are the Brown-Peterson theory, the Adams-Novikov spectral sequence, and the chromatic spectral sequence, a device for analyzing the global structure of the stable homotopy groups of spheres and relating them to the cohomology of the Morava stabilizer groups. These topics are described in detail in Chapters 4 to 6. The revamped Chapter 7 is the computational payoff of the book, yielding a lot of information about the stable homotopy group of spheres. Appendices follow, giving self-contained accounts of the theory of formal group laws and the homological algebra associated with Hopf algebras and Hopf algebroids. The book is intended for anyone wishing to study computational stable homotopy theory. It is accessible to graduate students with a knowledge of algebraic topology and recommended to anyone wishing to venture into the frontiers of the subject.
