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Constructible and perverse sheaves are the algebraic counterpart of the decomposition of a singular space into smooth manifolds, a great geometrical idea due to R. Thom and H. Whitney. These sheaves, generalizing the local systems that are so ubiquitous in mathematics, have powerful applications to the topology of such singular spaces (mainly algebraic and analytic complex varieties).

This introduction to the subject can be regarded as a textbook on modern algebraic topology, treating the cohomology of spaces with sheaf (as opposed to constant)coefficients.

The first 5 chapters introduce derived categories, direct and inverse images of sheaf complexes, Verdier duality, constructible and perverse sheaves, vanishing and characteristic cycles. They also discuss relations to D-modules and intersection cohomology. Later chapters apply this powerful tool to the study of the topology of singularities, polynomial functions and hyperplane arrangements.

Some fundamental results, for which excellent sources exist, are not proved but just stated and illustrated by examples and corollaries. In this way, the reader is guided rather quickly from the basic theory to current research questions, supported in this by examples and exercises.

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Algebraic topology is the study of the global properties of spaces by means of algebra. It is an important branch of modern mathematics with a wide degree of applicability to other fields, including geometric topology, differential geometry, functional analysis, differential equations, algebraic geometry, number theory, and theoretical physics. This book provides an introduction to the basic concepts and methods of algebraic topology for the beginner. It presents elements of both homology theory and homotopy theory, and includes various applications. The author's intention is to rely on the geometric approach by appealing to the reader's own intuition to help understanding. The numerous illustrations in the text also serve this purpose. Two features make the text different from the standard literature: first, special attention is given to providing explicit algorithms for calculating the homology groups and for manipulating the fundamental groups. Second, the book contains many exercises, all of which are supplied with hints or solutions. This makes the book suitable for both classroom use and for independent study. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. ... Read more

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amazing
S. Matveev wrote an excellent book for those who wish to quickly familiarize themselves with the basics of algebraic topology and be able to use it in other fields, such as analysis and differential geometry. Everything about this book is fresh and so different from other basic algebraic topology books. Firstly, it is reader-friendly and it gives plenty of geometric motivation. Secondly, unlike its 300 or 400 page counterparts, it does not engage in exhaustive treatment of topics. Thirdly, it contains plethora of lovely pictures that directly appeal to reader's geometric intuition. Finally, a superb translation by E. Pervova, smoothly brings the best features of author's style to the reader.

In only about 80 pages, Matveev leads the reader from the definition of the standard n-simplex to the classification of covering spaces. The starting point is a very careful and yet efficient treatment of simplicial homology. Here, on of the high points of Matveev's presentation is a clear expalantion of algorithms for computing simplicial homology groups. Additionally, the exact sequence of a pair, Meyer-Vietoris sequence and excision are all thoroughly explained and illustrated by well chosen examples.

Next, the author moves to cellular homology, and again does a marvelous job in presenting the essence of computations. Matveev assumes very little abstract algebra. To spare the reader from searching through exhaustive treatments of necessary algebra done elsewhere, he provides lovely algebra refreshers on the following topics: finitely generated abelian groups, tensor product and torsion product, and group presentation. Having allowed the reader to get the feel for fomology, Matveev turns to axioms of homology and indicates what singular homology is. Lastly, he gives careful attention to the universal coefficient theorem for homology.

In the homology part of the book, as an illustration of the covered material, Matveev skillfully injects related topics, such as the degree of a map and Lefschetz fixed point theorem.

Having explained homology, Matveev turns to cohomology, whose treatment culminates in the presentation of Kunneth formula and products in cohomology. The cohomology part ends with Poincare duality.

In the last part of the book, Matveev turns to the basics of homotopy theory and related topics: the fundamental group and van Kampen's theorem, higher homotopy groups and the corresponding exact sequence of higher homotopy groups for a locally trivial bundle, Hurewicz homomorphism, and classification of covering spaces. Again the reader will greatly profit from Matveev's masterful appeal to geometric intuition.

The book contains over 100 exercises, all supplied with hints, complete solutions or answers. This will make the book a great resource for self-study. Most importantly, the reader who embarks on the road of self-study guided by this book, will be able to achieve solid competence in the basics of algebraic topology in a relatively short time. Furthermore, the reader who has a good working knowledge of Matveev's book should be able to read Milnor's Characteristic Classes and Husemoller's Fibre Budles.

There are a lot of basic algebraic topology books on the market, such as Hatcher, Munkres, Vick, Rotman, Bredon, Massey, Greenberg, and Maunder. Most (if not all) of these books give lengthy treatments of topics from Matveev's books (not necessarily more topics but perhaps more in-depth treatment). Naturally, a reader who wishes to specialize in algebraic topology will need to completely digest at least one of the afore-mentioned books. In any event, it seems that having Matveev's book on the side will help. Perhaps getting an all-around picture of the subject from Matveev first and then, if necessary, plunging into one of those lengthier treatises would be a resonable way to approach algebraic topology.

In conclusion, I would like to invite readers to give serious consideration to this book. You will be captivated by Matveev's simplicity, efficiency, and clarity. Perhaps most importantly, you will gain a better appreciation of algebraic topology and acquire much needed skills to reach higher ground in the subject.



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''Intended mainly for physicists and mathematicians...its highquality will definitely attract a wider audience.''---Computational Mathematics and Mathematical Physics Thisworkacquaints the physicist with the mathematical principles ofalgebraictopology, group theory, and differential geometry, asapplicable toresearch in field theory and the theory of condensedmatter. Emphasisis placed on the topological structure of monopoleand instantonsolution to the Yang-Mills equations, the descriptionof phases insuperfluid <^>3He, and the topology of singularsolutions in<^>3He and liquid crystals. ... Read more

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Concise Ideas and Applications ofTopology
This essential book about Cartan`s Lie formalism and its applications is written concisely and with all the necessary theorems to understand the topic. Michael Monastyrsky is explaing some very interesting physical applications of Topology, f.e. Yang-Mills, Gauge field theories and the also very important theory of fluid Helium. It counts to read the book from the beginning to the end. I like chapter three, because it affected to me his explanations of Yang-Mills and the Higgs mechanism. This book is recommendable for every physicist, who needs an introduction to Topology besides receiving some essential modern physics; that`s because Topology is a famous theme in physics, which becomes comprehensible while reading this book. ... Read more

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contains sections on Algebraic $K$- and $L$-theory, Surgery and its applications, Group actions ... Read more

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This collection marks the recent resurgence of interest in combinatorial methods, resulting from their deep and diverse applications both in topology and algebraic geometry. Nearly thirty mathematicians met at the University of Rochester in 1982 to survey several of the areas where combinatorial methods are proving especially fruitful: topology and combinatorial group theory, knot theory, 3-manifolds, homotopy theory and infinite dimensional topology, and four manifolds and algebraic surfaces. This material is accessible to advanced graduate students with a general course in algebraic topology along with some work in combinatorial group theory and geometric topology, as well as to established mathematicians with interests in these areas. For both student and professional mathematicians, the book provides practical suggestions for research directions still to be explored, as well as the aesthetic pleasures of seeing the interplay between algebra and topology which is characteristic of this field.

In several areas the book contains the first general exposition published on the subject. In topology, for example, the editors have included M. Cohen, W. Metzler and K. Sauerman's article on "Collapses of $K\times I$ and group presentations" and Metzler's "On the Andrews-Curtis-Conjecture and related problems." In addition, J. M. Montesino has provided summary articles on both 3- and 4-manifolds. ... Read more

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This volume presents the proceedings from the AMS-IMS-SIAM Summer Research Conference on Homotopy Methods in Algebraic Topology held at the University of Colorado (Boulder). The conference coincided with the sixtieth birthday of J. Peter May. An article is included reflecting his wide-ranging and influential contributions to the subject area. Other articles in the book discuss the ordinary, elliptic and real-oriented Adams spectral sequences, mapping class groups, configuration spaces, extended powers, operads, the telescope conjecture, $p$-compact groups, algebraic K theory, stable and unstable splittings, the calculus of functors, the $E_{\infty}$ tensor product, and equivariant cohomology theories. The book offers a compendious source on modern aspects of homotopy theoretic methods in many algebraic settings. ... Read more

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Algebraic topology (also known as homotopy theory) is a flourishing branch of modern mathematics. It is very much an international subject and this is reflected in the background of the 36 leading experts who have contributed to the Handbook. Written for the reader who already has a grounding in the subject, the volume consists of 27 expository surveys covering the most active areas of research. They provide the researcher with an up-to-date overview of this exciting branch of mathematics. ... Read more

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This book consists of twenty-nine articles contributed by participants of an International Conference in Algebraic Topology held in July 1991 in Oaxtepec, Mexico. In addition to papers on current research, there are several surveys and expositions on the work of Mark Mahowald, whose sixtieth birthday was celebrated during the conference. The conference was truly international, with over 130 mathematicians from fifteen countries. It ended with a spectacular total eclipse of the sun, a photograph of which appears as the frontispiece. The papers range over much of algebraic topology and cross over into related areas, such as $K$-theory, representation theory, and Lie groups. Also included is a chart of the Adams spectral sequence and a bibliography of Mahowald's publications. ... Read more

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This book is intended to cover real algebraic varieties emphasizing the author's program to classify them topologically. The first chapter gives an overview of the classification program. The second chapter provides background material for the rest of the book. It covers subjects starting with the elementary properties of real algebraic sets and ending with the recent solution of the nash conjecture. Chapters three and four develop the theory of resolution towers, which are basic topologically defined objects generalizing the notion of manifold and enable us to study singular spaces in an organized way. Chapter five shows how to obtain algebraic sets from resolution towers. Chapter six explains how to put resolution tower structures on real or complex algebraic sets. Chapter seven applies this theory to real algebraic sets of dimensions less than four by giving their complete topological characterization. ... Read more

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This proceedings volume centers on new developments in rational homotopy and on their influence on algebra and algebraic topology. Most of the papers are original research papers dealing with rational homotopy and tame homotopy, cyclic homology, Moore conjectures on the exponents of the homotopy groups of a finite CW-c-complex and homology of loop spaces. Of particular interest for specialists are papers on construction of the minimal model in tame theory and computation of the Lusternik-Schnirelmann category by means articles on Moore conjectures, on tame homotopy and on the properties of Poincaré series of loop spaces. ... Read more