Algebraic topology Books

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 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 SynopsisGraduate students and researchers in modular representation theory, especially block theory, will find this systematic introduction indispensable. The two volumes include detailed treatments of classic material as well as more modern developments which have not appeared in any book before, giving readers a comprehensive overview of the subject.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; 1. Algebras and modules; 2. Functors between module categories; 3. Character theory; 4. Algebras over p-local rings; 5. Group algebras and modules over p-local rings; References; Index.

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Book SynopsisThis second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.Trade ReviewFrom the reviews of the second edition:“The book under review is an introduction to the theory of categories which, as the title suggests, is addressed to the (no-nonsense) working mathematician, thus presenting the ideas and concepts of Category Theory in a broad context of mainstream examples (primarily from algebra). … the book remains an authoritative source on the foundations of the theory and an accessible first introduction to categories. … It is very well-written, with plenty of interesting discussions and stimulating exercises.” (Ittay Weiss, MAA Reviews, July, 2014)Second EditionS.M. LaneCategories for the Working Mathematician"A very useful introduction to category theory."—INTERNATIONALE MATHEMATISCHE NACHRICHTENTable of Contents1: Categories, Functors and Natural Transformation. 2: Constructions on Categories. 3: Universals and Limits. 4: Adjoints. 5: Limits. 6: Monads and Algebras. 7: Monoids. 8: Abelian Categories. 9: Special Limits. 10: Kan Extensions. 11: Symmetry and Braiding in Monoidal Categories. 12: Structures in Categories. Tables of Categories. Bibliography.

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Book SynopsisIn most mathematics departments at major universities one of the three or four basic first-year graduate courses is in the subject of algebraic topology. This introductory textbook in algebraic topology is suitable for use in a course or for self-study, featuring broad coverage of the subject and a readable exposition, with many examples and exercises. The four main chapters present the basic material of the subject: fundamental group and covering spaces, homology and cohomology, higher homotopy groups, and homotopy theory generally. The author emphasizes the geometric aspects of the subject, which helps students gain intuition. A unique feature of the book is the inclusion of many optional topics which are not usually part of a first course due to time constraints, and for which elementary expositions are sometimes hard to find. Among these are: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James rTrade Review'… this is a marvellous tome, which is indeed a delight to read. This book is destined to become very popular amongst students and teachers alike.' Bulletin of the Belgian Mathematical Society'… clear and concise … makes the book useful both as a basis for courses and as a reference work.' Monatshefte für Mathematik'… the truly unusual abundance of instructive examples and complementing exercises is absolutely unique of such a kind … the distinctly circumspect, methodologically inductive, intuitive, descriptively elucidating and very detailed style of writing give evidence to the fact that the author's first priorities are exactly what students need when working with such a textbook, namely clarity, readability, steady motivation, guided inspiration, increasing demand, and as much self-containedness of the exposition as possible. No doubt, a very devoted and experienced teacher has been at work here, very much so to the benefit of beginners in the field of algebraic topology, instructors, and interested readers in general.' Zentralblatt MATHTable of ContentsPart I. Some Underlying Geometric Notions: 1. Homotopy and homotopy type; 2. Deformation retractions; 3. Homotopy of maps; 4. Homotopy equivalent spaces; 5. Contractible spaces; 6. Cell complexes definitions and examples; 7. Subcomplexes; 8. Some basic constructions; 9. Two criteria for homotopy equivalence; 10. The homotopy extension property; Part II. Fundamental Group and Covering Spaces: 11. The fundamental group, paths and homotopy; 12. The fundamental group of the circle; 13. Induced homomorphisms; 14. Van Kampen's theorem of free products of groups; 15. The van Kampen theorem; 16. Applications to cell complexes; 17. Covering spaces lifting properties; 18. The classification of covering spaces; 19. Deck transformations and group actions; 20. Additional topics: graphs and free groups; 21. K(G,1) spaces; 22. Graphs of groups; Part III. Homology: 23. Simplicial and singular homology delta-complexes; 24. Simplicial homology; 25. Singular homology; 26. Homotopy invariance; 27. Exact sequences and excision; 28. The equivalence of simplicial and singular homology; 29. Computations and applications degree; 30. Cellular homology; 31. Euler characteristic; 32. Split exact sequences; 33. Mayer–Vietoris sequences; 34. Homology with coefficients; 35. The formal viewpoint axioms for homology; 36. Categories and functors; 37. Additional topics homology and fundamental group; 38. Classical applications; 39. Simplicial approximation and the Lefschetz fixed point theorem; Part IV. Cohomology: 40. Cohomology groups: the universal coefficient theorem; 41. Cohomology of spaces; 42. Cup product the cohomology ring; 43. External cup product; 44. Poincaré duality orientations; 45. Cup product; 46. Cup product and duality; 47. Other forms of duality; 48. Additional topics the universal coefficient theorem for homology; 49. The Kunneth formula; 50. H-spaces and Hopf algebras; 51. The cohomology of SO(n); 52. Bockstein homomorphisms; 53. Limits; 54. More about ext; 55. Transfer homomorphisms; 56. Local coefficients; Part V. Homotopy Theory: 57. Homotopy groups; 58. The long exact sequence; 59. Whitehead's theorem; 60. The Hurewicz theorem; 61. Eilenberg–MacLane spaces; 62. Homotopy properties of CW complexes cellular approximation; 63. Cellular models; 64. Excision for homotopy groups; 65. Stable homotopy groups; 66. Fibrations the homotopy lifting property; 67. Fiber bundles; 68. Path fibrations and loopspaces; 69. Postnikov towers; 70. Obstruction theory; 71. Additional topics: basepoints and homotopy; 72. The Hopf invariant; 73. Minimal cell structures; 74. Cohomology of fiber bundles; 75. Cohomology theories and omega-spectra; 76. Spectra and homology theories; 77. Eckmann-Hilton duality; 78. Stable splittings of spaces; 79. The loopspace of a suspension; 80. Symmetric products and the Dold–Thom theorem; 81. Steenrod squares and powers; Appendix: topology of cell complexes; The compact-open topology.

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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 SynopsisDeveloped from a first-year graduate course in algebraic topology, this text is an informal introduction to some of the main ideas of contemporary homotopy and cohomology theory. The materials are structured around four core areas: de Rham theory, the Cech-de Rham complex, spectral sequences, and characteristic classes.Trade Review“Bott and Tu give us an introduction to algebraic topology via differential forms, imbued with the spirit of a master who knew differential forms way back when, yet written from a mature point of view which draws together the separate paths traversed by de Rham theory and homotopy theory. Indeed they assume "an audience with prior exposure to algebraic or differential topology". It would be interesting to use Bott and Tu as the text for a first graduate course in algebraic topology; it would certainly be a wonderful supplement to a standard text. “Bott and Tu write with a consistent point of view and a style which is very readable, flowing smoothly from topic to topic. Moreover, the differential forms and the general homotopy theory are well integrated so that the whole is more than the sum of its parts. "Not intended to be foundational", the book presents most key ideas, at least in sketch form, from scratch, but does not hesitate to quote as needed, without proof, major results of a technical nature, e.g., Sard's Theorem, Whitney's Embedding Theorem and the Morse Lemma on the form of a nondegenerate critical point.” —James D. Stasheff (Bulletin of the American Mathematical Society) “This book is an excellent presentation of algebraic topology via differential forms. The first chapter contains the de Rham theory, with stress on computability. Thus, the Mayer-Vietoris technique plays an important role in the exposition. The force of this technique is demonstrated by the fact that the authors at the end of this chapter arrive at a really comprehensive exposition of Poincaré duality, the Euler and Thom classes and the Thom isomorphism. “The second chapter develops and generalizes the Mayer-Vietoris technique to obtain in a very natural way the Čech-de Rham complex and the Čech cohomology for presheaves. The third chapter on spectral sequences is the most difficult one, but also the richest one by the various applications and digressions into other topics of algebraic topology: singular homology and cohomology with integer coefficients and an important part of homotopy theory, including the Hopf invariant, the Postnikov approximation, the Whitehead tower and Serre’s theorem on the homotopy of spheres. The last chapter is devoted to a brief and comprehensive description of the Chern and Pontryagin classes. “A book which covers such an interesting and important subject deserves some remarks on the style: On the back cover one can read “With its stress on concreteness, motivation, and readability, Differential forms in algebraic topology should be suitable for self-study.” This must not be misunderstood in the ense that it is always easy to read the book. The authors invite the reader to understand algebraic topology by completing himself proofs and examples in the exercises. The reader who seriously follows this invitation really learns a lot of algebraic topology and mathematics in general.” —Hansklaus Rummler (American Mathematical Society)Table of ContentsI De Rham Theory.- II The ?ech-de Rham Complex.- III Spectral Sequences and Applications.- IV Characteristic Classes.- References.- List of Notations.

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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 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 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 textbook on algebraic topology updates a popular textbook from the golden era of the Moscow school of I. M. Gelfand. The first English translation, done many decades ago, remains very much in demand, although it has been long out-of-print and is difficult to obtain. Therefore, this updated English edition will be much welcomed by the mathematical community. Distinctive features of this book include: a concise but fully rigorous presentation, supplemented by a plethora of illustrations of a high technical and artistic caliber; a huge number of nontrivial examples and computations done in detail; a deeper and broader treatment of topics in comparison to most beginning books on algebraic topology; an extensive, and very concrete, treatment of the machinery of spectral sequences. The second edition contains an entirely new chapter on K-theory and the Riemann-Roch theorem (after Hirzebruch and Grothendieck).Trade Review“This book is a treasure trove for every mathematician who has to deal with classical algebraic topology and homotopy theory on the research level. … Its style is refreshing and informative, and the reader can feel the authors’ joy at sharing their insight into algebraic topology. … will be a useful addition to any mathematical bookshelf.” (Thomas Hüttemann, Mathematical Reviews, March, 2017)“This book covers all the basic material necessary for complete understanding of the fundamentals of algebraic topology … . This increase in the number of topics has made the book more convenient for serious students not only to extend their knowledge but also to gain insight into the interplay between these three subjects. … This book is designed to help students to select the level of learning subjects they want to reach … .” (Haruo Minami, zbMATH 1346.55001, 2016)Table of ContentsIntroduction.- Homotopy.- Homology.- Spectral Sequences of Fibrations.- Cohomology Operations.- The Adams Spectral Sequence.- K-Theory and Other Extraordinary Cohomology Theories.

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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 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 SynopsisThis text puts together folklore results touching weak Asplund spaces. It presents a thorough examination of weak Asplund cases. Nonseparable Banach spaces, renorming, and differentiability are stressed throughout the text and all subclasses, including inferences and counterexamples are discussed. It also covers Stegall''s classes, fragmentability, and long sequences of linear projections. Notes, remarks and questions end each chapter.Table of ContentsCanonical Examples of Weak Asplund Spaces. Properties of Gateaux Differentiability Spaces and Weak AsplundSpaces. Stegall's Classes. Two More Concrete Classes of Banach Spaces that Lie in . Fragmentability. "Long Sequences" of Linear Projections. Vaak Spaces and Gul'ko Compacta. A Characterization of WCG Spaces and of Eberlein Compacta. Main Open Questions and Problems. References. Index.

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Book SynopsisA powerful introduction to one of the most active areas of theoretical and applied mathematics This distinctive introduction to one of the most far-reaching and beautiful areas of mathematics focuses on Banach spaces as the milieu in which most of the fundamental concepts are presented.Trade Review"This textbook for a two-semester course in functional analysis presents the basic ideas, techniques, and methods that form the underpinnings of the discipline." (SciTech Book News, Vol. 25, No. 3, September 2001) "...a useful book which helps the student to understand Banach space theory." (Mathematical Reviews, 2003a)Table of ContentsPreface. Introduction. Basic Definitions and Examples. Basic Principles with Applications. Weak Topologies and Applications. Operators on Banach Spaces. Bases in Banach Spaces. Sequences, Series, and a Little Geometry in Banach Spaces. Bibliography. Author/Name Index. Subject Index Symbol Index.

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Book SynopsisPioneered by David Kendall, the statistical theory of shape is an emerging area generating considerable interest for statisticians, engineers, and computer scientists. Co--written by Dr. Kendall, this volume presents a coherent theory of shape developed from Kendalla s own approach known as static and kinematic theory.Trade Review"This is a fascinating book mixing geometry, topology and probability theory..." (London Mathematical Society Bulletin, Vol 32, 2000) "The potential value that his volume should have to researchers in many areas for years to come." (Short Book Reviews, August 2000) "I would like to conclude this review by strongly recommending that geodists have this book on desk within ready reach of hands" (Journal of Geodesy, Vol. 75, 2001) "...a mathematical jewel..." (Mathematical Reviews, 2003g)Table of ContentsShapes and Shape Spaces. The Global Structure of Shape Spaces. Computing the Homology of Cell Complexes. A Chain Complex for Shape Spaces. The Homology Groups of Shape Spaces. Geodesics in Shape Spaces. The Riemannian Structure of Shape Spaces. Induced Shape-Measures. Mean Shapes and the Shape of the Means. Visualising the Higher Dimensional Shape Spaces. General Shape Spaces. Appendix. Bibliography. Index.

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Book SynopsisThis book contains all research papers published by the distinguished Brazilian mathematician Elon Lima. It includes the papers from his PhD thesis on homotopy theory, which are hard to find elsewhere. Elon Lima wrote more than 40 books in the field of topology and dynamical systems. He was a profound mathematician with a genuine vocation to teach and write mathematics.Table of ContentsComments on some mathematical contributions of Elon Lima.- The Spanier-Whitehead duality in new homotopy categories.- Stable Postnikov invariants and their duals.- Commuting vector fields on 2-manifolds.- On the local triviality of the restriction map for embeddings.- Commuting vector fields on S2.- Common singularities of commuting vector fields on 2-manifolds.- Commuting vector fields on S3.- Isometric immersions with semi-definite second quadratic forms.- Immersions of manifolds with non-negative sectional curvatures.- Orientability of smooth hypersurfaces and the Jordan-Brouwer separation theorem.- The Jordan-Brouwer separation theorem for smooth hypersurfaces.

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Book SynopsisThis is the second revised and extendededition of the successful book on the algebraic structure of the Stone-Čech compactification of a discrete semigroup and its combinatorial applications, primarily in the field known as Ramsey Theory. There has been very active research in the subject dealt with by the book in the 12 years which is now included in this edition. This book is a self-contained exposition of the theory of compact right semigroupsfor discrete semigroups and the algebraic properties of these objects. The methods applied in the book constitute a mosaic of infinite combinatorics, algebra, and topology. The reader will find numerous combinatorial applications of the theory, including the central sets theorem, partition regularity of matrices, multidimensional Ramsey theory, and many more.Trade ReviewThe present book is the first devoted to an extensive study of the algebraic structure of sS and the many applications thereof; it is an exciting book, written - and very well written - by two mathematicians who are eminently qualified two write it, and it is essentially self-contained, requiring only that the reader come to it with the basic concepts of first graduate courses in algebra, analysis and topology. I recommend this book highly; it will be very useful, both to researchers and to students. Its index, list of symbols and up-to-date bibliography are very helpful. Paul Milnes, Zentralblatt MATH / 1998 The authors present a self-contained exposition. The book under review is written by two mathematicians who have contributed in a decisive way to this rapidly expanding area and provides a unique opportunity to obtain a 'colorful' panoramic view of the subject. Michael Tkacenko, MathSciNet / 1999

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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 SynopsisThe ends of a topological space are the directions in which it becomes non-compact by tending to infinity. The tame ends of manifolds are particularly interesting, both for their own sake, and for their use in the classification of high-dimensional compact manifolds. The book is devoted to the related theory and practice of ends, dealing with manifolds and CW complexes in topology and chain complexes in algebra. The first part develops a homotopy model of the behaviour at infinity of a non-compact space. The second part studies tame ends in topology. Tame ends are shown to have a uniform structure, with a periodic shift map. Approximate fibrations are used to prove that tame manifold ends are the infinite cyclic covers of compact manifolds. The third part translates these topological considerations into an appropriate algebraic context, relating tameness to homological properties and algebraic K- and L-theory.Trade Review'The book gathers together the main strands of the theory of ends of manifolds from the last thirty years and presents a unified and coherent treatment of them. It also contains authoritative expositions of certain topics in topology such as mapping tori and telescopes, often omitted from textbooks. It is thus simultaneously a research monograph and a useful reference.' Proceedings of the Edinburgh Mathematical Society'This is a highly specialized monograph which is very clearly written and made as accessible for the reader as possible … It is absolutely indispensable for any specialist in the field.' European Mathematical SocietyTable of ContentsIntroduction; Chapter summaries; Part I. Topology at Infinity: 1. End spaces; 2. Limits; 3. Homology at infinity; 4. Cellular homology; 5. Homology of covers; 6. Projective class and torsion; 7. Forward tameness; 8. Reverse tameness; 9. Homotopy at infinity; 10. Projective class at infinity; 11. Infinite torsion; 12. Forward tameness is a homotopy pushout; Part II. Topology Over the Real Line: 13. Infinite cyclic covers; 14. The mapping torus; 15. Geometric ribbons and bands; 16. Approximate fibrations; 17. Geometric wrapping up; 18. Geometric relaxation; 19. Homotopy theoretic twist glueing; 20. Homotopy theoretic wrapping up and relaxation; Part III. The Algebraic Theory: 21. Polynomial extensions; 22. Algebraic bands; 23. Algebraic tameness; 24. Relaxation techniques; 25. Algebraic ribbons; 26. Algebraic twist glueing; 27. Wrapping up in algebraic K- and L-theory; Part IV. Appendices; References; Index.

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Book SynopsisThis book presents a definitive account of the applications of the algebraic L-theory to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincaré duality space with a local quadratic structure in the chain homotopy type of the universal cover.Trade Review"...develops lower K- and L-theory with a view to applications in topology....Apart from the obvious interest of this text both to topologists and to K-theorists, it also serves as an introduction to the field, since there is a comprehensive survey of previous results and applications." M.E. Keating, Bulletin of the London Mathematical SocietyTable of ContentsIntroduction; Summary; Part I. Algebra: 1. Algebraic Poincaré complexes; 2. Algebraic normal complexes; 3. Algebraic bordism categories; 4. Categories over complexes; 5. Duality; 6. Simply connected assembly; 7. Derived product and Hom; 8. Local Poincaré duality; 9. Universal assembly; 10. The algebraic π-π theorem; 11. ∆-sets; 12. Generalized homology theory; 13. Algebraic L-spectra; 14. The algebraic surgery exact sequence; 15. Connective L-theory; Part II. Topology: 16. The L-theory orientation of topology; 17. The total surgery obstruction; 18. The structure set; 19. Geometric Poincaré complexes; 20. The simply connected case; 21. Transfer; 22. Finite fundamental group; 23. Splitting; 24. Higher signatures; 25. The 4-periodic theory; 26. Surgery with coefficients; Appendices; Bibliography; Index.

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Book SynopsisThis book describes the construction and the properties of CW-complexes. These spaces are important because firstly they are the correct framework for homotopy theory, and secondly most spaces that arise in pure mathematics are of this type.Table of Contents1. The fundamental properties of CW-complexes; 2. Categories of CW-complexes; 3. Polyhedra and simplicial complexes; 4. Simplicial sets; 5. Spaces of the homotopy type of a CW-complex; Appendixes.

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Book SynopsisThe aim of this book is to promote a fibrewise perspective, particularly in topology, which is central to modern mathematics. Already this view is standard in the theory of fibre bundles and therefore in such subjects as global analysis. It has a role to play also in general and equivariant topology. There are strong links with equivariant topology, a topic which has latterly been subject to great research activity. It is to be hoped that this book will provide a solid and invigorating foundation for the increasing research interest in fibrewise topologyTable of ContentsPreface; Part I. Basic Fibrewise Topology: 1. Fibrewise topological spaces; 2. Fibrewise separation conditions; 3. Fibrewise compact spaces; 4. Tied filters; 5. Fibrewise quotient spaces; 6. Fibrewise pointed topological spaces; 7. Relation with equivariant topology; Part II. Further Fibrewise Topology: 8. Fibrewise compactification; 9. The fibrewise mapping-space; 10. Fibrewise compactly-generated spaces; 11. Naturality and naturalization; Part III. Fibrewise Uniform Spaces: 12. Fibrewise uniform structures; 13. Fibrewise uniform topology; 14. The Cauchy completion; 15. Fibrewise completion; 16. Functoriality; 17. Fibrewise compactness and precompactness; Part IV. Fibrewise Homotopy Theory: 18. Fibrewise homotopy; 19. Fibrewise pointed homotopy; 20. Fibrewise confibrations; 21. Fibrewise pointed confibrations; 22. Fibrewise non-degenerate spaces; 23. Fibrewise fibrations; 24. Relation with equivariant homotopy theory; Part V. Miscellaneous Topics: 25. Fibre bundles; 26. Numerable coverings; 27. Fibrewise connectedness; Note on the literature; References; Index.

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Book SynopsisJ. Frank Adams was one of the world's leading topologists. He solved a number of celebrated problems in algebraic topology, a subject in which he initiated many of the most active areas of research. This selection, in two volumes, brings together all his major research contributions.Table of Contents1. On the chain algebra of a loop space; 2. On the cobar construction; 3. The structure of the Steenrod algebra; 4. On the non-existence theory of elements of Hopf invariant one; 4. Applications of the Groethendieck–Atiyah–Hirzebruch functor K(X); 5. Vector fields on spheres; 6. On complex Stiefel manifolds; 7. On matrices whose real linear combinations are non-singular and correction; 8. On the groups J(X) I, II, III, and IV and correction; 9. K-theory and the Hopf invariant; 10. Geometric dimension of bundles over RPn; 11. Lectures on generalised cohomology; 12. Algebraic topology in the last decade.

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Book SynopsisJ. Frank Adams was one of the world's leading topologists. He solved a number of celebrated problems in algebraic topology, a subject in which he initiated many of the most active areas of research. This selection, in two volumes, brings together all his major research contributions.Table of Contents1. On formulae of Thom and Wu; 2. On Chern characters and the structure of the unitary group; 3. Chern characters revisited and the structure of the unitary group; 4. Chern characters revisited and addendum; 5. The Hurewicz homomorphism for MU and BP; 6. Hopf algebras of co-operators for real and complex K-theory; 7. Operations of the Nth kind in K-theory; 8. Operations on K-theory of torsion-free spaces; 9. Stable operations on complex K-theory; 10. Primitive elements in the K-theory of BSU; 11. A finiteness theorem in homological algebra; 12. A periodicity theorem in homological algebra; 13. Modules over the Steenrod algebra; 14. Sub-Hopf-algebras of the Steenrod algebra; 15. What we don't know about RP∞; 16. Calculations of Lin's Ext groups; 17. The Segal conjecture for elementary abelian p-groups; 18. The sphere considered as an H-space mod p; 19. H-spaces with few cells; 20. Finite H-spaces and Lie groups; 21. Spin(8) triality, F4 and all that; 22. The fundamental representations of E8; 23. 2-tori in E8; Maps between classifying spaces I, II, and III; 24. Maps between p-completed classifying spaces; 25. An example in homotopy theory; 26. A variant of E. H. Brown's representability theorem; 27. Idempotent functors in homotopy theory; 28. The Kahn–Priddy theorem; 29. Uniquenesss of BSO; 30. Graeme Segal's Burnsides ring conjecture; 31. A generalisation of the Segal conjecture; 32. A generalisation of the Atiyah–Segal completion theorem; 33. Atomic spaces and spectra; 34. Two theorems of J. Lannes; 35. The work of M. J. Hopkins.

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Book SynopsisProfessor Peter Hilton is one of the best known mathematicians of his generation. He has published almost 300 books and papers on various aspects of topology and algebra. The present volume is to celebrate the occasion of his sixtieth birthday. It begins with a bibliography of his work, followed by reviews of his contributions to topology and algebra. These are followed by eleven research papers concerned with various topics of current interest in algebra and topology. The articles are contributed by some of the many mathematicians with whom he has worked at one time or another. This book will be of interest to both topologists and algebraists, particularly those concerned with homotopy theory.Table of Contents1. Publications of P. J. Hilton; 2. Essay on Hilton's work in topology Guido Mislin; 3. The work of Peter Hilton in algebra Urs Stammbach; 4. The dual Whitehead theorems J. P. May; 5. Homotopy cocomplete classes of spaces and the realization of the singular complex D. Puppe; 6. Betti numbers of Hilbert modular varieties E. Thomas and A. T. Vasquez; 7. Homotopy pairs in Eckmann-Hilton duality K. A. Hardie; 8. Profinite Chern classes for group representations Beno Eckmann and Guido Mislin; 9. Automorphisms of surfaces and class numbers: an illustration of the G-index theorem John Ewing; 10. The real dimension of a vector bundle at the prime two Edgar H. Brown, Jr. and Franklin P. Peterson; 11. Maps between classifying spaces, III J. F. Adams and Z. Mahmud; 12. Idempotent codensity monads and the profinite completion of topological groups A. Deleanu; 13. Finitary automorphisms and integral homology J. Roitberg; 14. Finite group actions on Grassman manifolds Henry H. Glover and Guido Mislin.

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Book SynopsisThis is a memorial volume to the distinguished Canadian-born mathematician Hugh Dowker, one of the most highly regarded topologists in the United Kingdom and sometime Professor at Birkbeck College, London. The volume comprises specially written articles on various topological topics by experts in many countries who worked with Dowker at one time or another. These include survey, expository and research articles on general topology, algebraic topology and related subjects such as knot theory and graph theory. The volume will be of great interest to graduate students and professional mathematicians whose speciality is topology, in all its aspects.Table of Contents1. Obituary: Clifford Hugh Dowker Dona Strauss; 2. Knot tabulations and related topics Morwen B. Thistlethwaite; 3. How general is a generalized space? Peter T. Johnstone; 3. A survey of metrization theory J. Nagata; 4. Some thoughts on lattice valued functions and relations M. W. Warner; 5. General topology over a base I. M. James; 6. K-Dowker spaces M. E. Rudin; 7. Graduation and dimension in locales J. Isbell; 8. A geometrical approach to degree theory and the Leray-Schauder index J. Dugundji; 9. On dimension theory B. A. Pasynkov; 10. An equivariant theory of retracts Sergey Antonian; 11. P-embedding, LCn spaces and the homotopy extension property Kiiti Morita; 12. Special group automorphisms and special self-homotopy equivalences Peter Hilton; 13. Rational homotopy and torus actions Stephen Halperin; 14. Remarks on stars and independent sets P. Erdös and J. Pach; 15. Compact and compact Hausdorff A. H. Stone; 16. T1 - and T2 axioms for frames C. H. Dowker and D. Strauss.

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Book SynopsisThis book gives a general outlook on homotopy theory. The author describes powerful tools for homotopy classification problems, particularly for the classification of homotopy types and for the computation of the group homotopy equivalences. Applications and examples of such computations are given.Table of ContentsPreface; Introduction; List of symbols; 1. Axioms for homotopy theory and examples of cofibration categories; 2. Homotopy theory in a cofibration category; 3. The homotopy spectral sequences in a cofibration category; 4. Extensions, coverings and cohomology groups of a category; 5. Maps between mapping cones; 6. Homotopy theory of CW-complexes; 7. Homotopy theory of complexes in a cofibration category; 8. Homotopy theory of Postnikov towers and the Sullivan-de Rham equivalence of rational homotopy categories; 9. Homotopy theory of reduced complexes; Bibliography; Index.

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Book SynopsisThis is the first of two volumes which will provide an introduction to modern developments in the representation theory of finite groups and associative algebras. The subject is viewed from the perspective of homological algebra and the theory of representations of finite dimensional algebras; the author emphasises modular representations and the homological algebra associated with their categories. This volume is self-contained and independent of its successor, being primarily concerned with the exposition of the necessary background material. The heart of the book is a lengthy introduction to the (AuslanderâReiten) representation theory of finite dimensional algebras, in which the techniques of quivers with relations and almost split sequences are discussed in detail. Much of the material presented here has never appeared in book form. Consequently students and research workers studying group theory and indeed algebra in general will be grateful to Dr Benson for supplying an expositiTrade Review"...a very welcome addition to the existing classical literature on representation theory and cohomology of finite groups....fascinating reading for someone who already has a basic knowledge of representation theory and cohomology ...I can recommend this book highly to anyone who works with representation theory and cohomology of finite groups." K. W. Roggenkamp, Mathematical ReviewsTable of Contents1. Background material from rings and modules; 2. Homological algebra; 3. Modules for group algebra; 4. Methods from the representation of algebra; 5. Representation rings and Burnside rings; 6. Block theory.

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Book SynopsisTo attack certain problems in 4-dimensional knot theory the author draws on a variety of techniques, focusing on knots in S^T4, whose fundamental groups contain abelian normal subgroups. Their class contains the most geometrically appealing and best understood examples. Moreover, it is possible to apply work in algebraic methods to these problems. Work in four-dimensional topology is applied in later chapters to the problem of classifying 2-knots.Table of Contents1. Knots and Related Manifolds; 2. The Knot Group; 3. Localization and Asphericity; 4. The Rank 1 Case; 5. The Rank 2 Case; 6. Ascending Series and the Large Rank Cases; 7. The Homotopy Type of M(K); 8. Applying Surgery to Determine the Knot.

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Book SynopsisThis volume records the lectures given at a conference to celebrate Professor Ioan James' 60th birthday. Ioan James has made significant contributions to homotopy theory, highlighting problems and initiating new methods. The present volume contains papers from internationally distinguished researcher workers which lead on from recent and exciting breakthroughs. It will be important for all homotopy theorists as both a statement of current research and as a signpost for new directions.Table of ContentsPreface; Publications of I. M. James; 1. Homotopy fixed points in the algebraic K-theory of certain infinite discrete groups G. Carlsson; 2. Applications of loop spaces to some problems in topology F. R. Cohen; 3. Invariants spheres and critical points M. C. Crabb; 4. Bitangent spheres and critical points Duan H.-E. Rees; 5. The enveloping ring of a II-algebra W. G. Dwyer and D. M. Kan; 6. Cohomologie des groupes symétriques et application de Quillen J. H. Gunawardena, J. Lannes and S. Zarati; 7. Looping Massey-Peterson towers J. R. Harper and H. R. Miller; 8. Characters and elliptic cohomology M. J. Hopkins; 9. Self maps of H-spaces J. R. Hubbuck; 10. Character rings in algebraic topology N. J. Kuhn; 11. The rank filtration of Quillen's Q-construction S. A. Mitchell; 12. Obstruction theory and the strict associativity of Morava K-theories C. A. Robinson; 13. Loop groups and harmonic maps G. B. Segal; 14. An almost groupoid structure for the space of (open) strings and implications for string field theory J. D. Stasheff; 15. Steenrod squares of polynomials R. M. W. Wood.

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Book SynopsisThis book is based on a course taught to an audience of undergraduate and graduate students at Oxford, and can be viewed as a bridge between the study of metric spaces and general topological spaces. About half the book is devoted to relatively little-known results, much of which is published here for the first time. The author sketches a theory of uniform transformation groups, leading to the theory of uniform spaces over a base and hence to the theory of uniform covering spaces. Readers interested in general topology will find much to interest them here.Table of ContentsIntroduction; 1. Uniform structures; 2. Induced and coinduced uniform structures; 3. The uniform topology; 4. Completeness and completion; 5. Topological groups; 6. Uniform transformation groups; 7. Uniform spaces over a base; 8. Uniform covering spaces; Exercises; Bibliography.

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Book SynopsisThis book presents a definitive account of the applications of the algebraic L-theory to the surgery classification of topological manifolds. The central result is the identification of a manifold structure in the homotopy type of a Poincaré duality space with a local quadratic structure in the chain homotopy type of the universal cover.Trade Review"...develops lower K- and L-theory with a view to applications in topology....Apart from the obvious interest of this text both to topologists and to K-theorists, it also serves as an introduction to the field, since there is a comprehensive survey of previous results and applications." M.E. Keating, Bulletin of the London Mathematical SocietyTable of ContentsIntroduction; Summary; Part I. Algebra: 1. Algebraic Poincaré complexes; 2. Algebraic normal complexes; 3. Algebraic bordism categories; 4. Categories over complexes; 5. Duality; 6. Simply connected assembly; 7. Derived product and Hom; 8. Local Poincaré duality; 9. Universal assembly; 10. The algebraic π-π theorem; 11. ∆-sets; 12. Generalized homology theory; 13. Algebraic L-spectra; 14. The algebraic surgery exact sequence; 15. Connective L-theory; Part II. Topology: 16. The L-theory orientation of topology; 17. The total surgery obstruction; 18. The structure set; 19. Geometric Poincaré complexes; 20. The simply connected case; 21. Transfer; 22. Finite fundamental group; 23. Splitting; 24. Higher signatures; 25. The 4-periodic theory; 26. Surgery with coefficients; Appendices; Bibliography; Index.

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Book SynopsisJ. Frank Adams had a profound influence on algebraic topology, and his works continue to shape its development. The International Symposium on Algebraic Topology held in Manchester during July 1990 was dedicated to his memory, and virtually all of the world's leading experts took part.Table of ContentsThe work of J. F. Adams; Bibliography; Twisted tensor products of DGAs and the Adams-Hilton model for the total space of a fibration; Hochschild homology, cyclic homology and the cobar construction; Hermitian A∞-rings and their K-theory; A splitting result for the second homology of the general linear group; Low dimensional spinor representations, Adams maps and geometric dimension; The characteristic for the exceptional Lie groups; How can you tell two spaces apart when they have the same n-type for all n?; A generalised Grothendieck spectral sequence; Localisation of the axes of pairings; Fibrewise reduced product spaces; Computing homotopy types using crossed n-cubes of groups; On orthogonal pairs in categories and localization; A note on extensions of nil-potent groups; On the Swan subgroup of metacyclic groups; Fields of spaces; Maps between p-completed classifying spaces, III; Retracts of classifying spaces; On the dimension theory of dominant summands.

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Book SynopsisJ. Frank Adams had a profound influence on algebraic topology, and his works continue to shape its development. The International Symposium on Algebraic Topology held in Manchester during July 1990 was dedicated to his memory, and virtually all of the world's leading experts took part.Trade Review"...this exceptional three-volume history of Latin American Literature...surpasses its goal....Congatulations to contributors, editors, and publishers on what will become a classic in the field." F. Colecchia, ChoiceTable of ContentsProgress report on the telescope conjecture; On K* stable homotopy theory; Detruncating Morava K-theory; p-adic interpolation of stable homotopy groups; Some remarks on v1-periodic homotopy groups; The unstable Novikov spectral sequence for Sp(n), and the power series sinh-1(x); On a certain localisation of the stable homotopy of the space XG; Co-operations in elliptic homology; Completion of G-spectra at ideals of the Burnside ring, I & II; Theorems of Poisson, Euler and Bernouilli on the Adams spectral sequence; Algebras over the Steenrod algebra and finite H-spaces; The boundedness conjecture for the action of the Steenrod algebra on polynomials; Representations of the homology of BV and the Steenrod algebra, I; Generic representation theory and Lannes' T-functor; Some chromatic phenomena in the homotopy of MSp; On a conjecture of Mahowald concerning bordism with singularities; Topological gravity and algebraic topology.

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Book SynopsisAn updated and expanded second edition of a successful and well-reviewed text presenting a detailed exposition of the modern theory of supermanifolds, including a rigorous account of the super-analogs of all the basic structures of ordinary manifold theory. Many exercises are included to complement the text.Trade Review'Supermanifolds is destined to become the standard work for all serious study of super-symmetric theories of physics.' NatureTable of ContentsPreface to the first edition; Preface to the second editin; 1. Analysis over supernumbers; 2. Supermanifolds; 3. Super Lie groups: general theory; 4. Super Lie groups: examples; 5. Selected applications of supermanifold theory; 6. Applications involving topology; References; Index.

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Book SynopsisCohomology of Drinfeld Modular Varieties provides an introduction, in two volumes, to both the subject of the title and the Langlands correspondence for function fields. It is based on courses given by the author and will be welcomed by workers in number theory and representation theory.Table of ContentsPreface; 9. Trace of fA on the discrete spectrum; 10. Non-invariant Arthur trace formula: the geometric side; 11. Non-invariant Arthur trace formula: the spectral side; 12. Cohomology with compact supports of Drinfeld modular varieties; 13. Intersection cohomology of Drinfeld modular varieties; Appendix D. Representations of unimodular, locally compact, totally discontinuous, separated topological groups: addendum; Appendix E. Reduction theory and strong approximation; Appendix F. Proof of lemma 10. 6. 4; Appendix G. The decomposition of L2G following the cuspidal data.

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Book SynopsisThis work consists of two courses on the moduli spaces of vector bundles. The first is introductory, and assumes very little background; the second is more advanced and takes the reader into current areas of research. This a treatment of vector bundles that will be welcomed by experienced algebraic geometers and novices alike.Trade Review'The whole book is well written and is a valuable addition to the literature … It is essential purchase for all libraries maintaining a collection in algebraic geometry, and strongly recommended for individual researchers and graduate students with an interest in vector bundles.' Peter Newstead, Bulletin of the London Mathematical SocietyTable of ContentsPart I. Vector Bundles On Algebraic Curves: 1. Generalities; 2. The Riemann-Roch formula; 3. Topological; 4. The Hilbert scheme; 5. Semi-stability; 6. Invariant geometry; 7. The construction of M(r,d); 8. Study of M(r,d); Part II. Moduli Spaces Of Semi-Stable Sheaves On The Projective Plane; 9. Introduction; 10. Operations on semi-stable sheaves; 11. Restriction to curves; 12. Bogomolov's theorem; 13. Bounded families; 14. The construction of the moduli space; 15. Differential study of the Shatz stratification; 16. The conditions for existence; 17. The irreducibility; 18. The Picard group; Bibliography.

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Book SynopsisThese volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) on the subject of `Novikov conjectures, index theorems and rigidity'.Table of Contents1. Proper affine isometric actions of amenable groups M. E. Bekka, P. A. Cherix and A. Vakette; 2. Bounded K-theory and the assembly map in algebraic K-theory Gunnar Carlsson; 3. On the rigidity of the index on open manifolds J. Eichhorn; 4. Remarks on Steenrod Homology S. Ferry; 5. Epsilon surgery Steven C. Ferry and E. Pederson; 6. On the coarse Baum-connes conjecture N. Higson and J. Roe; 7. Exotic index theory and the Novikov conjecture S. Hurder; 8. Bounded and continuous control Eric Pedersen; 9. On the homotopy invariance of the boundedly controlled signature of a manifold over an open cone E. K. Pederson, J. Roe and S. Weinberger; 10. 3-manifolds and PD(3)-groups Charles B. Thomas; 11. Orthogonal complements and endomorphisms of Hilbert Modules and C*-Elliptic Complexes E. V. Troitsky; 12. Assembly Michael Weiss and Bruce Williams; 13. Pro-exclusive Functions Michael Weiss and Bruce Williams.

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Book SynopsisThese volumes are the outgrowth of a conference held at the Mathematisches Forschungsinstitut Oberwolfach (Germany) on the subject of 'Novikov Conjectures, Index Theorems and Rigidity'.Trade Review'These proceedings will be useful for all who are interested in related fields of mathematics such as topology, geometry and analysis and especially for those who are attracted by problems related to the topology of manifolds.' European Mathematical Society Newsletter' … this two volume collection ofers a very nice way to see both the historical development and the current activities in the field.' Monatshefte für MathematikTable of Contents1. Homotopy type of differentiable manifolds William Browder; 2. A history and survey of the Novikov conjecture S. C. Ferry, A. RanickI and J. Rosenberg; 3. A coarse approach to the Novikov conjecture S. C. Ferry and Shmuel Weinberger; 4. Reflections on the Novikov conjecture Mikhael Gromov; 5. K-theory, group C*-algebras and higher signatures (Conspectus) Gennadi Kasparov; 6. On the Steenrod homology theory John Milnor; 7. Controlled Fredholm Representations A. S. Mishchenko; 8. Assembly maps in bordism-type theories Frank Quin; 9. On the Novikov conjecture Andrew Ranicki.

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Book SynopsisThe study of vector bundles over algebraic varieties has been stimulated over the last few years by successive waves of migrant concepts, largely from mathematical physics, whilst retaining its roots in old questions concerning subvarieties of projective space. The 1993 Durham Symposium on Vector Bundles in Algebraic Geometry brought together some of the leading researchers in the field to explore further these interactions. This book is a collection of survey articles by the main speakers at the symposium and presents to the mathematical world an overview of the key areas of research involving vector bundles. Topics covered include those linking gauge theory and geometric invariant theory such as augmented bundles and coherent systems; Donaldson invariants of algebraic surfaces; Floer homology and quantum cohomology; conformal field theory and the moduli spaces of bundles on curves; the HorrocksâMumford bundle and codimension 2 subvarieties in P4 and P5; exceptional bundles and stableTable of Contents1. On the deformation theory of moduli spaces of vector bundles V. Balaji and P. Vishwanath; 2. Stable augmented bundles over Riemann surfaces S. Bradlow, G. Daskapoulos, O. Garcia-Pradia and R. Wentworth; 3. On surfaces in P4 and threefolds in P5 W. Decker and S. Popescu; 4. Exceptional bundles and moduli spaces of stable sheaves on Pn J. M. Drezet; 5. Floer homology and algebraic geometry S. Donaldson; 6. The Horrocks-Mumford bundle K. Hulek; 7. Faisaux semi-stable et systems coherent J. Le Potier; 8. Combinatorics of the Verlinde formula A. Szenes; 9. Canonical and almost canonical spin polynomials of an algebraic surface A. Tyurin; 10. On conformal field theory K. Ueno.

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Book SynopsisStarting with a friendly account of the theory of compact Riemann surfaces, this 2011 book then introduces the Belyi–Grothendieck theory of dessins d'enfants, taking the reader with no previous knowledge of the subject to the forefront of current research. Many worked examples and illustrations are provided.Trade Review"Overall the text is very well written and easy to follow, partly due to the abundance of good concrete examples in every single section illustrating concepts from the very basic to the very technical." Aaron D. Wootton, Mathematical ReviewsTable of Contents1. Riemann surfaces and algebraic curves; 2. Riemann surfaces and Fuchsian groups; 3. Belyi's theorem; 4. Dessins d'enfants; References; Index.

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Book SynopsisThe book makes the topology of non-compact spaces accessible to both geometric and algebraic topologists, and algebraists. Recent developments are explained, and tools for further research are provided. In short, this book provides a systematic exposition of the theory and practice of ends of manifolds and CW complexes, along with their algebraic analogues for chain complexes.Trade Review'The book gathers together the main strands of the theory of ends of manifolds from the last thirty years and presents a unified and coherent treatment of them. It also contains authoritative expositions of certain topics in topology such as mapping tori and telescopes, often omitted from textbooks. It is thus simultaneously a research monograph and a useful reference.' Proceedings of the Edinburgh Mathematical Society'This is a highly specialized monograph which is very clearly written and made as accessible for the reader as possible … It is absolutely indispensable for any specialist in the field.' European Mathematical SocietyTable of ContentsIntroduction; Chapter summaries; Part I. Topology at Infinity: 1. End spaces; 2. Limits; 3. Homology at infinity; 4. Cellular homology; 5. Homology of covers; 6. Projective class and torsion; 7. Forward tameness; 8. Reverse tameness; 9. Homotopy at infinity; 10. Projective class at infinity; 11. Infinite torsion; 12. Forward tameness is a homotopy pushout; Part II. Topology Over the Real Line: 13. Infinite cyclic covers; 14. The mapping torus; 15. Geometric ribbons and bands; 16. Approximate fibrations; 17. Geometric wrapping up; 18. Geometric relaxation; 19. Homotopy theoretic twist glueing; 20. Homotopy theoretic wrapping up and relaxation; Part III. The Algebraic Theory: 21. Polynomial extensions; 22. Algebraic bands; 23. Algebraic tameness; 24. Relaxation techniques; 25. Algebraic ribbons; 26. Algebraic twist glueing; 27. Wrapping up in algebraic K- and L-theory; Part IV. Appendices; References; Index.

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