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This is a modern introduction to Kaehlerian geometry and Hodge structure. Coverage begins with variables, complex manifolds, holomorphic vector bundles, sheaves and cohomology theory (with the latter being treated in a more theoretical way than is usual in geometry). The book culminates with the Hodge decomposition theorem. In between, the author proves the Kaehler identities, which leads to the hard Lefschetz theorem and the Hodge index theorem.The second part of the book investigates the meaning of these results in several directions. ... Read more

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Excellent text for upper-level undergraduate and graduate students shows how geometric and algebraic ideas met and grew together into an important branch of mathematics. Lucid coverage of vector fields, surfaces, homology of complexes, much more. Some knowledge of differential equations and multivariate calculus required. Many problems and exercises (some solutions) integrated into the text. 1979 edition. Bibliography.
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Customer Reviews (7)

A reader's opinion
This is the second time I have bought this book since I offered
the first one to my son. An excellent introduction to the topic!

A good start
Historically, combinatorial topology was a precursor to what is now the field of algebraic topology, and this book gives an elementary introduction to the subject, directed towards the beginning student of topology or geometry. Due to its importance in applications, the physicist reader who is intending eventually to specialize in elementary particle physics will gain much in the perusal of this book.

Combinatorial topology can be viewed first as an attempt to study the properties of polyhedra and how they fit together to form more complicated objects. Conversely, one can view it as a way of studying complicated objects by breaking them up into elementary polyhedral pieces. The author takes the former view in this book, and he restricts his attention to the study of objects that are built up from polygons, with the proviso that vertices are joined to vertices and (whole) edges are joined to (whole) edges.

He begins the book with a consideration of the Euler formula, and as one example considers the Euler number of the Platonic solids, resulting in a Diophantine equation. This equation only has five solutions, the Platonic solids. The author then motivates the concept of a homeomorphism (he calls them "topological equivalences") by considering topological transformations in the plane. Using the notion of topological equivalence he defines the notions of cell, path, and Jordan curve. Compactness and connectedness are then defined, along with the general notion of a topological space.

Elementary notions from differential topology are then considered in chapter 2, with the reader encountering for the first time the connections between analysis and topology, via the consideration of the phase portraits of differential equations. Brouwer's fixed point theorem is proved via Sperner's lemma, the latter being a combinatorial result which deals with the labeling of vertices in a triangulation of the cell. Gradient vector fields, the Poincare index theorem, and dual vector fields,which are some elementary notions in Morse theory, are treated here briefly.

An excellent introduction to some elementary notions from algebraic topology is done in chapter 3. The author treats the case of plane homology (mod 2), which is discussed via the use of polygonal chains on a grating in the plane. Beginning students will find the presentation very understandable, and the formalism that is developed is used to give a proof of the Jordan curve theorem. Then in chapter 4, the author proves the classification theorem for surfaces, using a combinatorial definition of a surface.

The author raises the level of complication in chapter 5, wherein he studies the (mod 2) homology of complexes. A complex is defined somewhat loosely as a topological space that is constructed out of vertices, edges, and polygons via topological identification. He proves the invariance theorem for triangulations of surfaces by showing that the homology groups of the triangulation are same as the homology groups of the plane model of the surface. This is an example of the invariance principle, and the author briefly details some of the history of invariance principles, such as the Hauptvermutung, its counterexample due to the mathematician John Milnor, and Heawood's conjecture, the latter of which deals with the minimum number of colors needed to color all maps on a surface with a given Euler characteristic. Integral homology is also introduced by the author, and he shows the origin of torsion in the consideration of the "twist" in a surface.

In the last part of the book, the author returns to the consideration of continuous transformations, tackling first the idea of a universal covering space. Algebraic topology again makes its appearance via the consideration of transformations of triangulated topological spaces, i.e. simplicial transformations. He shows how these transformations induce transformations in the homology groups, thus introducing the reader to some notions from category theory. The elaboration of the invariance theorem for homology leads the author to studying the properties of the group homomorphisms via matrix algebra, and then to a proof of the Lefschetz fixed point theorem. The book ends with a brief discussion of homotopy, topological dynamics, and alternative homology theories.

The beginning student of topology will thus be well prepared to move on to more rigorous and advanced treatments of differential, algebraic, and geometric topology after the reading of this book. There are still many unsolved problems in these areas, and each one of these will require a deep understanding and intuition of the underlying concepts in topology. This book is a good start.

Splendidly intuitive yet rigorous
This covers the basics of algebraic topology with simplexes, covering in essence the fundamental ideas behind of the work of Poincare, Brouwer, and Alexander. He proves the Jordan curve theorem, classifies all compact surfaces, and the relationship with vector fields. The homology groups are defined and used.

There are excellent examples, clear writing, and humour. An outstanding introduction.

One nice feature is that he bases his notions of continuity on "nearness" not epsilon-delta.

An excellent read
Ignore those that suggest this book is too "elementary". This is a wonderful text that concretizes the more abstract notions of algebraic topology. True, it should not be your only text on algebraic topology, andthe proofs are not as rigorous as a pedant might want, but it clearlyconveys the geometric underpinnings of topology and deserves a space on anytopologist's bookshelf.

Not for resolute students of algebraci/diff. topology.
I believe the two existing reviews are over-ratng. True, the book is accessible to anyone without prior knowledge of topology/algebra, but the treatment is too "elementary".For example, the author doesn'teven introduce the word "mod 2 homology".If you are resolutelyto study algebraic (or differential) topology, this is NOT the book to"study". Try Bredon or Fomenko-Novikov or May. For the subjectcovered, look for the book by Stillwell. ... Read more

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How many dimensions does our universe require for a comprehensive physical description? In 1905, Poincaré argued philosophically about the necessity of the three familiar dimensions, while recent research is based on 11 dimensions or even 23 dimensions. The notion of dimension itself presented a basic problem to the pioneers of topology. Cantor asked if dimension was a topological feature of Euclidean space. To answer this question, some important topological ideas were introduced by Brouwer, giving shape to a subject whose development dominated the twentieth century. The basic notions in topology are varied and a comprehensive grounding in point-set topology, the definition and use of the fundamental group, and the beginnings of homology theory requires considerable time. The goal of this book is a focused introduction through these classical topics, aiming throughout at the classical result of the Invariance of Dimension. This text is based on the author's course given at Vassar College and is intended for advanced undergraduate students. It is suitable for a semester-long course on topology for students who have studied real analysis and linear algebra. It is also a good choice for a capstone course, senior seminar, or independent study. ... Read more

Customer Reviews (3)

My First Book in Topology
John McCleary, the author of "A First Course in Topology: Continuity and Dimension", attempted to answer the question -- What is topology?--in his book. According to Wikipedia, topology is the study of those properties of objects that do not change when homeomorphisms are applied. John McCleary thinks that the central concept of topology is continuity, defined for the functions between the sets equipped with a notion of nearness (topological spaces). Since a coffee mug can be continuously deformed into a donut, the topological spaces of the coffee mug and donut are homeomorphic. The two spaces are the same from a topological view point.

John McCleary, the author, revealed that he spent too much time on developing the definitions when he first taught his undergraduate courses in topology. In fact, his book contains many definitions but a few examples. His strategy on presenting topology is that: "The first chapter reviews the set theory, so that the problem of topological invariance of dimension can be posed. The next five chapters treat the basic point-set notions of topology. The next two chapters treat the fundamental group of a space (equivalent spaces lead to isomorphic groups). The last two chapters focus on the combinatorial theme (simplicial complexes). We then associate the homology of the simplicial complexes, a sequence of vector spaces. This eventually leads to a proof of the topological invariance of dimension through homology."

There are hardly examples on the book. Thus the readers have hard time on applying the concepts. On the other hand, the book comes with proofs on well-known results. For example, the well-known proofs include (1) the fundamental theorem of algebra and (2) merely five Platonic solids exist.

$35 - This book is FREE online!
This is an awesome book, however, $35 dollars is a bit high.You can download the book online for free.If the book was $20 or less, I would say sure, but $35 for a thin soft cover......download it and review it yourself.

Elegant introduction to topology.
Being a mathematics student, I find McCleary as one of the finest introduction to topology for a beginner. ... Read more

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Alexander Grothendieck's concepts turned out to be astoundingly powerful and productive, truly revolutionizing algebraic geometry. He sketched his new theories in talks given at the Séminaire Bourbaki between 1957 and 1962. He then collected these lectures in a series of articles in Fondements de la géométrie algébrique (commonly known as FGA). Much of FGA is now common knowledge. However, some of it is less well known, and only a few geometers are familiar with its full scope. The goal of the current book, which resulted from the 2003 Advanced School in Basic Algebraic Geometry (Trieste, Italy), is to fill in the gaps in Grothendieck's very condensed outline of his theories. The four main themes discussed in the book are descent theory, Hilbert and Quot schemes, the formal existence theorem, and the Picard scheme. The authors present complete proofs of the main results, using newer ideas to promote understanding whenever necessary, and drawing connections to later developments. With the main prerequisite being a thorough acquaintance with basic scheme theory, this book is a valuable resource for anyone working in algebraic geometry. ... Read more

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This text examines topological methods in algebraic geometry. ... Read more

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This is the first authored book to be dedicated to the new field of directed algebraic topology that arose in the 1990s, in homotopy theory and in the theory of concurrent processes. Its general aim can be stated as 'modelling non-reversible phenomena' and its domain should be distinguished from that of classical algebraic topology by the principle that directed spaces have privileged directions and directed paths therein need not be reversible. Its homotopical tools (corresponding in the classical case to ordinary homotopies, fundamental group and fundamental groupoid) should be similarly 'non-reversible': directed homotopies, fundamental monoid and fundamental category. Homotopy constructions occur here in a directed version, which gives rise to new 'shapes', like directed cones and directed spheres. Applications will deal with domains where privileged directions appear, including rewrite systems, traffic networks and biological systems. The most developed examples can be found in the area of concurrency. ... Read more

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From the reviews of the 1st edition:

"This book provides a comprehensive and detailed account of different topics in algorithmic 3-dimensional topology, culminating with the recognition procedure for Haken manifolds and including the up-to-date results in computer enumeration of 3-manifolds. Originating from lecture notes of various courses given by the author over a decade, the book is intended to combine the pedagogical approach of a graduate textbook (without exercises) with the completeness and reliability of a research monograph…

All the material, with few exceptions, is presented from the peculiar point of view of special polyhedra and special spines of 3-manifolds. This choice contributes to keep the level of the exposition really elementary.

In conclusion, the reviewer subscribes to the quotation from the back cover: "the book fills a gap in the existing literature and will become a standard reference for algorithmic 3-dimensional topology both for graduate students and researchers".

Zentralblatt für Mathematik 2004

For this 2^nd edition, new results, new proofs, and commentaries for a better orientation of the reader have been added. In particular, in Chapter 7 several new sections concerning applications of the computer program "3-Manifold Recognizer" have been included.

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Brings Readers Up to Speed in This Important and Rapidly Growing Area

Supported by many examples in mathematics, physics, economics, engineering, and other disciplines, Essentials of Topology with Applications provides a clear, insightful, and thorough introduction to the basics of modern topology. It presents the traditional concepts of topological space, open and closed sets, separation axioms, and more, along with applications of the ideas in Morse, manifold, homotopy, and homology theories.

After discussing the key ideas of topology, the author examines the more advanced topics of algebraic topology and manifold theory. He also explores meaningful applications in a number of areas, including the traveling salesman problem, digital imaging, mathematical economics, and dynamical systems. The appendices offer background material on logic, set theory, the properties of real numbers, the axiom of choice, and basic algebraic structures.

Taking a fresh and accessible approach to a venerable subject, this text provides excellent representations of topological ideas. It forms the foundation for further mathematical study in real analysis, abstract algebra, and beyond.

Customer Reviews (1)

Poorly conceived and executed
I was looking forward to reading this book and had high hopes that it would be a suitable text for our first year graduate course sequence in topology, since Professor Krantz has a reputation for being an excellent expositor of mathematics.

After going through about half the book however, I must say that I am seriously disappointed in this book. It seems to be poorly conceived and poorly executed.There are just too many topics covered in this book and most of them are covered very superficially.

Let me just give a small random sample of flaws I found.

1) The section of Morse theory in Chapter 2 is very cursory, hardly enough to give any useful insight into this topic. Moreover it refers to notions which are not defined (e.g. 1-cell, 2-cell) or to notions which are defined later (e.g. homotopy, defined in Chapter 3)
2) In Chapter 3 the author purports to cover singular homology theory without any mention of excision (or equivalently Mayer-Vietoris)!
3) In Chapter 4 the torus and Klein bottle are described as being obtained by pasting the sides of a rectangle.This however is never related to the notion of quotient space, which is treated very briefly in Chapter 2, and which notion is needed to rigorously describe what "pasting" really means.

This is by no means an exhaustive list, but should be enough to indicate the issues which plague this book.

I am not sure what audience this is aimed at.It seems to be a thoughtless jumble of formal arguments interspersed with a lot of informal "hand-waving".

So I am sorry to say that I can not recommend this book as a text in any of our courses.
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This seminal text on Fourier-Mukai Transforms in Algebraic Geometry by a leading researcher and expositor is based on a course given at the Institut de Mathematiques de Jussieu in 2004 and 2005.Aimed at postgraduate students with a basic knowledge of algebraic geometry, the key aspect of this book is the derived category of coherent sheaves on a smooth projective variety.Including notions from other areas, e.g. singular cohomology, Hodge theory, abelian varieties, K3 surfaces; full proofs are given and exercises aid the reader throughout. ... Read more

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Topology is a relatively young and very important branch of mathematics, which studies the properties of objects that are preserved through deformations, twistings, and stretchings. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. This book is well suited for readers who are interested in finding out what topology is all about. ... Read more

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Gentle topology book without heart
This short, elementary survey of topology is meant to be accessible for the most part to high school students and beginning undergraduates. I hope that such unspoiled souls will have the courage to be dissatisfied at least with the first chapter, since it cares only about concepts (continuity, homeomorphism, etc.) while offering little substance, and also there is the usual overemphasis of the Jordan curve theorem and pathological curves. Young people should not be tricked into thinking that topology has been built around such silly things. But in the other two chapters we get to actual topology, and all the usual stuff is here: Euler characteristic, classification of surfaces, knots, the fundamental group, homology, etc. Each topic is treated in a relatively sensible, swift manner; rather too swift towards the end, I think, when there seems to be a race to include as many topological concepts as possible, with little concern for what would be the most natural or interesting way to proceed (of course this does not have to be a bad thing if one is using the book as a down-to-earth complement to a formal textbook). ... Read more

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This book is dedicated to the memory of the outstanding Russian mathematician, V. A. Rokhlin (1919--1984). It is a collection of research papers written by his former students and followers, who are now experts in their fields. The topics in this volume include topology (the Morse-Novikov theory, spin bordisms in dimension 6, and skein modules of links), real algebraic geometry (real algebraic curves, plane algebraic surfaces, algebraic links, and complex orientations), dynamics (ergodicity, amenability, and random bundle transformations), geometry of Riemannian manifolds, theory of Teichmüller spaces, measure theory, etc. The book also includes a biography of Rokhlin by Vershik and two articles of historical interest. ... Read more

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An advanced textbook in mathematics, offering a coherent and thorough treatment of the configuration spaces of Euclidian spaces and spheres. Requires a minimal background in classical homotopy theory and algebraic topology. Covers a variety of advanced topics, including a geometric presentation of the classical pure braid group. DLC: Configuration space. ... Read more

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Several recent investigations have focused attention on spaces and manifolds which are non-compact but where the problems studied have some kind of "control near infinity". This monograph introduces the category of spaces that are "boundedly controlled" over the (usually non-compact) metric space Z. It sets out to develop the algebraic and geometric tools needed to formulate and to prove boundedly controlled analogues of many of the standard results of algebraic topology and simple homotopy theory. One of the themes of the book is to show that in many cases the proof of a standard result can be easily adapted to prove the boundedly controlled analogue and to provide the details, often omitted in other treatments, of this adaptation. For this reason, the book does not require of the reader an extensive background. In the last chapter it is shown that special cases of the boundedly controlled Whitehead group are strongly related to lower K-theoretic groups, and the boundedly controlled theory is compared to Siebenmann's proper simple homotopy theory when Z = IR or IR2. ... Read more

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This book is a general introduction to Higher AlgebraicK-groups of rings and algebraic varieties, which were firstdefined by Quillen at the beginning of the 70's.These K-groups happen to be useful in many different fields,including topology, algebraic geometry, algebra and numbertheory.The goal of this volume is to provide graduate students,teachers and researchers with basic definitions, conceptsand results, and to give a sampling of current directions ofresearch.Written by five specialists of different parts ofthesubject, each set of lectures reflects the particularperspective ofits author. As such, this volume can serve asa primer (if not as a technical basic textbook) formathematicians from many different fields ofinterest. ... Read more

Customer Reviews (1)

A quick overview of a highly developed branch of mathematics
Algebraic K-theory may be viewed loosely as a theory of large matrices and how to define invariants for them. In ordinary linear algebra, the trace and determinant are elementary examples of these invariants and are straightforward to calculate. And in that same context, if one has M equations in N unknowns over some field K, then the solutions of this system of equations form a vector space S over K. If R is the subspace spanned by column vectors of length M, then Dim S + Dim R = N. As H. Gillet explains in one of the articles in the book, this result can be cast into the language of short exact sequences, and attempts to generalize this for the case when K is a ring is one of the tasks of algebraic K-theory. In this connection algebraic K-theory is useful in the following way: If K is either the integers or a finite field over an indeterminate, then for a polynomial ring A with D -1 variables over K, the group GL(n, A) is finitely generated when N is greater than equal to D + 2. This is an example where things look more stable when the dimension is high, a theme that occurs over and over again in K-theory. Indeed, when A is a finitely generated commutative regular ring, the question as to whether GL(n, A) is finitely generated for all sufficiently large n is equivalent to the question as to whether the K-group K1(A) is finitely generated.

This book goes considerably further then these relatively elementary considerations, in that it treats the higher K-groups and the connection with topological K-theory. Readers will need an extensive background in algebra and topology to appreciate the constructions in this book, which are mostly formal and thus there is the canonical inverse relationship between rigor and understanding. There are many places in the book though where readers can gain useful insights into a mature and highly developed branch of mathematics.

As was hinted above, for a ring R, K0 gives a measure of the failure of finitely generated projective R-modules from having a dimension theory like that of vector spaces.The first algebraic K-group K1 of a ring R is the quotient group of the infinite general linear group GL(R) modulo the infinite elementary group E(R) (the infinite elementary group comes from considering those matrices which differ from the identity only by an off diagonal element). Whitehead's Lemma shows that E(R) is a normal subgroup of GL(R). One can show that K1 of the integers is just {-1, 1}, and, for more general commutative rings R, that the determinant on GL(R) to the units R* of R induces a universal homomorphism and K1(R) is equal to these units. Thus the determinant gives in this case a universal invariant as was noted above. The second algebraic K-group of a ring R is then defined by generalizing the elementary group to the `Steinberg group' and then taking limits. There is an epimorphism from the Steinberg group to the elementary group and after passing to the infinite limit, the kernel of this epimorphism is defined as the second algebraic K-group of the ring R. K2(R) measures to what extent the Steinberg relations do not define the relations for the elementary group. One can show that K2 of the integers Z is Z/2, and that K2 of the direct product of two rings is the direct sum of their K2 groups.

Topological K-theory, also discussed in detail throughout the book, has its origins in the theory of vector bundles. Two vector bundles are called `stably-equivalent' if they are isomorphic after taking their direct sum with trivial bundles. Stable equivalence forms an equivalence relation and the stable classes form a ring under direct sum and tensor product. This ring is called the K-ring K(X) of the space X on which the vector bundles are defined. If X is compact and E is a vector bundle over X, then the sheaf of sections of this vector bundle is a finitely generated projective module over the ring C(X) of continuous functions on X. This result is known as the Serre-Swan theorem and allows one to discuss the K-theory of the space X in terms of the K-theory of C(X). The properties of this K-theory satisfy those needed to make it a cohomology theory, except for the dimension axiom. Topological K-theory also has the property of Bott periodicity, wherein the K-groups at one dimension are isomorphic to those of two dimensions less.

The higher topological K-theory groups have a counterpart in algebraic K-theory. This can be shown in several different ways, but this book discusses the Quillen or `Q-construction' of higher algebraic K-theory. Dependent on the notion of a `nerve' of a category and its classifying space, the Q-construction involves starting with an exact category M and defining a new category QM with the same objects but with morphisms satisfying certain properties of admissibility and composition. For a small exact category M, the ith higher algebraic K-group of this category is defined as the (i+1)-th homotopy group of the classifying space of M. The book also discusses, for a ring R, the `+-construction' of Quillen, which was the first definition of higher algebraic K-theory, and is considerably less esoteric than the Q-construction since it involves the well-known result that GL(R) is the first homotopy group of the classifying space of GL(R) and the intuitive geometric construction of adding cells to the classifying space to form a new space that has certain useful properties. The ith-higher algebraic K-group is then defined as the ith-homotopy group of this space. Although it is not done in this book, this definition coincides with the Q-construction when the latter is applied to the category of finitely generated projective R-modules. ... Read more