Algebraic topology Books

Book SynopsisEtale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ℓ-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.Table of ContentsDescent Theory; Etale Morphisms and Smooth Morphisms; Etale Fundamental Groups; Group Cohomology and Galois Cohomology; Etale Cohomology; Derived Categories and Derived Functors; Base Change Theorems; Duality; Finiteness Theorems; -Adic Cohomology;

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Book SynopsisSheaves also appear in logic as carriers for models of set theory. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.Trade ReviewFrom the reviews: "A beautifully written book, a long and well motivated book packed with well chosen clearly explained examples. … authors have a rare gift for conveying an insider’s view of the subject from the start. This book is written in the best Mac Lane style, very clear and very well organized. … it gives very explicit descriptions of many advanced topics--you can learn a great deal from this book that, before it was published, you could only learn by knowing researchers in the field." (Wordtrade, 2008)Table of ContentsPreface; Prologue; Categorical Preliminaries; 1. Categories of Functors; 2. Sheaves of Sets; 3. Grothendieck Topologies and Sheaves; 4. First Properties of Elementary Topoi; 5. Basic Constructions of Topoi; 6. Topoi and Logic; 7. Geometric Morphisms; 8. Classifying Topoi; 9. Localic Topoi; 10. Geometric Logic and Classifying Topoi; Appendix: Sites for Topoi; Epilogue; Bibliography; Index of Notations; Index

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Book SynopsisAlgebraic Topology is an introductory textbook based on a class for advanced high-school students at the Stanford University Mathematics Camp (SUMaC) that the authors have taught for many years. Each chapter, or lecture, corresponds to one day of class at SUMaC. The book begins with the preliminaries needed for the formal definition of a surface. Other topics covered in the book include the classification of surfaces, group theory, the fundamental group, and homology. This book assumes no background in abstract algebra or real analysis, and the material from those subjects is presented as needed in the text. This makes the book readable to undergraduates or high-school students who do not have the background typically assumed in an algebraic topology book or class. The book contains many examples and exercises, allowing it to be used for both self-study and for an introductory undergraduate topology course.Trade Review“Algebraic topology provides a self-contained introduction to the field … . the book thus provides a particularly well-organized, interesting, and smooth exposition of its subject. … This particular book unique is that it provides a clear, elementary, but mathematically solid introduction to algebraic topology that keeps the subject interesting throughout. … provides a clear, readable, and detailed treatment of the ideas and proofs in the subject … .” (Thomas Mack, Mathematical Reviews, July, 2022)“The book could easily be used in an undergraduate course or read by a bright high school student. It should certainly be in any high school library.” (Jonathan Hodgson, zbMATH 1481.55001, 2022)Table of ContentsIntroduction.- 1. Surface Preliminaries.- 2. Surfaces.- 3. The Euler Characteristic and Identification Spaces.- 4. Classification Theorem of Compact Surfaces.- 5. Introduction to Group Theory.- 6. Structure of Groups.- 7. Cosets, Normal Subgroups, and Quotient Groups.- 8. The Fundamental Group.- 9. Computing the Fundamental Group.- 10. Tools for Fundamental Groups.- 11. Applications of Fundamental Groups.- 12. The Seifert-Van Kampen Theorem.- 13. Introduction to Homology.- 14. The Mayer-Vietoris Sequence.- A. Topological Notions.- Bibliography.- Index.

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Book SynopsisThis book provides an accessible introduction to algebraic topology, a field at the intersection of topology, geometry and algebra, together with its applications. Moreover, it covers several related topics that are in fact important in the overall scheme of algebraic topology. Comprising eighteen chapters and two appendices, the book integrates various concepts of algebraic topology, supported by examples, exercises, applications and historical notes. Primarily intended as a textbook, the book offers a valuable resource for undergraduate, postgraduate and advanced mathematics students alike.Focusing more on the geometric than on algebraic aspects of the subject, as well as its natural development, the book conveys the basic language of modern algebraic topology by exploring homotopy, homology and cohomology theories, and examines a variety of spaces: spheres, projective spaces, classical groups and their quotient spaces, function spaces, polyhedra, topological groups, Lie groups and cell complexes, etc. The book studies a variety of maps, which are continuous functions between spaces. It also reveals the importance of algebraic topology in contemporary mathematics, theoretical physics, computer science, chemistry, economics, and the biological and medical sciences, and encourages students to engage in further study.Trade Review“Adhikari’s work is an excellent resource for any individual seeking to learn more about algebraic topology. By no means will this text feel like an introduction to algebraic topology, but it does offer much for both beginners and experts. … the text will be a valuable reference on the bookshelf of any reader with an interest in algebraic topology. Summing Up: Recommended. Upper-division undergraduates and above; researchers and faculty.” (A. Misseldine, Choice, Vol. 54 (9), May, 2017)“I am pretty enthusiastic about this book. … it shows very good taste on the author’s part as far as what he’s chosen to do and how he’s chosen to do it. … Wow! What a nice book. I’m glad I have a copy.” (Michael Berg, MAA Reviews, maa.org, February, 2017)“This is a comprehensive textbook on algebraic topology. … accessible to students of all levels of mathematics, so suitable for anyone wanting and needing to learn about algebraic topology. It can also offer a valuable resource for advanced students with a specialized knowledge in other areas who want to pursue their interest in this area. … further readings are provided at the end of each of them, which also enables students to study the subject discussed therein in more depth.” (Haruo Minami, zbMATH 1354.55001, 2017)Table of ContentsPrerequisite Concepts and Notations.- Basic Homotopy.- The Fundamental Groups.-Covering Spaces.- Fibre Bundles, Vector Bundles and K-theory.- Geometry of Simplicial Complexes and Fundamental Groups.- Higher Homotopy Groups.- Products in Higher Homotopy Groups.- CW-complexes and Homotopy.- Eilenberg-MacLane Spaces.- Homology and Cohomology Theories.- Eilenberg-Steenrod Axioms for Homology and Cohomology Theories.- Consequences of the Eilenberg-Steenrod Axioms.- Some Applications of Homology Theory.- Spectral Homology and Cohomology Theories.- Obstruction Theory.- More Relations Between Homotopy and Homology Groups.- A Brief Historical Note.

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Book SynopsisThis book is intended as a one-semester course in general topology, a.k.a. point-set topology, for undergraduate students as well as first-year graduate students. Such a course is considered a prerequisite for further studying analysis, geometry, manifolds, and certainly, for a career of mathematical research. Researchers may find it helpful especially from the comprehensive indices.General topology resembles a language in modern mathematics. Because of this, the book is with a concentration on basic concepts in general topology, and the presentation is of a brief style, both concise and precise. Though it is hard to determine exactly which concepts therein are basic and which are not, the author makes efforts in the selection according to personal experience on the occurrence frequency of notions in advanced mathematics, and to related books that have received admirable reviews.This book also contains exercises for each chapter with selected solutions. Interrelationships among concepts are taken into account frequently. Twelve particular topological spaces are repeatedly exploited, which serve as examples to learn new concepts based on old ones.Table of ContentsPreface; Introduction; Topological Spaces; Continuous Maps and Homeomorphisms; Connectedness; Separation Axioms and Quotient Axioms; Compactness; Product Spaces and Quotient Spaces; Appendix: Some Elementary Inequalities;

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Book SynopsisAlgebraic Topology and basic homotopy theory form a fundamental building block for much of modern mathematics. These lecture notes represent a culmination of many years of leading a two-semester course in this subject at MIT. The style is engaging and student-friendly, but precise. Every lecture is accompanied by exercises. It begins slowly in order to gather up students with a variety of backgrounds, but gains pace as the course progresses, and by the end the student has a command of all the basic techniques of classical homotopy theory.

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Book SynopsisThis is the second (revised and enlarged) edition of the book originally published in 2003. It introduces the first concepts of Algebraic Topology like general simplicial complexes, simplicial homology theory, fundamental groups, covering spaces and singular homology theory in detail. The text has been designed for undergraduate and beginning graduate students of Mathematics. It assumes a minimal background of linear algebra, group theory and topological spaces. The author has dealt with the basic concepts and ideas in a very lucid manner giving suitable motivations and illustrations. As an application of the tools developed in this book, some classical theorems like Brouwer’s fixed point theorem, the Lefschetz fixed point theorem, the Borsuk-Ulam theorem, Brouwer’s separation theorem and the theorem on invariance of domain have been proved and illustrated. Most of the exercises are elementary but some are more challenging and will help the readers in their understanding of the subject.Table of Contents 1 Basic Topology: A review 2 The Fundamental Group 3 Simplicial Complexes 4 Simplicial Homology 5 Covering Projections 6 Singular Homology 7 Appendix References Index

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Book SynopsisWith firm foundations dating only from the 1950s, algebraic topology is a relatively young area of mathematics. This title addresses the course material, such as fundamental groups, covering spaces, the basics of homotopy theory, and homology and cohomology. It covers topics that are useful for algebraic topologists.Trade Review"All researchers in algebraic topology should have at least a passing acquaintance with the material treated in this book, much of which does not appear in any of the standard texts." (Kathryn Hess, Ecole Polytechnique Federale de Lausanne)"

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Book SynopsisThese two volumes provide a self-contained account of research on algebraic cycles and motives. Twenty-two contributions from leading figures survey the key research strands, including: Abel-Jacobi/regulator maps and normal functions; Voevodsky's triangulated category of mixed motives; conjectures of Bloch-Beilinson and Murre on filtrations on Chow groups.Table of ContentsForeword; Part I. Survey Articles: 1. The motivic vanishing cycles and the conservation conjecture J. Ayoub; 2. On the theory of 1-motives L. Barbieri-Viale; 3. Motivic decomposition for resolutions of threefolds M. de Cataldo and L. Migliorini; 4. Correspondences and transfers F. D´eglise; 5. Algebraic cycles and singularities of normal functions M. Green and Ph. Griffiths; 6. Zero cycles on singular varieties A. Krishna and V. Srinivas; 7. Modular curves, modular surfaces and modular fourfolds D. Ramakrishnan.

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Book SynopsisThis is the first book in English on BruhatTits theory, an important topic in number theory, representation theory, and algebraic geometry. A comprehensive account of the theory, it can serve both as a reference for researchers in the field and as a thorough introduction for graduate students and early career mathematicians.Table of ContentsIntroduction; Part I. Background and Review: 1. Affine root systems and abstract buildings; 2. Algebraic groups; Part II. Bruhat–Tits theory: 3. Examples: Quasi-split groups of rank 1; 4. Overview and summary of Bruhat–Tits theory; 5. Bruhat, Cartan, and Iwasawa decompositions; 6. The apartment; 7. The Bruhat–Tits building for a valuation of the root datum; 8. Integral models; 9. Unramified descent; Part III. Additional Developments: 10. Residue field f of dimension ≤ 1; 11. The buildings of classical groups via lattice chains; 12. Component groups of integral models; 13. Finite group actions and tamely ramified descent; 14. Moy–Prasad filtrations; 15. Functorial properties; Part IV. Applications: 16. Classification of maximal unramified tori (d'après DeBacker); 17. Classification of tamely ramified maximal tori; 18. The volume formula; Part V. Appendices: A. Operations on integral models; B. Integral models of tori; C. Integral models of root subgroups; References; Index.

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Book SynopsisThis book gives a brief treatment of the equivariant cohomology of the classical configuration space F(ℝ^d,n) from its beginnings to recent developments. This subject has been studied intensively, starting with the classical papers of Artin (1925/1947) on the theory of braids, and progressing through the work of Fox and Neuwirth (1962), Fadell and Neuwirth (1962), and Arnol'd (1969). The focus of this book is on the mod 2 equivariant cohomology algebras of F(ℝ^d,n), whose additive structure was described by Cohen (1976) and whose algebra structure was studied in an influential paper by Hung (1990). A detailed new proof of Hung's main theorem is given, however it is shown that some of the arguments given by him on the way to his result are incorrect, as are some of the intermediate results in his paper.This invalidates a paper by three of the authors, Blagojević, Lück and Ziegler (2016), who used a claimed intermediate result in order to derive lower bounds for the existence of k-regular and ℓ-skew embeddings. Using the new proof of Hung's main theorem, new lower bounds for the existence of highly regular embeddings are obtained: Some of them agree with the previously claimed bounds, some are weaker.Assuming only a standard graduate background in algebraic topology, this book carefully guides the reader on the way into the subject. It is aimed at graduate students and researchers interested in the development of algebraic topology in its applications in geometry.Trade Review“The book is well written. … The book will be important for those who study the cohomology rings of configuration spaces.” (Shintarô Kuroki, Mathematical Reviews, November, 2022)Table of Contents​- 1. Snapshots from the History. - Part I Mod 2 Cohomology of Configuration Spaces. - 2. The Ptolemaic Epicycles Embedding. - 3. The Equivariant Cohomology of Pe(Rd, 2m). - 4. Hu’ng’s Injectivity Theorem. - Part II Applications to the (Non-)Existence of Regular and SkewEmbeddings. - 5. On Highly Regular Embeddings: Revised. - 6. More Bounds for Highly Regular Embeddings. - Part III Technical Tools. - 7. Operads. - 8. The Dickson Algebra. - 9. The Stiefel–Whitney Classes of the Wreath Square of a Vector Bundle. - 10. Miscellaneous Calculations.

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Book SynopsisThis third volume in Vladimir Tkachuk's series on Cp-theory problems applies all modern methods of Cp-theory to study compactness-like properties in function spaces and introduces the reader to the theory of compact spaces widely used in Functional Analysis. The text is designed to bring a dedicated reader from basic topological principles to the frontiers of modern research covering a wide variety of topics in Cp-theory and general topology at the professional level. The first volume, Topological and Function Spaces © 2011, provided an introduction from scratch to Cp-theory and general topology, preparing the reader for a professional understanding of Cp-theory in the last section of its main text. The second volume, Special Features of Function Spaces © 2014, continued from the first, giving reasonably complete coverage of Cp-theory, systematically introducing each of the major topics and providing 500 carefully selected problems and exercises with complete solutions. This third volume is self-contained and works in tandem with the other two, containing five hundred carefully selected problems and solutions. It can also be considered as an introduction to advanced set theory and descriptive set theory, presenting diverse topics of the theory of function spaces with the topology of point wise convergence, or Cp-theory which exists at the intersection of topological algebra, functional analysis and general topology.Trade Review“This volume … is a very useful book for all researchers working in Cp-theory (also in general topology) and its relationships with other mathematical disciplines, especially with functional analysis. The problems in chapter 4 can attract young mathematicians to work in this field and to solve some of quite difficult problems.” (Ljubiša D. Kočinac, zbMATH, Vol. 1325.54001, 2016)From the Reviews of Topological and Function Spaces: “…It is designed to bring a dedicated reader from the basic topological principles to the frontiers of modern research. Any reasonable course in calculus covers everything needed to understand this book. This volume can also be used as a reference for mathematicians working in or outside the field of topology (functional analysis) wanting to use results or methods of Cp-theory...On the whole, the book provides a useful addition to the literature on Cp-theory, especially at the instructional level." (Mathematical Reviews)Table of ContentsPreface.- Contents.- Detailed summary of exercise sections.- Introduction.- 1. Behavior of Compactness in Function Spaces.- 2. Solutions of Problems 001-0500.- 3. Bonus Results: Some Hidden Statements.- 4. Open Problems.- Bibliography.- List of Special Symbols.- Index.

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Book SynopsisThe main goal of the CIME Summer School on "Algebraic Cycles and Hodge Theory" has been to gather the most active mathematicians in this area to make the point on the present state of the art. Thus the papers included in the proceedings are surveys and notes on the most important topics of this area of research. They include infinitesimal methods in Hodge theory; algebraic cycles and algebraic aspects of cohomology and k-theory, transcendental methods in the study of algebraic cycles.Table of ContentsContents: M. Green: Infinitesimal methods in Hodge theory.- J.P. Murre: Algebraic cycles and algebraic aspects of cohomology and k-theory.- C. Voisin: Transcendental methods in the study of algebraic cycles.- P. Pirola: The infinitesimal invariant of C(+)-C(-).- B. van Geemen: An introduction to the Hodge conjecture for abelian varieties.- S. Müller-Stach: A remark on height pairings.

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Book SynopsisThis is the proceedings of the meeting entitled “The 12th MSJ International Research Institute of the Mathematical Society of Japan 2003”. The papers cover several important topics in Singularity theory. Especially some of them are survey on motivic integrations, Thom polynomials, complex analytic singularity theory, generic differential geometry etc.Published by Mathematical Society of Japan and distributed by World Scientific Publishing Co. for all markets except North AmericaTable of ContentsInvariants of combinatorial line arrangements and Rybnikov's example by E. A. Bartolo, J. C. Ruber, J. I. Cogolludo-Agustin, and M. A. Marco-Buzunariz On time averaged optimization of dynamic inequalities on a circle by A. Davydov Thom polynomial computing strategies. A survey by L. M. Feher and R. Rimanyi The complex crystallographic groups and symmetries of $J_{10}$ by V. Goryunov and S. H. Man $tt^*$ geometry and mixed Hodge structures by C. Hertling Thom polynomials by M. Kazarian Quasi-convex decomposition in o-minimal structures. Application to the gradient conjecture by K. Kurdyka and A. Parusinski Homotopy groups of complements to ample divisors by A. Libgober Massey products of complex hypersurface complements by D. Matei On degree of mobility for complete metrics by V. S. Matveev Valuations and moduli of Goursat distributions by P. Mormul Semidifferentiabilite et version lisse de la conjecture de fibration de Whitney by C. Murolo and D. Trotman Submanifolds with a nondegenerate parallel normal vector field in euclidean spaces by J. J. Nuno-Ballesteros Weighted homogeneous polynomials and blow-analytic equivalence by O. M. Abderrahmane Characteristic classes of singular varieties by A. Parusinski On the classification of 7th degree real decomposable curves by G. M. Polotovskiy $\mathcal A$-topological triviality of map germs and Newton filtrations by M. J. Saia and L. M. Soares On the topology of symmetry sets of smooth submanifolds in $\mathbb{R}^k$ by V. D. Sedyh An infinitesimal criterion for topological triviality of families of sections of analytic varieties by M. A. S. Ruas and J. N. Tomazella Lines of principal curvature near singular end points of surfaces in $\mathbb{R}^3$ by J. Sotomayor and R. Garcia $r$ does not imply $n$ or $(npf)$ for definable sets in non polynomially bounded o-minimal structures by D. Trotman and L. Wilson Valuations and local uniformization by M. Vaquie Arc spaces, motivic integration and stringy invariants by W. Veys Finite Dehn surgery along A'Campo's divide knots by Y. Yamada.

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Book SynopsisThis book is an introduction to manifolds at the beginning graduate level, and accessible to any student who has completed a solid undergraduate degree in mathematics. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields.Although this second edition has the same basic structure as the first edition, it has been extensively revised and clarified; not a single page has been left untouched. The major changes include a new introduction to CW complexes (replacing most of the material on simplicial complexes in Chapter 5); expanded treatments of manifolds with boundary, local compactness, group actions, and proper maps; and a new section on paracompactness.Trade ReviewFrom the reviews of the second edition:“An excellent introduction to both point-set and algebraic topology at the early-graduate level, using manifolds as a primary source of examples and motivation. … The author has … fulfilled his objective of integrating a study of manifolds into an introductory course in general and algebraic topology. This text is well-organized and clearly written, with a good blend of motivational discussion and mathematical rigor. … Any student who has gone through this book should be well-prepared to pursue the study of differential geometry … .” (Mark Hunacek, The Mathematical Association of America, March, 2011)“This book is designed for first year graduate students as an introduction to the topology of manifolds. … The book can be read with advantage by any graduate student with a good undergraduate background, and indeed by many upper class undergraduates. It can be used for self study or as a text book for a fine geometrically flavored introduction to manifolds. One which provides excellent motivation for studying the machinery needed for more advanced work.” (Jonathan Hodgson, Zentralblatt MATH, Vol. 1209, 2011)Table of ContentsPreface.- 1 Introduction.- 2 Topological Spaces.- 3 New Spaces from Old.- 4 Connectedness and Compactness.- 5 Cell Complexes.- 6 Compact Surfaces.- 7 Homotopy and the Fundamental Group.- 8 The Circle.- 9 Some Group Theory.- 10 The Seifert-Van Kampen Theorem.- 11 Covering Maps.- 12 Group Actions and Covering Maps.- 13 Homology.- Appendix A: Review of Set Theory.- Appendix B: Review of Metric Spaces.- Appendix C: Review of Group Theory.- References.- Notation Index.- Subject Index.

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Book SynopsisThis is a comprehensive introduction to the modular representation theory of finite groups, with an emphasis on block theory. The two volumes take into account classical results and concepts as well as some of the modern developments in the area. Volume 1 introduces the broader context, starting with general properties of finite group algebras over commutative rings, moving on to some basics in character theory and the structure theory of algebras over complete discrete valuation rings. In Volume 2, blocks of finite group algebras over complete p-local rings take centre stage, and many key results which have not appeared in a book before are treated in detail. In order to illustrate the wide range of techniques in block theory, the book concludes with chapters classifying the source algebras of blocks with cyclic and Klein four defect groups, and relating these classifications to the open conjectures that drive block theory.Trade Review'This 2-volume book is a very welcome addition to the existing literature in modular representation theory. It contains a wealth of material much of which is here presented in textbook form for the first time. It gives an excellent overview of the state of the art in this fascinating subject and also of the many challenging and fundamental open problems. It is well written and will certainly become a standard reference.' Burkhard Kűlshammer, MathSciNetTable of ContentsIntroduction; 6. Blocks and source algebras; 7. Modules over finite p-groups; 8. Local structure; 9. Isometries and bimodules; 10. Structural results in block theory; 11. Blocks with cyclic defect groups; 12. Blocks with Klein four defect groups; Appendix; References; Index.

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Book SynopsisThis book provides the first lucid and accessible account to the modern study of the geometry of four-manifolds. It has become required reading for postgraduates and research workers whose research touches on this topic. Pre-requisites are a firm grounding in differential topology, and geometry as may be gained from the first year of a graduate course. The subject matter of this book is the most significant breakthrough in mathematics of the last fifty years, and Professor Donaldson won a Fields medal for his work in the area. The authors start from the standpoint that the fundamental group and intersection form of a four-manifold provides information about its homology and characteristic classes, but little of its differential topology. It turns out that the classification up to diffeomorphism of four-manifolds is very different from the classification of unimodular forms and that the study of this question leads naturally to the new Donaldson invariants of four-manifolds. A central tTrade Review... authoritative and comprehensive... it must be regarded as compulsory reading for any young researcher approaching this difficult but fascinating area. * Bulletin of the London Mathematical Society *Table of Contents1. Four-manifolds ; 2. Connections ; 3. The Fourier transform and ADHM construction ; 4. Yang-Mills moduli spaces ; 5. Topology and connections ; 6. Stable holomorphic bundles over Kahler surfaces ; 7. Excision and glueing ; 8. Non-existence results ; 9. Invariants of smooth four-manifolds ; 10. The differential topology of algebraic surfaces ; Appendix ; References ; Index

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Book Synopsis1. K0 of Rings.- 1. Defining K0.- 2. K0 from idempotents.- 3. K0 of PIDs and local rings.- 4. K0 of Dedekind domains.- 5. Relative K0 and excision.- 6. An application: Swan's Theorem and topological K- theory.- 7. Another application: Euler characteristics and the Wall finiteness obstruction.- 2.K1 of Rings.- 1. Defining K1.- 2. K1 of division rings and local rings.- 3. 1 of PIDs and Dedekind domains.- 4. Whitehead groups and Whitehead torsion.- 5. Relative K1 and the exact sequence.- 3. K0 and K1 of Categories, Negative K-Theory.- 1. K0 and K1 of categories, Go and G1 of rings.- 2. The Grothendieck and Bass-Heller-Swan Theorems.- 3. Negative K-theory.- 4. Milnor's K2.- 1. Universal central extensions and H2.- 2. The Steinberg group.- 3. Milnor's K2.- 4. Applications of K2.- 5. The +?Construction and Quillen K-Theory.- 1. An introduction to classifying spaces.- 2. Quillen's +?construction and its basic properties.- 3. A survey of higher K-theory.- 6. Cyclic homology and its relation to K-Theory.- 1. Basics of cyclic homology.- 2. The Chern character.- 3. Some applications.- References.- Books and Monographs on Related Areas of Algebra, Analysis, Number Theory, and Topology.- Books and Monographs on Algebraic K-Theory.- Specialized References.- Notational Index.Table of Contents1. K0 of Rings.- 1. Defining K0.- 2. K0 from idempotents.- 3. K0 of PIDs and local rings.- 4. K0 of Dedekind domains.- 5. Relative K0 and excision.- 6. An application: Swan’s Theorem and topological K- theory.- 7. Another application: Euler characteristics and the Wall finiteness obstruction.- 2.K1 of Rings.- 1. Defining K1.- 2. K1 of division rings and local rings.- 3. 1 of PIDs and Dedekind domains.- 4. Whitehead groups and Whitehead torsion.- 5. Relative K1 and the exact sequence.- 3. K0 and K1 of Categories, Negative K-Theory.- 1. K0 and K1 of categories, Go and G1 of rings.- 2. The Grothendieck and Bass-Heller-Swan Theorems.- 3. Negative K-theory.- 4. Milnor’s K2.- 1. Universal central extensions and H2.- Universal central extensions.- Homology of groups.- 2. The Steinberg group.- 3. Milnor’s K2.- 4. Applications of K2.- Computing certain relative K1 groups.- K2 of fields and number theory.- Almost commuting operators.- Pseudo-isotopy.- 5. The +?Construction and Quillen K-Theory.- 1. An introduction to classifying spaces.- 2. Quillen’s +?construction and its basic properties.- 3. A survey of higher K-theory.- Products.- K-theory of fields and of rings of integers.- The Q-construction and results proved with it.- Applications.- 6. Cyclic homology and its relation to K-Theory.- 1. Basics of cyclic homology.- Hochschild homology.- Cyclic homology.- Connections with “non-commutative de Rham theory”.- 2. The Chern character.- The classical Chern character.- The Chern character on K0.- The Chern character on higher K-theory.- 3. Some applications.- Non-vanishing of class groups and Whitehead groups.- Idempotents in C*-algebras.- Group rings and assembly maps.- References.- Books and Monographs on Related Areas of Algebra, Analysis, Number Theory, and Topology.- Books and Monographs on Algebraic K-Theory.- Specialized References.- Notational Index.

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Book SynopsisA clear exposition, with exercises, of the basic ideas of algebraic topology. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.Table of Contents0 Introduction.- Notation.- Brouwer Fixed Point Theorem.- Categories and Functors.- 1.Some Basic Topological Notions.- Homotopy.- Convexity, Contractibility, and Cones.- Paths and Path Connectedness.- 2 Simplexes.- Affine Spaces.- Affine Maps.- 3 The Fundamental Group.- The Fundamental Groupoid.- The Functor ?1.- ?1(S1).- 4 Singular Homology.- Holes and Green’s Theorem.- Free Abelian Groups.- The Singular Complex and Homology Functors.- Dimension Axiom and Compact Supports.- The Homotopy Axiom.- The Hurewicz Theorem.- 5 Long Exact Sequences.- The Category Comp.- Exact Homology Sequences.- Reduced Homology.- 6 Excision and Applications.- Excision and Mayer-Vietoris.- Homology of Spheres and Some Applications.- Barycentric Subdivision and the Proof of Excision.- More Applications to Euclidean Space.- 7 Simplicial Complexes.- Definitions.- Simplicial Approximation.- Abstract Simplicial Complexes.- Simplicial Homology.- Comparison with Singular Homology.- Calculations.- Fundamental Groups of Polyhedra.- The Seifert-van Kampen Theorem.- 8 CW Complexes.- Hausdorff Quotient Spaces.- Attaching Cells.- Homology and Attaching Cells.- CW Complexes.- Cellular Homology.- 9 Natural Transformations.- Definitions and Examples.- Eilenberg-Steenrod Axioms.- Chain Equivalences.- Acyclic Models.- Lefschetz Fixed Point Theorem.- Tensor Products.- Universal Coefficients.- Eilenberg-Zilber Theorem and the Künneth Formula.- 10 Covering Spaces.- Basic Properties.- Covering Transformations.- Existence.- Orbit Spaces.- 11 Homotopy Groups.- Function Spaces.- Group Objects and Cogroup Objects.- Loop Space and Suspension.- Homotopy Groups.- Exact Sequences.- Fibrations.- A Glimpse Ahead.- 12 Cohomology.- Differential Forms.- Cohomology Groups.- Universal Coefficients Theorems for Cohomology.- Cohomology Rings.- Computations and Applications.- Notation.

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Book SynopsisThese conjectures can be viewed as a vast generalization of Dirichlet’s class number formula and Kronecker’s limit formula. They provide an unexpected contribution to Hilbert’s 12th problem on the generalization of class fields by the values of transcendental functions.Table of ContentsIntroduction.-Fonctions L D’Artin.-La Conjecture Principale de Stark.-Caracteres a Valeurs Rationnelles.-Les Cas r(x)=0 et r(x)=1.-La Conjecture Plus Fine Dans le Cas Abelien.-Le Cas Des Corps de Fonctions.-Analogues p-Adiques des Conjectures de Stark.-Bibliographie.

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Book Synopsis1 Categories.- 2 Modules.- 2.1 Generalities.- 2.2 Tensor Products.- 2.3 Exactness of Functors.- 2.4 Projectives, Injectives, and Flats.- 3 Ext and Tor.- 3.1 Complexes and Projective Resolutions.- 3.2 Long Exact Sequences.- 3.3 Flat Resolutions and Injective Resolutions.- 3.4 Consequences.- 4 Dimension Theory.- 4.1 Dimension Shifting.- 4.2 When Flats are Projective.- 4.3 Dimension Zero.- 4.4 An Example.- 5 Change of Rings.- 5.1 Computational Considerations.- 5.2 Matrix Rings.- 5.3 Polynomials.- 5.4 Quotients and Localization.- 6 Derived Functors.- 6.1 Additive Functors.- 6.2 Derived Functors.- 6.3 Long Exact SequencesI. Existence.- 6.4 Long Exact SequencesII. Naturality.- 6.5 Long Exact SequencesIII. Weirdness.- 6.6 Universality of Ext.- 7 Abstract Homologieal Algebra.- 7.1 Living Without Elements.- 7.2 Additive Categories.- 7.3 Kernels and Cokernels.- 7.4 Cheating with Projectives.- 7.5 (Interlude) Arrow Categories.- 7.6 Homology in Abelian Categories.- 7.7 Long Exact Sequences.- 7.8 An Alternative for Unbalanced Categories.- 8 Colimits and Tor.- 8.1 Limits and Colimits.- 8.2 Adjoint Functors.- 8.3 Directed Colimits, ?, and Tor.- 8.4 Lazard's Theorem.- 8.5 Weak Dimension Revisited.- 9 Odds and Ends.- 9.1 Injective Envelopes.- 9.2 Universal Coefficients.- 9.3 The Künneth Theorems.- 9.4 Do Connecting Homomorphisms Commute?.- 9.5 The Ext Product.- 9.6 The Jacobson Radical, Nakayama's Lemma, and Quasilocal Rings.- 9.7 Local Rings and Localization Revisited (Expository).- A GCDs, LCMs, PIDs, and UFDs.- B The Ring of Entire Functions.- C The MitchellFreyd Theorem and Cheating in Abelian Categories.- D Noether Correspondences in Abelian Categories.- Solution Outlines.- References.- Symbol Index.Trade Review“Each chapter contains a reasonable selection of exercises. … its intended audience is second or third year graduate students in algebra, algebraic topology, or other fields that use homological algebra. … the author’s style is both readable and entertaining … . All in all, this book is a very welcome addition to the literature.” (T.W.Hungerford, zbMATH 0948.18001, 2022)"The book is well written. We find here many examples. Each chapter is followed by exercises, and at the end of the book there are outline solutions to some of them. ... I especially appreciated the lively style of the book; compared with some other books on homological algebra, one has here the good feeling that one understands why a notion is defined in this way,that one can easily remember at least the structure of the theory, and that one is quickly able to find necessary details. The prerequisite for this book is a graduate course on algebra, but one get quite far with a modest knowledge of algebra. The book can be strongly recommended as a textbook for a course on homological algebra."EMS Newsletter, June 2001Table of Contents1 Categories.- 2 Modules.- 2.1 Generalities.- 2.2 Tensor Products.- 2.3 Exactness of Functors.- 2.4 Projectives, Injectives, and Flats.- 3 Ext and Tor.- 3.1 Complexes and Projective Resolutions.- 3.2 Long Exact Sequences.- 3.3 Flat Resolutions and Injective Resolutions.- 3.4 Consequences.- 4 Dimension Theory.- 4.1 Dimension Shifting.- 4.2 When Flats are Projective.- 4.3 Dimension Zero.- 4.4 An Example.- 5 Change of Rings.- 5.1 Computational Considerations.- 5.2 Matrix Rings.- 5.3 Polynomials.- 5.4 Quotients and Localization.- 6 Derived Functors.- 6.1 Additive Functors.- 6.2 Derived Functors.- 6.3 Long Exact Sequences—I. Existence.- 6.4 Long Exact Sequences—II. Naturality.- 6.5 Long Exact Sequences—III. Weirdness.- 6.6 Universality of Ext.- 7 Abstract Homologieal Algebra.- 7.1 Living Without Elements.- 7.2 Additive Categories.- 7.3 Kernels and Cokernels.- 7.4 Cheating with Projectives.- 7.5 (Interlude) Arrow Categories.- 7.6 Homology in Abelian Categories.- 7.7 Long Exact Sequences.- 7.8 An Alternative for Unbalanced Categories.- 8 Colimits and Tor.- 8.1 Limits and Colimits.- 8.2 Adjoint Functors.- 8.3 Directed Colimits, ?, and Tor.- 8.4 Lazard’s Theorem.- 8.5 Weak Dimension Revisited.- 9 Odds and Ends.- 9.1 Injective Envelopes.- 9.2 Universal Coefficients.- 9.3 The Künneth Theorems.- 9.4 Do Connecting Homomorphisms Commute?.- 9.5 The Ext Product.- 9.6 The Jacobson Radical, Nakayama’s Lemma, and Quasilocal Rings.- 9.7 Local Rings and Localization Revisited (Expository).- A GCDs, LCMs, PIDs, and UFDs.- B The Ring of Entire Functions.- C The Mitchell—Freyd Theorem and Cheating in Abelian Categories.- D Noether Correspondences in Abelian Categories.- Solution Outlines.- References.- Symbol Index.

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Book SynopsisWe will develop the K -theory of Banach algebras, the theory of extensions of C*-algebras, and the operator K -theory of Kasparov from scratch to its most advanced aspects.Table of ContentsI. Introduction To K-Theory.- 1. Survey of topological K-theory.- 2. Overview of operator K-theory.- II. Preliminaries.- 3. Local Banach algebras and inductive limits.- 4. Idempotents and equivalence.- III. K0-Theory and Order.- 5. Basi K0-theory.- 6. Order structure on K0.- 7. Theory of AF algebras.- IV. K1-Theory and Bott Periodicity.- 8. Higher K-groups.- 9. Bott Periodicity.- V. K-Theory of Crossed Products.- 10. The Pimsner-Voiculescu exact sequence and Connes’ Thorn isomorphism.- 11. Equivariant K-theory.- VI. More Preliminaries.- 12. Multiplier algebras.- 13. Hilbert modules.- 14. Graded C*-algebras.- VII. Theory of Extensions.- 15. Basic theory of extensions.- 16. Brown-Douglas-Fillmore theory and other applications.- VIII. Kasparov’s KK-Theory.- 17. Basic theory.- 18. Intersection product.- 19. Further structure in KK-theory.- 20. Equivariant KK-theory.- IX. Further Topics.- 21. Homology and cohomology theories on C*-algebras.- 22. Axiomatic K-theory.- 23. Universal coefficient theorems and Künneth theorems.- 24. Survey of applications to geometry and topology.

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Trade Review“The present book gives a detailed treatment of the standard monomial theory (SMT) for the Grassmannians and their Schubert subvarieties along with several applications of SMT. It can be used as a reference book by experts and graduate students who study varieties with a reductive group action such as flag and toric varieties.” (Valentina Kiritchenko, zbMATH 1343.14001, 2016)“The book under review is more elementary; it is exclusively devoted to Grassmannians and their Schubert subvarieties. The book is divided into three parts. … This is a nicely written book, one that may appeal to students and researchers in related areas.” (Felipe Zaldivar, MAA Reviews, maa.org, December, 2015)Table of ContentsPreface.- 1. Introduction.- Part I. Algebraic Geometry—A Brief Recollection - 2. Preliminary Material.- 3. Cohomology Theory.- 4. Gröbner Bases.- Part II. Grassmannian and Schubert Varieties.- 5. The Grassmannian and Its Schubert Varieties.- 6. Further Geometric Properties of Schubert Varieties.- 7. Flat Degenerations.- Part III. Flag Varieties and Related Varieties.- 8. The Flag Variety: Geometric and Representation-Theoretic Aspects.- 9. Relationship to Classical Invariant Theory.- 10. Determinantal Varieties.- 11. Related Topics.- References.- List of Symbols.- Index.

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Table of ContentsLocally compact groupoids.- The C*-algebra of a groupoid.- Some examples.

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Table of ContentsContinuum mechanics and geometric integration theory.- Structure of continuum physics.- On differentiable spaces.- Cartesian closed categories and analysis of smooth maps.- to synthetic differential geometry, and a synthetic theory of dislocations.- Synthetic reasoning and variable sets.- Recent research on the foundations of thermodynamics.- Global and local versions of the second law of thermodynamics.- Thermodynamics and the hahn-banach theorem.- What is the length of a potato?.

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Book SynopsisFreeness of an action of a compact Lie group on a compact Hausdorff space is equivalent to a simple condition on the corresponding equivariant K-theory. This fact can be regarded as a theorem on actions on a commutative C*-algebra, namely the algebra of continuous complex-valued functions on the space. The successes of "noncommutative topology" suggest that one should try to generalize this result to actions on arbitrary C*-algebras. Lacking an appropriate definition of a free action on a C*-algebra, one is led instead to the study of actions satisfying conditions on equivariant K-theory - in the cases of spaces, simply freeness. The first third of this book is a detailed exposition of equivariant K-theory and KK-theory, assuming only a general knowledge of C*-algebras and some ordinary K-theory. It continues with the author's research on K-theoretic freeness of actions. It is shown that many properties of freeness generalize, while others do not, and that certain forms of K-theoretic freeness are related to other noncommutative measures of freeness, such as the Connes spectrum. The implications of K-theoretic freeness for actions on type I and AF algebras are also examined, and in these cases K-theoretic freeness is characterized analytically.Table of ContentsIntroduction: The commutative case.- Equivariant K-theory of C*-algebras.- to equivariant KK-theory.- Basic properties of K-freeness.- Subgroups.- Tensor products.- K-freeness, saturation, and the strong connes spectrum.- Type I algebras.- AF algebras.

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Book SynopsisFollowing Quillen's approach to complex cobordism, the authors introduce the notion of oriented cohomology theory on the category of smooth varieties over a fixed field. They prove the existence of a universal such theory (in characteristic 0) called Algebraic Cobordism. The book also contains some examples of computations and applications.Table of ContentsCobordism and oriented cohomology.- The definition of algebraic cobordism.- Fundamental properties of algebraic cobordism.- Algebraic cobordism and the Lazard ring.- Oriented Borel-Moore homology.- Functoriality.- The universality of algebraic cobordism.

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Book SynopsisFrom the reviews: "The author has attempted an ambitious and most commendable project. […] The book contains much material that has not previously appeared in this format. The writing is clean and clear and the exposition is well motivated. [...] This book is, all in all, a very admirable work and a valuable addition to the literature." Mathematical Reviews Trade ReviewFrom the reviews: "This book contains much impressive mathematics, namely the achievements by algebraic topologists in obtaining extensive information on the stable homotopy groups of spheres, and the computation of various cobordism groups. It is a long book, and for the major part a very advanced book. ... (It is) suitable for specialists, or for those who already know what algebraic topology is for, and want a guide to the principal methods of stable homotopy theory."R. Brown in Bulletin of the London Mathematical Society, 1980 "In the more than twenty five years since its first appearance, the book has met with favorable response, both in its use as a text and as reference. It is a good course which leads the reader systematically to the point at which he can begin to tackle problems in algebraic topology. … This book remains one of the best sources for the material which every young algebraic topologist should know." (Corina Mohorianu, Zentralblatt MATH, Vol. 1003 (3), 2003)Table of Contentso. Some Facts from General Topology 1. Categories, Functors and Natural Transformations 2. Homotopy Sets and Groups 3. Properties of the Homotopy Groups 4. Fibrations 5. CW-Complexes 6. Homotopy Properties of CW-Complexes 7. Homology and Cohomology Theories 8. Spectra 9. Representation Theorems 10. Ordinary Homology Theory 11. Vector Bundles and K-Theory 12. Manifolds and Bordism 13. Products 14. Orientation and Duality 15. Spectral Swquences 16. Characteristic Classes 17. Cohomology Operations and Homology Cooperations 18. The Steenrod Algebra and its Dual 19. The Adams Spectral Sequence and the e-Invariant 20. Calculation of the Corbordism Groups Bibliography Subject Index

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Book SynopsisSheaf Theory is modern, active field of mathematics at the intersection of algebraic topology, algebraic geometry and partial differential equations. This volume offers a comprehensive and self-contained treatment of Sheaf Theory from the basis up, with emphasis on the microlocal point of view. From the reviews: "Clearly and precisely written, and contains many interesting ideas: it describes a whole, largely new branch of mathematics." –Bulletin of the L.M.S.Table of ContentsA Short History: Les débuts de la théorie des faisceaux.- I. Homological algebra.- II. Sheaves.- III. Poincaré-Verdier duality and Fourier-Sato transformation.- IV. Specialization and microlocalization.- V. Micro-support of sheaves.- VI. Micro-support and microlocalization.- VII. Contact transformations and pure sheaves.- VIII. Constructible sheaves.- IX. Characteristic cycles.- X. Perverse sheaves.- XI. Applications to O-modules and D-modules.- Appendix: Symplectic geometry.- Summary.- A.1. Symplectic vector spaces.- A.2. Homogeneous symplectic manifolds.- A.3. Inertia index.- Exercises to the Appendix.- Notes.- List of notations and conventions.

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Book SynopsisFrom the Preface: K-theory was introduced by A. Grothendieck in his formulation of the Riemann- Roch theorem. For each projective algebraic variety, Grothendieck constructed a group from the category of coherent algebraic sheaves, and showed that it had many nice properties. Atiyah and Hirzebruch con­sidered a topological analog defined for any compact space X, a group K{X) constructed from the category of vector bundles on X. It is this ''topological K-theory" that this book will study. Topological K-theory has become an important tool in topology. Using K- theory, Adams and Atiyah were able to give a simple proof that the only spheres which can be provided with H-space structures are S1, S3 and S7. Moreover, it is possible to derive a substantial part of stable homotopy theory from K-theory.The purpose of this book is to provide advanced students and mathematicians in other fields with the fundamental material in this subject. In addition, several applications of the type described above are included. In general we have tried to make this book self-contained, beginning with elementary concepts wherever possible; however, we assume that the reader is familiar with the basic definitions of homotopy theory: homotopy classes of maps and homotopy groups.Thus this book might be regarded as a fairly self-contained introduction to a "generalized cohomology theory".Trade ReviewFrom the reviews: "Karoubi’s classic K-Theory, An Introduction … is ‘to provide advanced students and mathematicians in other fields with the fundamental material in this subject’. … K-Theory, An Introduction is a phenomenally attractive book: a fantastic introduction and then some. … serve as a fundamental reference and source of instruction for outsiders who would be fellow travelers." (Michael Berg, MAA Online, December, 2008)Table of ContentsVector Bundles.- First Notions of K-Theory.- Bott Periodicity.- Computation of Some K-Groups.- Some Applications of K-Theory.- Vector Bundles.- First Notions of K-Theory.- Bott Periodicity.- Computation of Some K-Groups.

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Book SynopsisFrom the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coming from algebraic topology. The bibliographic comments at the end of each chapter offer good suggestions for further reading and research. The book can be strongly recommended to anybody interested in noncommutative geometry, contemporary algebraic topology and related topics." European Mathematical Society Newsletter In this second edition the authors have added a chapter 13 on MacLane (co)homology.Trade ReviewFrom the reviews: "This is a very interesting book containing material for a comprehensive study of the cyclid homological theory of algebras, cyclic sets and S1-spaces. Lie algebras and algebraic K-theory and (in the last chapter) an introduction to Connes'work and recent results on the Novikov conjecture. The book requires a knowledge of homological algebra and Lie algebra theory as well as basic technics coTable of Contents1. Hochschild Homology.- 2. Cyclic Homology of Algebras.- 3. Smooth Algebras and Other Examples.- 4. Operations on Hochschild and Cyclic Homology.- 5. Variations on Cyclic Homology.- 6. The Cyclic Category, Tor and Ext Interpretation.- 7. Cyclic Spaces and Sl-Equivariant Homology.- 8. Chern Character.- 9. Classical Invariant Theory.- 10. Homology of Lie Algebras of Matrices.- 11. Algebraic K-Theory.- 12. Non-commutative Differential Geometry.- 13. Mac Lane (co)homology.- Appendices.- A. Hopf Algebras.- B. Simplicial.- C. Homology of Discrete Groups and Small Categories.- D. Spectral Sequences.- E. Smooth Algebras.- References.- References 1992–1996.- Symbols.

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Book SynopsisThis volume deals with the theory of finite topological spaces and its relationship with the homotopy and simple homotopy theory of polyhedra. The interaction between their intrinsic combinatorial and topological structures makes finite spaces a useful tool for studying problems in Topology, Algebra and Geometry from a new perspective. In particular, the methods developed in this manuscript are used to study Quillen's conjecture on the poset of p-subgroups of a finite group and the Andrews-Curtis conjecture on the 3-deformability of contractible two-dimensional complexes. This self-contained work constitutes the first detailed exposition on the algebraic topology of finite spaces. It is intended for topologists and combinatorialists, but it is also recommended for advanced undergraduate students and graduate students with a modest knowledge of Algebraic Topology.Trade ReviewFrom the reviews:“This book deals with the algebraic topology of finite topological spaces and its applications, and includes well-known results on finite spaces and original results developed by the author. The book is self-contained and well written. It is understandable and enjoyable to read. It contains a lot of examples and figures which help the readers to understand the theory.” (Fumihiro Ushitaki, Mathematical Reviews, March, 2014)“This book illustrates convincingly the idea that the study of finite non-Hausdorff spaces from a homotopical point of view is useful in many areas and can even be used to study well-known problems in classical algebraic topology. … This book is a revised version of the PhD Thesis of the author. … All the concepts introduced with the chapters are usefully illustrated by examples and the recollection of all these results gives a very nice introduction to a domain of growing interest.” (Etienne Fieux, Zentralblatt MATH, Vol. 1235, 2012)Table of Contents1 Preliminaries.- 2 Basic topological properties of finite spaces.- 3 Minimal finite models.- 4 Simple homotopy types and finite spaces.- 5 Strong homotopy types.- 6 Methods of reduction.- 7 h-regular complexes and quotients.- 8 Group actions and a conjecture of Quillen.- 9 Reduced lattices.- 10 Fixed points and the Lefschetz number.- 11 The Andrews-Curtis conjecture.

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