Product Description
Three million high school students and 172, 000 college students enroll in geometry classes every year. Schaum's Outline of Geometry, Third Edition, is fully updated to reflect the many changes in geometry curriculum, including new terminology and notation and a new chapter on how to use the graphing calculator. ... Read more

Customer Reviews (12)

Good review of concepts - Poor Editing!
I am using this book as a refresher while studying for an upcoming qualification exam. In just the opening chapters covering Algebra review, I was appalled at the number of errors made. These are not typographical errors as is the case where the book says that something weighs "11 km", but flagrant syntactical errors that produce a muddled understanding of the underlying concepts.

Luckily, I am still fresh enough with Algebra to catch these errors, however, I have to question their ability to clearly convey geometry material without confusion.

This is not to say that the text is a total wash, but I do expect a certain level of accuracy from outline materials such as this.

Geometry review on schaum's outline
For the most part this book is great as a tutorial with ample exercises that provide critical thinking pertaining to proving triangles congruent and reasoing in general. Thats the way geometry should be taught critical thinking unlike most books that regurgitate and are watered down. Even though this book is modern, its not as great as plane geometry munro 1959, theres not ample exercises on tedious proofs. My favorite sections are proving quadrilaterals are parallelograms and areas of polygons because these problems force the individual (student) to think outside the box. For example, there are problems where circles are inscribed in circles, sectors inscribed in equilateral triangles, right triangles part of circles, etc.

The weakness is the fact that this book does not provide surface area of hexagonal prisms, polygonal pyramids involve apothems, radii, surface area of cones, no composite figures, just surface area of rectangular prisms and cubes, its the same like schaums outline for algebra in regards to this section and the same regarding reflections and translations. I therefore give this 3 stars because of some redundancies.

Not horrible, not great...
I got this book as review of basic geometry.Mostly because they never made a Forgotten Geometry text.It's really not bad.It covers quite a bit of ground, from simple geometry to a taste of analytical and transformational geometry.

Conversely, this book didn't wow me either.It covers some basic algebra in the beginning, which is fine for the easier formulas like area and perimeter, but not nearly enough for a comprehensive study of geometry.One should study Schaum's outline of Intermediate Algebra or College Algebra, or a text of their choosing, before tackling any geometry.Especially if your preparing for a Calculus/Analytical Geometry Course.

I haven't found a good intro geometry book yet(I haven't even looked!), but when I do, I'll be sure to update this review.As always, good luck!

Good for basic refresher
I bought this book as a recommended book for a masters Geometry class I was taking.While the concepts of my class were far more advanced than this book, the content of the book made remembering about the geometry and basics (probably good through hs geometry) more clear.I good book to take you through high school geometry.

Caution - There are Mistakes
I bought this book to use with my daughter for added drill and review of Geometry and was shocked to find an inaccurate mathematical statement after the Commutative Law of Multiplication (3a X 5 = 5 X 2a = 10a) and three errors in the solutions provided to practice questions in the first eight pages.I haven't ventured further, but based on my experience thus far, I will need to review every example and problem for accuracy, something I hadn't planned to do and shouldn't need to do. ... Read more

Product Description
This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised. ... Read more

Customer Reviews (2)

Written for the physicist in mind
This book, written by some of the master expositors of modern mathematics, is an introduction to modern differential geometry with emphasis on concrete examples and concepts, and it is also targeted to a physics audience. Each topic is motivated with examples that help the reader appreciate the essentials of the subject, but rigor is not sacrificed in the book.

In the first chapter the reader gets a taste of differentiable manifolds and Lie groups, the later gving rise to a discussion of Lie algebras by considering, as usual, the tangent space at the identity of the Lie group. Projective space is shown to be a manifold and the transition functions explicitly written down. The authors give a neat example of a Lie group that is not a matrix group. A rather quick introduction to complex manifolds and Riemann surfaces is given, perhaps too quick for the reader requiring more details. Homogeneous and symmetric spaces are also discussed, and the authors plunge right into the theory of vector bundles on manifolds. Thus there is a lot packed into this chapter, and the authors should have considered spreading out the discussion more, as it leaves the reader wanting for more detail.

The authors consider more fundamental questions in smooth manifolds in chapter 3, with partitions of unity used to prove the existence of Riemannian metrics and connections on manifolds. They also prove Stokes formula, and prove the existence of a smooth embedding of any compact manifold into Euclidean space of dimension 2n + 1. Properties of smooth maps, such as the ability to approximate a continuous mapping by a smooth mapping, are also discussed. A proof of Sard's theorem is given, thus enabling the study of singularities of a mapping. The reader does get a taste of Morse theory here also, along with transversality, and thus a look at some elementary notions of differential topology. An interesting discussion is given on how to obtain Morse functions on smooth manifolds by using focal points.

Notions of homotopy are introduced in chapter 3, along with more concepts from differential topology, such as the degree of a map. A very interesting discussion is given on the relation between the Whitney number of a plane closed curve and the degree of the Gauss map. This leads to a proof of the important Gauss-Bonnet theorem. Degree theory is also applied to vector fields and then to an application for differential equations, namely the Poincare-Bendixson theorem. The index theory of vector fields is also shown to lead to the Hopf result on the Euler characteristic of a closed orientable surface and to the Brouwer fixed-point theorem.

Chapter 4 considers the orientability of manifolds, with the authors showing how orientation can be transported along a path, thus giving a non-traditional characterization as to when a connected manifold is orientable, namely if this transport around any closed path preserves the orientation class. More homotopy theory, via the fundamental group, is also discussed, with a few examples being computed and the connection of the fundamental group with orientability. It is shown that the fundamental group of a non-orientable manifold is homomorphic onto the cyclic group of order 2. Fiber bundles with discrete fiber, also known as covering spaces, are also discussed, along with their connections to the theory of Riemann surfaces via branched coverings. The authors show the utility of covering maps in the calculation of the fundamental group, and use this connection to introduce homology groups. A very detailed discussion of the action of the discrete group on the Lobachevskian plane is given.

Absolute and relative homotopy groups are introduced in chapter 5,and many examples are given of their calculation. The idea of a covering homotopy leads to a discussion of fiber spaces. The most interesting discussion in this chapter is the one on Whitehead multiplication, as this is usually not covered in introductory books such as this one, and since it has become important in physics applications. The authors do take a stab at the problem of computing homotopy groups of spheres, and the discussion is a bit unorthodox since it depends on using framed normal bundles.

The theory of smooth fiber bundles is considered in the next chapter. The physicist reader should pay close attention to this chapter is it gives many insights into the homotopy theory of fiber bundles that cannot be found in the usual books on the subject. The discussion of the classification theory of fiber bundles is very dense but worth the time reading. Interestingly, the authors include a discussion of the Picard-Lefschetz formula, as an example of a class of "fiber bundles with singularities". Those interested in the geometry of gauge field theories will appreciate the discussion on the differential geometry of fiber bundles.

Dynamical systems are introduced in chapter 7, first as defined over manifolds, and then in the context of symplectic manifolds via Hamaltonian mechanics. Liouville's theorem is proven, and a few examples are given from relativistic point mechanics. The theory of foliations is also discussed, although the discussion is too brief to be of much use. The authors also consider variational problems, and given its importance in physics, they continue the treatment in the last chapter of the book, giving several examples in general relativity, and in gauge theory via a consideration of the vacuum solutions of the Yang-Mills equation. The physicist reader will appreciate this discussion of the classical theory of gauge fields, as it is good preparation for further reading on instantons and the eventual quantization of gauge fields.

A masterful sequel!
Novikov et al's first volume was the defining book on differential geometry (S-V 93). The second volume picks up on the detailed theory of manifolds and topology and other advanced theories of differentialgeometry, including homotopy groups, Lie algebras and digressing intophysical theories as well (eg.Yang-Mills) giving one of the juciest bookson the subject - an utter delight! ... Read more

Product Description

These lecture notes contain a guided tour to the Novikov Conjecture and related conjectures due to Baum-Connes, Borel and Farrell-Jones. They begin with basics about higher signatures, Whitehead torsion and the s-Cobordism Theorem. Then an introduction to surgery theory and a version of the assembly map is presented. Using the solution of the Novikov conjecture for special groups some applications to the classification of low dimensional manifolds are given. Finally, the most recent developments concerning these conjectures are surveyed, including a detailed status report.
The prerequisites consist of a solid knowledge of the basics about manifolds, vector bundles, (co-) homology and characteristic classes.

Customer Reviews (1)

An effective overview
In its simplest form, the Novikov conjecture asserts that if there is a map f from a closed smooth manifold M and a classifying space of a group, and if g is a homotopy equivalence from a closed smooth manifold N to M, then the `higher signatures' of (M, f) and (N, fg) agree. The goal of this book is to introduce the reader to the precise notion of `higher signature' and to discuss various concepts and tools used in attempted resolutions of the conjecture. Also discussed in some details are conjectures that are related to the Novikov conjecture. Readers will needto have a strong background in algebraic and differential topology in order to appreciate the content of the book, but the authors develop some of the needed material in it, such as h- and s-cobordism, simple homotopy, surgery theory, and the classification problem for manifolds via characteristic classes. Without too many exceptions the authors motivate the concepts exceedingly well, especially in chapter 12 where they give one of the best explanations in print for the surgery obstruction groups.

When reading the book it becomes apparent that the Novikov conjecture has many guises, and attempts to resolve it have involved some quite esoteric constructions. The main strategy used in its resolution involves a generalization of the Hirzebruch signature, called a `higher signature' and the notion of an `assembly map.' The assembly map, as the name implies, collects all the higher signatures into a single invariant: essentially the image of the Poincare dual of the L-class under the map induced from f. One then constructs a homomorphism (the assembly map) from the Poincare duals of the Pontrjagin classes to a particular Abelian group L(G), such that the value of the assembly map on the image is a homotopy invariant. The Novikov conjecture is the assertion that the assembly map is an isomorphism. Much of the first part of the book discusses how to make these notions meaningful and how to interpret them geometrically via the surgery obstruction groups.

The authors also discuss them in a purely algebraic context, constructing an algebraic notion of bordism in the context of chain complexes and the notions of symmetric and quadratic forms over chain complexes. Algebraic cobordism allows the definition of a symmetric and quadratic algebraic L-group. The nth symmetric algebraic L-group of a ring R with involution is defined as the collection of cobordism classes of n-dimensional symmetric algebraic Poincare complexes, and the quadratic L-group of R, with a similar definition for the quadratic case. From these constructions the reader is introduced to the subject of L-theory, which has been the subject of intense research in the last two decades.

Central to the research into the Novikov conjecture is the category of `spectra', which is usually encountered in any treatment of algebraic topology but is discussed here with examples given in K- and L-theory and the famous Thom spectrum of a stable vector bundle. The discussion of spectra involves the important notion of a `homotopy pushout', which are defined so as to commute with the suspension with the unit circle, and the `homotopy pullback', which commutes with the loop functor. Both homotopy pushouts and pullbacks are homotopy equivalences.

Given a discrete group G, a family of subgroups of G, and an equivariant homology theory with respect to G, after constructing the classifying space of the family of subgroups, the authors want to show that the assembly map induced from the projection of the family to the one-point space is an isomorphism. To understand for which groups this is true, the authors must first define the notion of a G-homology theory. They use the notion of a G-CW-complex that they defined when discussing classifying spaces of families of subgroups to define this homology theory. For a group G and an associative commutative ring Q with unit it consists of a collection of covariant functors from the category of G-CW-pairs to the category of Q-modules indexed by the integers that satisfies the usual properties such as G-homotopy invariance, the existence of a long exact sequences for pairs, and excision. An equivariant homology theory is then a G-homology theory that has a `induction structure', the latter of which is a collection of isomorphisms between the nth G-homology groups and nth homology groups of a group that has a homomorphism into G. The authors then show how to obtain an equivariant homology from a spectrum. Central to their construction is the `orbit category Or(G)' of a group G. The objects of this category are homogeneous G-spaces and the morphisms are G-maps. For a small category C they define a `C-space' to be a functor from C to the set of compactly generated spaces. A `C-spectrum' is a functor from C to the category of spectra. After defining a notion of smash product for a C-space and a C-spectrum, the authors then quote a lemma that illustrates how one can obtain a G-homology theory from an Or(G)-spectrum. In order to obtain an induction structure, the Or(G)-spectrum must be obtained from a spectrum of groupoids. The authors show how to do this and thus obtain an equivariant homology theory. Therefore the K- and L-theory spectra over groupoids that were constructed earlier thus give rise to equivariant homology theories.

These G-homology theories reduce to the K- and L-theory of the group ring when evaluated on a one-point space, and the topological K-theory of the reduced C*-algebra. The Farell-Jones conjecture claims that the assembly maps from the equivariant homology groups for K and L-theory to the one-point homology are isomorphisms. The Baum-Connes conjecture does the same for topological K-theory. If these conjectures were answered positively then they would allow the computation of the K- and L-groups from the K- and L- finite or virtually cyclic subgroups of G. The authors spend two chapters discussing for what groups these conjectures have been found to be true, and also a chapter on how the Novikov conjecture follows from these conjectures. ... Read more

Product Description
Intended for use both as a text and a reference, this book is an exposition of the fundamental ideas of algebraic topology. The first third of the book covers the fundamental group, its definition and its application in the study of covering spaces. The focus then turns to homology theory, including cohomology, cup products, cohomology operations, and topological manifolds. The remaining third of the book is devoted to Homotropy theory, covering basic facts about homotropy groups, applications to obstruction theory, and computations of homotropy groups of spheres. In the later parts, the main emphasis is on the application to geometry of the algebraic tools developed earlier. ... Read more

Customer Reviews (6)

Pioneering text
This book was an incredible step forward when it was written (1962-1963). Lefschetz's Algebraic Topology (Colloquium Pbns. Series, Vol 27) was the main text at the time. A large number of other good to great books on the subject have appeared since then, so a review for current readers needs to address two separate issues: its suitability as a textbook and its mathematical content.
I took the course from Mr. Spanier at Berkeley a decade after the text was written.He was a fantastic teacher - one of the two best I've ever had (the other taught nonlinear circuit theory). We did NOT use this text, except as a reference and problem source. He had pretty much abandonded the extreme abstract categorical approach by then.The notes I have follow the topical pattern of the book, but are so modified as to be essentially a different book, especially after covering spaces and the first homotopy group. His statement was that his treatment had changed since the subject had changed significantly.So much more has changed since then that I would not recommend this book as a primary text these days. Bredon's Topology and Geometry (Graduate Texts in Mathematics) is much better suited to today's student.
So, why did I give it four stars?First, notice that it splits stylewise into three segments, corresponding the treatment of its material in a three quarter academic year.The first three chapters (intro, covering spaces, polyhedral) have really not been superceded in a beginning text.Topics are covered very thoroughly, aiding the student new to the subject.The next three chapters (homology) are written much with much less explanation included - indeed, some areas leave much to the reader to discover and, consequently, aren't very helpful if the instructor doesn't fill in the details (the text expects a rather rapid mathematical maturation from the first part - too much of a ramp in my opinion), but the text is comprehensive.The last section (homotopy theory, obstruction theory and spectral sequences) should just be treated as a reference - it'd be hard to find all this material in such a compact form elsewhere and the obstruction theory section has fantastic coverage of what was known as of the writing of this book.It's way too terse for a novice to learn from and there are some great books out there these days on the material.

For reference ONLY
This book is a highly advanced and very formal treatment of algebraic topology and meant for researchers who already have considerable background in the subject. A category-theoretic functorial point of view is stressed throughout the book, and the author himself states that the title of the book could have been "Functorial Topology". It serves best as a reference book, although there are problem sets at the end of each chapter.

After a brief introduction to set theory, general topology, and algebra, homotopy and the fundamental group are covered in Chapter 1. Categories and functors are defined, and some examples are given, but the reader will have to consult the literature for an in-depth discussion. Homotopy is introduced as an equivalence class of maps between topological pairs. Fixing a base point allows the author to define H-spaces, but he does not motivate the real need for using pointed spaces, namely as a way of obtaining the composition law for the loops in the fundamental group. By suitable use of the reduced join, reduced product, and reduced suspension, the author shows how to obtain H-groups and H co-groups. The fundamental group is defined in the last section of the chapter, and the author does clarify the non-uniqueness of the fundamental group based at different points of a path-connected space.

Covering spaces and fibrations are discussed in the next chapter. The author does a fairly good job of discussing these, and does a very good job of motivating the definition of a fiber bundle as a generalized covering space where the "fiber" is not discrete. The fundamental group is used to classify covering spaces.

In chapter 3 the author gets down to the task of computing the fundamental group of a space using polyhedra. Although this subject is intensely geometrical. only six diagrams are included in the discussion.

Homology is introduced via a categorical approach in the next chapter. Singular homology on the category of topological pairs and simplicial homology on the category of simplicial pairs. The author begins the chapter with a nice intuitive discussion, but then quickly runs off to an extremely abstract definition-theorem-proof treatment of homology theory. The discussion reads like one straight out of a book on homological algebra.

This approach is even more apparent in the next chapter, where homology theory is extended to general coefficient groups. The Steenrod squaring operations, which have a beautiful geometric interpretation, are instead treated in this chapter as cohomology operations. The logic used is impeccable but the real understanding gained is severely lacking.

General cohomology theory is treated in the next chapter with the duality between homology and cohomology investigated via the slant product. Characteristic classes, so important in applications, are discussed using algebraic constructions via the cup product and Steenrod squares. Characteristic classes do have a nice geometric interpretation, but this is totally masked in the discussion here.

The higher homotopy groups and CW complexesare discussed in Chapter 7, but again, the functorial approach used here totally obscures the underlying geometrical constructions.

Obstruction theory is the subject of Chapter8, with Eilenberg-Maclane spaces leading off the discussion. The author does give some motivation in the first few paragraphs on how obstructions arise as an impediment to a lifting of a map, but an explicit, concrete example is what is needed here.

The last chapter covers spectral sequences as applied to homotopy groups of spheres. More homological algebra again, and the same material could be obtained (and in more detail) in a book on that subject.

Definitely not for beginners
I gave Spanier only three stars not because I think it is a bad book: as the previous two reviewers have pointed out, Spanier is a comprehensive (and still good) account of the subject, but is by no means for beginners. Now that more user-frinedly ones like Bredon, Fomenko-Novikov, and Hatcher (forthcoming) are available,it would hardly justify giving it four or five stars.And for reference purposes, there is a small (and sometimes too terse) but attractive account by May that covers topics not touched by Spanier.

Excellent reference, poor textbook
This book is terrific as a reference for those who already know the subject, but if you teach algebraic topology it would be dangerous to use it as a graduate text (unless you're willing to supplement it extensively).The basic problem is that Spanier does not teach students how to computeeffectively because his abstract, high-powered algebraic approach obscuresthe underlying geometry, which is not developed at all. Here I'd recommendthe books by Munkres, or Greenberg; even the old-fashioned treatment ofLefschetz, with its explicit and rather cumbersome treatment of cohomology,could serve as an antidote to Spanier. Somewhere, the student has toacquire a good intuitive feeling for the geometry underlying the subject(the same can be said of algebraic geometry -- here earlier work (e.g., ofthe Italian school, Weil's old book on intersection theory, ...) should notbe neglected entirely in favor of Grothendieck et al., for somethingessential is lost)

That said, if you already know the subject Spanier'sbook is an excellent reference. Even here, though, you'll need to providesome details toward the ends of the later chapters. Each chapter starts outrelatively easily and works up to a crescendo, the treatment becomingterser and more advanced.

I give it four stars (5 for mathematicalquality, 3 for usefulness as a text). The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences. Some of Serre's classical results on finitenesstheorems for homotopy groups are presented.

Excellent reference, poor textbook
This book is terrific as a reference for those who already know thesubject, but if you teach algebraic topology it would be dangerous to useit as a graduate text (unless you're willing to supplement it extensively). The basic problem is that Spanier does not teach students how to computeeffectively because his abstract, high-powered algebraic approach obscuresthe underlying geometry, which is not developed at all.Here I'd recommendthe books by Munkres, or Greenberg; even the old-fashioned treatment ofLefschetz, with its explicit and rather cumbersome treatment of cohomology,could serve as an antidote to Spanier.Somewhere, the student has toacquire a good intuitive feeling for the geometry underlying the subject(the same can be said of algebraic geometry -- here earlier work (e.g., ofthe Italian school, Weil's old book on intersection theory, ...) should notbe neglected entirely in favor of Grothendieck et al., for somethingessential is lost)

That said, if you already know the subject Spanier'sbook is an excellent reference.Even here, though, you'll need to providesome details toward the ends of the later chapters.Each chapter startsout relatively easily and works up to a crescendo, the treatment becomingterser and more advanced.

I give it four stars (5 for mathematicalquality, 3 for usefulness as a text).The first three chapters deal withcovering spaces and fibrations; the middle three with (co)homology andduality; the last three with general homotopy theory, obstruction theory,and spectral sequences.Some of Serre's classical results on finitenesstheorems for homotopy groups are presented. ... Read more

Product Description
In most major universities one of the three or four basic first-year graduate mathematics courses is algebraic topology. This introductory text is suitable for use in a course on the subject or for self-study, featuring broad coverage and a readable exposition, with many examples and exercises.The four main chapters present the basics: 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 is the inclusion of many optional topics not usually part of a first course due to time constraints: Bockstein and transfer homomorphisms, direct and inverse limits, H-spaces and Hopf algebras, the Brown representability theorem, the James reduced product, the Dold-Thom theorem, and Steenrod squares and powers. ... Read more

Customer Reviews (19)

Terrible textbook
This book is horrible if regarded as a mathematics book. Like previous reviewers I feel there is a total lack of clarity and rigor. Definitions are lacking, perhaps the author feels it is better to provide a "intuitive" feel for the material, than just definingg things. He fails miserably. The fact that what we are really dealing with in this subject are functors((co)homology, homotopy ) is nearly absent from the text. Instead drawings and pictures that are meant to provide "geometric" feel are supplanted.
I would state that this book attempts to teach how to compute in and use the theory than have you understand how the theory is built. It is a book for using the oven, not understanding how it works.

More Hand-Waving Than an Orchestral Conductor
In the TV series "Babylon 5" the Minbari had a saying: "Faith manages."If you are willing to take many small, some medium and a few very substantial details on faith, you will find Hatcher an agreeable fellow to hang out with in the pub and talk beer-coaster mathematics, you will be happy taking a picture as a proof, and you will have no qualms with tossing around words like "attach", "collapse", "twist", "embed", "identify", "glue" and so on as if making macaroni art.

To be sure, the book bills itself as being "geometrically flavored", which over the years I have gathered is code in the mathematical community for there being a lot of cavalier hand-waving and prose that reads more like instructions for building a kite than the logical discourse of serious mathematics.Some folks really like that kind of stuff, I guess (judging from other reviews).Professors do, because they already know their stuff so the wand-waving doesn't bother them any more than it would bother the faculty at Hogwarts.When it comes to Hatcher some students do as well, I think because so often Hatcher's style of proof is similar to that of an undergrad:inconvenient details just "disappear" by the wayside if they're even brought up at all, and every other sentence features a leap in logic or an unremarked gap in reasoning that facilitates completion of an assignment by the due date.

Some will say this is a book for mature math students, so any gaps should be filled in by the reader en route with pen and paper.I concede this, but only to a point.The gaps here are so numerous that, to fill them all in, a reader would be spending a couple of days on each page of prose.It is not realistic.Some have charged that this text reads like a pop science book, while others have said it is extremely difficult.Both charges are true.Never have I encountered such rigorous beer-coaster explanations of mathematical concepts.Yet this book seems to get a free ride with many reviewers, I think because it is offered for free.In the final analysis is it a good book or a bad book?Well, it depends on your background, what you hope to gain from it, how much time you have, and (if your available time is not measured in years) how willing you are to take many things on faith as you press forward through homology, cohomology and homotopy theory.

First, one year of graduate algebra is not enough, you should take two. Otherwise while you may be able to fool yourself and even your professor into thinking you know what the hell is going on, you won't really.Not right away.Ignore this admonishment only if you enjoy applying chaos theory to your learning regimen.

Second, you better have a well-stocked library nearby, because as others have observed Hatcher rarely descends from his cloud city of lens spaces, mind-boggling torus knots and pathological horned spheres to answer the prayers of mortals to provide clear definitions of the terms he is using.Sometimes when the definition of a term is supplied (such as for "open simplex"), it will be immediately abused and applied to other things without comment that are not really the same thing (such as happens with "open simplex") -- thus causing countless hours of needless confusion.

Third: yes, the diagram is commutative.Believe it.It just is.Hatcher will not explain why, so make the best of it by turning it into a drinking game.The more shots you take, the easier things are to accept.

In terms of notation, if A is a subspace of X, Hatcher just assumes in Chapter 0 that you know what X/A is supposed to mean (the cryptic mutterings in the user-hostile language of CW complexes on page 8 don't help).It flummoxed me for a long while.The books I learned my point-set topology and modern algebra from did not prepare me for this "expanded" use of the notation usually reserved for quotient groups and the like.Munkres does not use it.Massey does not use it.No other topology text I got my hands on uses it.How did I figure it out?Wikipedia.Now that's just sad.Like I said earlier:one year of algebra won't necessarily prepare you for these routine abuses by the pros; you'll need two, or else tons of free time.

Now, there are usually a lot of examples in each section of the text, but only a small minority of them actually help illuminate the central concepts.Many are pathological, being either extremely convoluted or torturously long-winded -- they usually can be safely skipped.

One specific gripe. The development of the Mayer-Vietoris sequence in homology is shoddy.It's then followed by Example 2.46, which is trivial and uncovers nothing new, and then Example 2.47, which is flimsy because it begins with the wisdom of the burning bush: "We can decompose the Klein bottle as the union of two Mobius bands glued together by a homeomorphism between their boundary circles." Oh really?(Cue clapping back-up chorus: "Well, ya gotta have faith...")That's the end of the "useful" examples at the Church of Hatcher on this important topic.

Another gripe. The write-up for delta-complexes is absolutely abominable. There is not a SINGLE EXAMPLE illustrating a delta-complex structure.No, the pictures on p. 102 don't cut it -- I'm talking about the definition as given at the bottom of p. 103.A delta-complex is a collection of maps, but never once is this idea explicitly developed.

A final gripe.The definition of the suspension of a map...?Anyone?Lip service is paid on page 9, but an explicit definition isn't actually in evidence.I have no bloody idea what "the quotient map of fx1" is supposed to mean. I can make a good guess, but it would only be a guess. Here's an idea for the 2nd edition, Allen: Sf([x,t]) := [f(x),t]. This is called an explicit definition, and if it had been included in the text it would have saved me half an hour of aggravation that, once again, only ended with Wikipedia.

But still, at the end of the day, even though it's often the case that when I add the details to a one page proof by Hatcher it becomes a five page proof (such as for Theorem 2.27 -- singular and simplicial homology groups of delta-complexes are isomorphic), I have to grant that Hatcher does leave just enough breadcrumbs to enable me to figure things out on my own if given enough time.I took one course that used this text and it was hell, but now I'm studying it on my own at a more leisurely pace.It's so worn from use it's falling apart.Another good thing about the book is that it doesn't muck up the gears with pervasive category theory, which in my opinion serves no use whatsoever at this level (and I swear it seems many books cram ad hoc category crapola into their treatments just for the sake of looking cool and sophisticated).My recommendation for a 2nd edition:throw out half of the "additional topics" and for the core material increase attention to detail by 50%.Oh, and rewrite Chapter 0 entirely. Less geometry, more algebra.

Really bad as a "readable" texbookbut good reference
I am not able to understad why people seems to love this book my feelings, beeing mixed, are perhaps closer to hate.

The book is OK if (and only if) you previously know the matter but the lack of clear definitions, the excessive reliance in reader geometrical intuition, the conversational style of demos the long paragraphs describing obscure geometric objects, etc make it very difficult to follow if it is your first approach to AT.

On the other hand has useful insigths if you already know the matter.

If the purpouse of the author has really been to write a "readable" book (as he told us repeatedly) I think the attemp is a complete failure.

On the other handthe "Table of contents" is excellent and is a very good book for teachers,I think this is the reason of itspopularity.

If you can afford the cost, purchase J Rotman "An introduction to Algebraic Topology" and you really will get a "readable" book

amazing book, but caveat emptor
I think that Allen Hatcher has given us all something very valuable in this book.If you are like me, you've had those moments when reading in a math book when you read a sentence, and your eyes shoot open and you suddenly feel like someone has been standing behind you that you never knew was there.There are lots of those kinds of sentences in this book.On the other hand, I view it as a supplement to a book like Munkres or Bredon that provides the rigor necessary to allow the learner to figure out the topologist's geometric language.I have used these three and found them to compliment one another well.

excellent modern introduction
This is an excellent introduction to the subject. It's affordable, well-written, and the topics are well chosen. The presentation is modern, but includes enough intuition that the fairly naive reader (e.g., me) can see the point of things. I needed to (re)learn topology for a research project I was part of in the intersection of math/CS/statistics and this book was a big help. I wish that he had included simplicial sets in the topics, because I like the way he writes and would like to have a more elementary exposition tied to the rest of the book (I eventually found an expository paper that did a pretty good job, but worked out examples would still help with that topic), but it can't include everything. I highly recommend this book to anyone trying to get started in this fascinating subject. It will just scratch the surface, but it does a good job of that. ... Read more

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In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry. ... Read more

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Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introduction to Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristics and Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. ... Read more

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This second edition, divided into fourteen chapters, presents a comprehensive treatment of contact and symplectic manifolds from the Riemannian point of view. The monograph examines the basic ideas in detail and provides many illustrative examples for the reader.

Riemannian Geometry of Contact and Symplectic Manifolds, Second Edition provides new material in most chapters, but a particular emphasis remains on contact manifolds. New principal topics include a complex geodesic flow and the accompanying geometry of the projectivized holomorphic tangent bundle and a complex version of the special directions discussed in Chapter 11 for the real case. Both of these topics make use of Étienne Ghys's attractive notion of a holomorphic Anosov flow.

Researchers, mathematicians, and graduate students in contact and symplectic manifold theory and in Riemannian geometry will benefit from this work. A basic course in Riemannian geometry is a prerequisite.

Reviews from the First Edition:

"The book . . . can be used either as an introduction to the subject or as a reference for students and researchers . . . [it] gives a clear and complete account of the main ideas . . . and studies a vast amount of related subjects such as integral sub-manifolds, symplectic structure of tangent bundles, curvature of contact metric manifolds and curvature functionals on spaces of associated metrics."   —Mathematical Reviews

"…this is a pleasant and useful book and all geometers will profit [from] reading it. They can use it for advanced courses, for thesis topics as well as for references. Beginners will find in it an attractive [table of] contents and useful ideas for pursuing their studies."   —Memoriile Sectiilor Stiintifice

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This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development.The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane.While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. ... Read more

Customer Reviews (5)

An excellent transition for the beginning grad student
I agree that this work is bit to terse for those completely uninitiated with the subject. However, for a first semester graduate student (or anyone with about this level of maturity) who has at least a minimal acquaintance with curves/surfaces this is a wonderful book which fulfills a long standing gap. The divide between undergraduate differential geometry and graduate geometry is too great. A beginning graduate student walk into a course on differentiable manifolds and Riemannian Geometry with no undergraduate diff geometry and be fine (as I did). There simply doesn't seem to be anymore that a superficial connection (no pun intended) between the two. This may not trouble the student at first, but as Spivak notes in his tome: "this ignorance of the roots of the subject has its price." Eventually one needs to assimilate the intuition of classical geometry with the technical language of manifolds. This book very elegantly leads the student from his undergraduate education to the doorstep of modern global geometry. As an added bonus, the author also endeavors to bridge the gap between Euclid and differential geometry.

The only other successful attempt at this is in Spivak, but unfortunatelyhe goes backwards. Volume 1 is entirely devoted to manifolds. Then in volume 2 he explains the classical point of view and then builds the bridge. While these are beautiful books, this is not efficient for the beginning student. The prospective geometer should read this before a class on manifold theory.

I give 4 stars only because the author advertises this as a "first exposure", and this book is simply not suited for this purpose.

great history of geometry book, terrible introductory differential geometry book
Do not buy this inappropriately titled book if you are seeking an introductory text to learn differential geometry. It's not that the concepts in the book are so advanced, so much as not that much space is actually devoted to the subject. The author's real objective is to trace the development of geometry from Euclid to the (relatively) modern formulation of differential geometry, and as a book on that topic it succeeds admirably.

The core theme of the book is that efforts to prove the parallel postulate, or, equivalently, show that non-Euclidean geometries are impossible, inadvertently, through their failure, led to the discovery of many fascinating areas of mathematics, such as hyperbolic and Riemannian geometries, and to the development of philosophical ideas about what actually constitutes mathematics and how it is independent from physical reality. The book culminates with the results of Beltrami and Poincare that showed that hyperbolic and Euclidean geometries are logically equivalent, in the sense that if there is a self-contradiction in one then the other is also impossible, thus putting an end to all attempts to disprove hyperbolic geometry. (Unfortunately, Marilyn vos Savant is unaware of this, or at least she was when she wrote an article some years back criticising Andrew Wiles's proof of Fermat's last theorem because it used hyperbolic geometry.)

As an appendix, McCleary adds a translation of Riemann's lecture "On the hypothesis which lie at the foundations of geometry," perhaps the most influential single lecture in the history of mathematics (and physics), in which, in the mid-1860s, he presented to a general faculty a talk (involving only a single equation) on the foundations of geometry that anticipated the concepts of a manifold and Riemannian geometry as well as general relativity and even hinted at quantum mechanics.

I used this text as a primary reference when conducting an undergraduate seminar on the history of hyperbolic geometry 12 years ago. For this purpose it was suited perfectly, but if you want to learn differential geometry by all means buy one of do Carmo's books or Gallot, Hulin, and LaFontaine.

not for the uninitiated
I'm a master's student in math. I bought the book thinking I'd use it for an independent study. I was wrong.

The book has interesting historical tidbits and some classical proofs, including material I hadn't seen elsewhere. However, it takes little time to explain to the novice exactly what's going on. It comes off more as a set of lecture notes than as a text for self-study.

For instance, in ch. 8 McCleary breezes through the basics of regular surfaces--coordinate charts, differentiability, implicit/inverse function theorem, the tangent space, orientability, the first fundamental form in about 19 pages. This is the same foundational material that folk like do Carmo or O'Neill rightfully spend 60-70 pages to cover.

His treatment of the Gauss map and the second fundamental form is even more schematic.

If I hadn't already worked the other books, when I got to McCleary's treatment of surfaces I would've been completely lost.

This book is best for people who know basic differential geometry already but are curious about certain historical aspects of it, not for people who are trying to learn differential geometry.

a great book!
This is a great book.The author develops the differential geometry of curves and surfaces.The endpoint is the vindication of Euclid's parallel postulate.I thoroughly enjoyed reading this book.Very readable.

Excellent text connecting classical to differential geometry
This book is ideal for those with a long time interest in mathematics or the student just becoming interested in advanced topics.It successfully takes the concepts of classic geometry (Euclidean), clearly explains how the parallel postulate interacts with the other postulates and then introduces differential geometry as a natural outgrowth of hyperbolic geometry.McLeary's book succeeds by demonstrating the connection of modern differential geometry to the concepts in which we were educated.This is not a book for the casual reader, but includes many problems and solutions to the more interesting of them ... Read more

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A sweeping yet uniquely accessible introduction to a variety of central geometrical topics

Covering over two centuries of innovations in many of the central geometrical disciplines, Introduction to Topology and Geometry is the most comprehensive introductory-level presentation of modern geometry currently available.

Unique in both style and scope, the book covers an unparalleled range of topics, yet strikes a welcome balance between academic rigor and accessibility. By including subject matter previously relegated to higher-level graduate courses in mathematics and making it both interesting and accessible, the author presents a complete and cohesive picture of the science for students just entering the field.Historical notes throughout provide readers with a feel for how mathematical disciplines and theorems come into being.

Students and teachers will benefit from a uniquely unified treatment of such topics as:

• Homeomorphism
• Graph theory
• Surface topology
• Knot theory
• Differential geometry
• Riemannian geometry
• Hyperbolic geometry
• Algebraic topology
• General topology

Using a variety of theorems to tie these seemingly disparate topics together, the author demonstrates the essential unity of mathematics.

A logical yet flexible organization makes the text useful for courses in basic geometry as well as those with a more topological focus, while exercises ranging from the routine to the challenging make the material accessible at varying levels of study. ... Read more

Customer Reviews (1)

A great introductory book
This is a great introductory book for students (or non-students) interested in the subject. The book does not delve into rigorous discussions, nor does it get too involved in proofs of theorems, however it produces a nice overall picture of the principles involved in Topology.

Anyone, looking for a relatively smooth first-time study of the subject should find this book very helpful.
... Read more

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This book brings the beauty and fun of mathematics to the classroom. It offers serious mathematics in a lively, reader-friendly style. Included are exercises and many figures illustrating the main concepts.The first chapter talks about the theory of manifolds. It includes discussion of smoothness, differentiability, and analyticity, the idea of local coordinates and coordinate transformation, and a detailed explanation of the Whitney imbedding theorem (both in weak and in strong form). The second chapter discusses the notion of the area of a figure on the plane and the volume of a solid body in space. It includes the proof of the Bolyai-Gerwien theorem about scissors-congruent polynomials and Dehn's solution of the Third Hilbert Problem.This is the third volume originating from a series of lectures given at Kyoto University (Japan). It is suitable for classroom use for high school mathematics teachers and for undergraduate mathematics courses in the sciences and liberal arts. ... Read more

Customer Reviews (2)

Great book
Simply put, this is a great book.

It is definitely not a 'serious' math book (no definitions, no proofs) but it is very well written and provides a nice, intuitive introduction to many advanced topics.

Should be required reading for math majors at the freshman or sophomore level!

for the love of geometry and topology
I'm a masters student who recently completed an introductory course in differential geometry, and another in algebraic topology. I enjoyed the courses and did well, but if I'd read this book beforehand, the going would've been even better.

The authors' love of the subject and love of teaching shines through. The book is written so that even advanced high school students can follow it, and yet it's deep enough to spark the imagination of this grad student and remind me of the magic of manifolds and space.

Provides excellent and intuitive overviews and proofs on the Euler characteristic, Poincare-Hopf theorem, Gauss-Bonnet, different notions of dimension, higher dimensional manifolds, and various related topics.

When I teach geometry or topology, I will definitely include "A Mathematical Gift" in the syllabus.

I look forward to checking out volumes II and III. ... Read more

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Customer Reviews (7)

Decent book with flaws
The book has its virtues, sure enough. But there are some downsides
to it as well that I feel are underrepresented in the other reviews so far.

Let me first note that, contrary to the statement of one other reviewer, there are exercises in this book, and not too few. However, I found that I did not need them, since thinking deeply about all the little flaws and omissions that are scattered through the text allowed me to mature faster than going through these exercises. Needless to say, though, that this type of exercise can be a bit frustrating. I often found myself wondering if it was my lack of maturity that made me struggle, or if the authors actually made their life too simple at various points. Luckily, I found amply evidence for the latter. For example, the reader familiar with homotopy may open the book on page 164 and inspect their proof that the curve given by f(1-x) is the inverse of that given by f(x) in the fundamental group. While this is a true statement of course, their constructed homotopy to prove this is not really continuous, and a slight modification of it could be used as a "proof" that every curve is homotopy equivalent to a constant one. A useful review of the book by a professional can be found at the following URL,

http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183524657

where similar shortcomings are noted. I agree that the latter will probably not slow down an expert who chooses this book as a reference. For beginners, however, they are unnecessary obstacles. I bought this book because I got attracted by the balanced selection of topics ranging from point set topology to algebraic topology. I wanted to learn the latter, but first needed to become proficient in the former. Having now read only the first part of the book devoted to point set topology, I can say that the book did its job, and did it quite well. However, I cannot shake off the feeling that I could have learned the same material in a fraction of the time from a different book. Feeling that I do now have a solid enough background in point set topology, I am considering to not read the second half of the book, and instead learn algebraic topology from a more modern text.

Very Impressed
I am teaching myself topology with this book right now, and I must say it has an excellent balance of motivation and rigor. The very first definition in the book reveals the implications of topology to anyone who has studied limit pts (and how connectedness is defined in terms of same). After less than a week of study, I understood the big picture better than most people I know who have taken a full course. The exercises are a little sparse, perhaps, but they generally make up for their small number with increased difficulty. I have only encountered a few exercises that I could call trivial. My only gripe is that the exercises are sometimes a little tricky to find.

A good start
Very clearly written, full of examples and counterexamples, making use of pictures but never sacrificing rigor, the authors of this book have given students of topology a superb introduction to the field. Many students have been educated in topology by using this book, and it is sure to remain a classic in the field. It builds a solid understanding of the basic rudiments and intuition behind point-set, geometric, and algebraic topology. There is a lot of material covered in the book, and some very specialized subjects, such as Cech and Vietoris homology and some dimension theory, but with some preserverance and concentration, the entire book can be grasped within reasonable time constraints. Probably the only minus to the book is the lack of exercises. This is a quite serious omission, for the only way to master a subject is to work problems that require careful thought for their solution.

The beginning student of topology should probably read this book with the following mindset: try to think of ways and techniques that you would devise to study the structure of a topological space. Homotopy and homology (in various forms) are the standard techniques for doing this. These strategies have varying degrees of success, but their use in topology now seems to be reaching a saturation limit, even though the explicit calculation of homotopy groups is still a very active area. New techniques and concepts, representing sort of a "large deviation" from the standard ones discussed in this book, will be needed to make further progress in the study of complicated topological spaces. Something more is needed now, that is completely different than homology and homotopy theory, that will make more transparent the properties of these spaces. These new techniques will be somewhat radical from the standpoint of current ones, but they will be more effective from a conceptual (and computational) point of view.

A Professional Topologist loves this book.
When I was a graduate student 40 years ago there were very few texts in topology.The only two that I recall being in use were Hocking and Young and the book by Kelley.Over the years my copy of Hocking and Young has become quite worn.It is a wonderful book that gives the true flavor of topology.It is also contains a large number of topics that one can refer to later on.It becomes quite apparent very earlier that no one will be able to fully appreciate the book in the time span of one course.It is a book that must be read and reread over and over again.It is a real classic.I do not believe that it is the type of book that would be of much or any general interest but to a point set topologist it is a classic and must for his bookself.I am quite surprised over its low price.I can not help but compare it with the newer book by Munkres.I recall seeing Munkres book many years ago and disliking it.But the current edition seems much closer in flavor to HY and Munkres book is quite good.Munkres style is much clearer than HY, but both books target a very specialized group of people.Neither book is for the faint of heart and will take many years to absorb.Considering that Munkres book is 9 times as expensive as HY, HY seems to be the better buy.

Theoretical Dictionary
An excellent book, not for those persons unfamiliar with the topic of topolgy; yet, combined with simpler texts this book is a goldmine of topological theorems and their proofs. ... Read more

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Product Description
The book contains a clear exposition of two contemporary topics in modern differential geometry:

* Distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature is applied to study the Laplace operator on minimal submanifolds.

* The application of the Lichnerowicz formula for Dirac operators to the study of Gromov's invariants to measure the K-theoretic size of a Riemannian manifold.

It is intended for both graduate students and researchers who want to get a quick and modern introduction to these topics. ... Read more

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This book, the inaugural volume in the University Lecture Series, is based on lectures presented at Pennsylvania State University in February 1987. The lectures attempt to give a taste of the accomplishments of manifold topology over the last 30 years. By the late 1950s, algebra and topology had produced a successful and beautiful fusion. Geometric methods and insight, now vitally important in topology, encompass analytic objects such as instantons and minimal surfaces, as well as nondifferentiable constructions. Keeping technical details to a minimum, the authors lead the reader on a fascinating exploration of several developments in geometric topology. They begin with the notions of manifold and smooth structures and the Gauss-Bonnet theorem, and proceed to the topology and geometry of foliated 3-manifolds. They also explain, in terms of general position, why four-dimensional space has special attributes, and they examine the insight Donaldson theory brings. The book ends with a chapter on exotic structures on $\mathbf R^4$, with a discussion of the two competing theories of four-dimensional manifolds, one topological and one smooth. Background material was added to clarify the discussions in the lectures, and references for more detailed study are included. Suitable for graduate students and researchers in mathematics and the physical sciences, the book requires only background in undergraduate mathematics. It should prove valuable for those wishing a not-too-technical introduction to this vital area of current research. ... 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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This monograph deals with the geometrical and topological aspects related to quantum field theory with special reference to the electroweak theory and skyrmions. This book is unique in its emphasis on the topological aspects of a fermion manifested through chiral anomaly which is responsible for the generation of mass. This has its relevance in electroweak theory where it is observed that weak interaction gauge bosons attain mass topologically. These geometrical and topological features help us to consider a massive fermion as a skyrmion and for a composite state we can realise the internal symmetry of hadrons from reflection group. Also, an overview of noncommutative geometry has been presented and it is observed that the manifold M 4 x Z2 has its relevance in the description of a massive fermion as skyrmion when the discrete space is considered as the internal space and the symmetry breaking gives rise to chiral anomaly leading to topological features.
... Read more