国外优秀数学教材选评（第九章 拓扑与微分几何）

9 拓扑与微分几何

拓扑学是现代数学的基础之一，国内外很多大学数学系都把它列为本科生高年级基础课，甚至是必修棵。按大数学家嘉当的观点，数学中最基本的结构就是代数结构和拓扑结构。
拓扑又分点集拓扑（也称一般拓扑）、代数拓扑和微分拓扑等。点集拓扑讨论拓扑空间的基本概念和拓扑空间的连通性、分离性、紧致性等重要性质。代数拓扑则是用抽象代数中的工具来研究拓扑空间进一步的性质，其内容包括基本群、覆盖空间、同调、上同调、示性类等。一般在本科生的拓扑课程中学习点集拓扑、基本群和覆盖空间。其余是研究生课程的内容。微分拓扑的对象是微分流形的拓扑性质，也属于研究生课程的范围。
初等微分几何主要讨论欧氏空间中曲线和曲面的几何学，需要的数学工具比较少，基本上有数学分析、线性代数和常微分方程的预备知识就够了，它也可称为 古典微分几何，大学本科的微分几何课程一般学初等微分几何。近代微分几何的对象是微分流形、向量丛、李群等，它和拓扑学的关系异常密切。微分几何里又有各种分支如黎曼几何、复几何、辛几何等，这里不一一介绍了。

书名： Elements of Algebraic Topology
作者： James R. Munkres
出版商： Addison-Wesley Publishing Company
页数： 454
适用范围：基础数学研究生一学年的教材和数学系高年级本科生
预备知识：一般拓扑学和线性代数
习题数量：适中
习题难度： 适中
推荐强度： 9.6
书评：本书是美国 MIT 的 James R. Munkres 教授的力作，从 80 年代开始就作为麻省理工学院数学系一年级研究生教材， 是一本极具特色的优秀教材。本书以拓扑中的重要实例为线索，引导出抽象的概念， 对同调论的基本思想做到了深入浅出。本书以现代数学的语言来阐述代数拓扑中同调理论的， 内容安排上由浅入深，即从单纯同调开始，逐渐地过渡到奇异同调及上同调。 虽然教材的主要内容是基本的代数结构，但几何动机、背景和应用贯穿始终。 全书分 8 章，从单纯复形引入同调群概念，到如何定义一般拓扑空间的同调群； 到最后一章又转到流形的同调理论。遵循了从特殊到一般，再回到特殊的哲学规律。 本书的习题和教材的衔接处理的非常好。
本书比较适合我国综合性大学数学系研究生的外文教材，尤其适合一学年。 本教材自封性非常好，故此还特别适合程度较好的数学系本科生进行自学。 （吕志）

国外评论选摘
1) This well written text is one of the standard references in algebraic topology courses because of its conciseness, and I find it very useful as a reference text. However I think it is a little incomplete because of several reasons.
(1)It pays no attention to one basic concept of algebraic topology: the fundamental group.
(2) It doesn't cover Cech homology, important in other areas, like dimension theory for example.
(3) It doesn't stress the most important feature of algebraic topology: its connection to other areas of mathematics (analysis, differential geometry, etc.).
(4) Its list of references is too short, and lacks almost completely HISTORICAL references which are always important to become an expert in any field.
Conclusion: a good reference on homology and cohomology essentials, but not the reference on algebraic topology as a whole

2) Algebraic topology is a tough subject to teach, and this book does a very good job. Some prerequisites, however, are essential:

point set topology (e.g. in Munkres' Topology)

Abstract algebra

Mathematical maturity to be willing to follow a definition and argument even when it seems like a weird side-track

In addition, this would not be the first book I would recommend to those interested in algebraic topology. First might be Massey's Algebraic Topology: and Introduction that introduces the fundamental group (conceptually easier than homology and cohomology).
At some point, however, a prospective student in topology will have to learn homological algebra and this provides the most concrete approach I know to the subject.
Algebraic topology is a lot of fun, but many of the previous textbooks had not given that impression. This one does.

书名： Lecture Notes on Elementary Topology and Geometry
作者： Singer & Thorpe
出版商：Springer Verlag (1967)
页数：232
适用范围：大学数学系本科高年级教材或参考读物
预备知识：微积分、线性代数、抽象代数
习题数量：少
习题难度：中等
推荐强度：9.2
书评： 本书是大数学家为本科生写教材的又一典范，在 60-70 年代曾用作麻省理工学院数学系本科高年级一学年的课程的教材。 时隔几十年，行家们仍然认为这是一本不可多得的拓扑和几何的优秀入门读物。
本书篇幅不大，包含的内容不少，深入浅出，引人入深。作者不追求完整性，比如前两章的点集拓扑的基本知识不拘泥于一些 公理的仔细探讨，而是简明实用地把几何中最常用的拓扑空间讲清楚。第三章只化 20 几页就把基本群和复叠空间及其关系写清楚的， 值得指出的是前三章看似枯燥的内容中时而出现非常有趣的例子。第四、五、六章是全书的第一重点，讲述拓扑空间的同调群及微分流形 的概念，高潮是 de Rham 定理的证明。这里充分体现了数学中各不同分支间的渗透。最后两章是黎曼几何的导引，讲述了曲面上的 Gauss-Bonnet 公式这一深刻的定理。即使从现在的角度去看，这本书的选材仍然是反映现代数学主流的。
从内容来看，本书是点集拓扑、微分拓扑、微分几何三合一，这正是作者开这门课的宗旨。我国综合性大学的数学系一般对本科生也设有 拓扑和微分几何的课程，一般各占一个学期，这些学校的学生在学完这些课程后常常不知道 de Rham 定理和 Gauss-Bonnet 公式。 Singer-Thorpe 的这本教材也许可以启发我们在拓扑和几何的本科生教学方面作些改革。 （杨劲根）

书名： Topology from the differentiable viewpoint
作者： John Milnor
出版商：The University Press of Virginia
页数：61
适用范围：大学数学系本科高年级参考读物
预备知识：微积分、线性代数、基础拓扑
习题数量：17 题
习题难度：从易到难
推荐强度：9.5
书评： 华罗庚前辈写过好几本题为《从 ... 谈起》的数学小册子，其中一本是 《从单位圆谈起》。我们可以把 Milnor 的这本小书起名为《从单位球面谈起》，事实上本书自始至终不离单位球面。
拓扑学家 John Milnor 是费尔滋奖得主，在 Princeton 执教多年，他在拓扑方面的很多系列讲座笔记被整理出版，成为脍炙人口的数学读物， 这本书就是其中一本。
一本可以让大学三年级学生能看懂的 60 页的小册子，却包含如此多的深刻的定理（从 Sard 定理直到 Hopf 定理）以及完整的证明，这是何等的不可思议！ 这是学拓扑或几何的学生的必读书。 （杨劲根）

国外评论选摘

i) Perfect for a first-year graduate or advanced undergraduate course, Milnor takes us on a brief stroll through elementary differential topology. Elegant and self-contained, this book serves as an excellent first taste of the subject. Milnor is a master expositor, and is at his best in this book.
ii) One of the best points of this little book is its brevity and clear exposition of the basic ideas. It makes a great reference guide because it's so short and well-organized. Written by a distinguished mathematician, it's no wonder that other graduate-level texts such as Guillemin & Pollacks Differential Topology highly recommend reading it alongside their book. Milnor's booklet is a classic, whose style and ideas surely pervade other texts.

书名： Algebraic topology
作者： Hatcher
出版商： Cambridge University Press (2002)
页数： 542
适用范围：大学数学系基础数学研究生教材
预备知识：点集拓扑、抽象代数
习题数量：中等
习题难度：中等
推荐强度： 8.5

国外评论选摘
i) No serious introductory text on basic algebraic topology has ever achieved this level of clarity, readability and depth. Its richness in examples (in both the main text and the problems) exposes a beginner to the underlying mechanisms of geometry in algebraic topology; its choice and arrangement of topics strike a perfect balance between accesibility and substantiveness; its lively and motivating exposition makes a student reluctant to attend the often boring topology classes. For a novice, this should be the first reading on the subject before (s)he is ruined by the many existing daunting texts; for a veteran, this can be very nourishing, especially if (s)he is already ruined by those either unreadable or shallow 'introduction's.
ii) Allen Hatcher has gone to great length's in order to create a text which, albeit overly verbose, can be used as a gentle introduction to modern Algebraic Topology. Why 'modern'? Compare this text with the tried and tested texts of Spanier, Munkres as well as May and, almost immediately, you will see what I mean. The obvious example is Hatcher's use of CW-complexes as opposed to the more traditional build up beginning with simplices. For the die-hard mathematician who enjoys less fluff, this book is not for you and, in particular, if this is your first venture in Algebriac Topology, you enjoy the theorem-proof-theorem style with a light sprinkling of explaination, then I would recommend J.J. Rotman's text. Whereas, if you enjoy filler, background information, and lots of side-notes or examples, then Hatcher's text would be a perfect fit. Myself, I fall into the category of those who enjoy the more terse texts but, I purchased Hatcher's (the hardcover) because of the clarity and percision found in the proofs. The majority of other texts have a tendancy to obfuscate the underlying meaning that should be unerstood by the up-and-coming mathematician. Of course this approach has it's merits since, in particular, it forces the reader to fill in the blanks but, as a matter of insight, Hatcher's approach is also beneficial. Another positive strength of Hatcher's text lies in the fact that he effectively breaks the subject into it's prime sub-categories in such a way that the reader can begin with either of the four parts of the text without having to rely too much on previous sections. This novel feature allows someone interested in, say, Cohomology to pick up an begin learning about Cohomology without having to waste time making their way through material they are not interested in. Finally, yes you can get the book for free via Hatcher's website but I highly recommend purchasing the hardback text. It is well made, it will last for years, and it becomes truely mobile as compared to burning your eyes out while reading the text on your computer. Moreover, why waste the time printing it out.

书名： Differential forms in algebraic topology
作者： Bott & Tu
出版商： Springer GTM 82 (1982)
页数： 331
适用范围：大学数学系基础数学研究生教材
预备知识：基础数学本科生的大部分知识
习题数量：少
习题难度：中等
推荐强度： 9.3

国外评论摘选
1) The authors of this book, through clever examples and in-depth discussion, give the reader a rare accounting of some of the important concepts of algebraic topology. The introduction motivates the subject nicely, and the authors succeed in giving the reader an appreciation of where the concepts of algebraic topology come from, how they do their jobs, and their limitations. The de Rham cohomology, which is the main subject of the book, is explained in here in a way that gives the reader an intuitive and geometric understanding, which is sorely needed, especially for physicists who are interested in applications. As an example, they give a neat argument as to why de Rham cohomology cannot detect torsion. In chapter 1, the authors get down to the task of constructing de Rham cohomology, starting with the de Rham complex on R(n). The de Rham complex is then specialized to the case where only C-infinity functions with compact support are used, giving the de Rham complex with compact supports on R(n). The de Rham complex is then generalized to any differentiable manifold and the de Rham cohomology computed using the Mayer-Vietoris sequence.
The discussion gets a little more involved when the authors characterize the cohomology of a fiber bundle. The all-important Thom isomorphism for vector bundles, is treated in detail. The authors give several good examples of the Poincare duals of submanifolds. The connection to ideas in differential topology is readily apparent in this chapter, namely transversality and the degree of a map. In addition, the first construction of a characteristic class, the Euler class, is done in this chapter.
The Mayer-Vietoris sequence is generalized to the case of countably many open sets in chapter 2, and shown to be isomorphic to the Cech cohomology for a good cover of a manifold. Good examples are given for computing the de Rham cohomology from the combinatorics of a good cover. The authors then characterize Cech cohomology groups in more detail, introducing the important concept of a presheaf. Presheaves are usually introduced abstractly in most books, so it is a real treat to see them described here in such an understandable way. Computations of the case of a sphere bundle are given, and the role of orientability and the Euler class in giving the existence of a global form on the total space is detailed. The Thom isomorphism theorem and Poincare duality are generalized to the cases where the manifold does not have a finite good cover and the vector bundle is not orientable. A very concrete introduction to monodromy is given and nice examples of presheaves that are not constant are given.
The authors treat spectral sequences in chapter 4, and as usual with this topic, the level of abstraction can be a stumbling block for the newcomer. The authors though explain that the spectral sequence is nothing other than a generalization of the double complex of differential forms that was considered in chapter 2. The crucial step in the chapter is the transition to cohomology with integer coefficients, which is necessary if one is to study torsion phenomena. The De Rham theory is then extended to singular cohomology and the Mayer-Vietoris sequence studied for singular cochains. The authors show that the singular cohomology of a triangularizable space is isomorphic to its Cech cohomology with the constant presheaf the integers. After a fairly detailed review of homotopy theory (including a discussion of Morse theory) the authors compute the fourth and fifth homotopy groups of S(3). The last section of the chapter discusses the rational homotopy theory of Sullivan as applied to differentiable manifolds. The authors discussion is illuminating, and shows how eliminating any torsion information allows one to prove some interesting results on the homotopy groups of spheres. One such result is Serre's theorem, the other being the computation of some low-dimensional homotopy groups of the wedge product of S(2) with itself.
The last chapter of the book considers the theory of characteristic classes, with Chern classes of complex vector bundles being treated first. The theory of characteristic classes is usually treated formally, and this book is no exception, wherein the authors formulate it using ideas of Grothendieck. They do however give one nice example of the computation of the first Chern class of a tautological bundle over a projective space. The Pontryagin class is defined in terms of a complexification of a real vector bundle and computed for spheres and complex manifolds. A superb discussion is given of the construction of the universal bundle and the representation of any bundle as the pullback map over this bundle.
2) This book is almost unique among mathematics books in that it strives to ensure that you have the clearest picture possible of the topics under discussion. For example almost every text that discusses spectral sequences introduces them as a completely abstract machine that pumps out theorems in a mysterious way. But it turns out that all those maps actually have a clear meaning and Bott and Tu get right in there with clear diagrams showing exactly what those maps mean and where the generators of the various groups get mapped. It's clear enough that you can almost reach out and touch the things :-) And the same is true of all of the other constructions in the book - you always have a concrete example in mind with which to test out your understanding. That makes this one of my all time favourite mathematics texts.

书名：Knot thoery
作者：Livingston
出版商：Mathematical Association of America (1996)
页数：258
适用范围：大学数学系本科生自学读物
预备知识：线性代数，群论
习题数量：多
习题难度：中等
推荐强度：8.8

书评：本书是美国数学协会出版的大学生系列丛书“Carus Mathematical Monographs 中的一册，是拓扑学中纽结理论的优秀入门书。
本书的预备知识非常少，只要少量的线性代数知识。如果知道一些群论更好，但作者在用到群和二次型时都从头讲起。本书从纽结的历史和直观形象开始讲述纽结的分类，分别从组合、几何和代数三个方面引入各种重要不变量，如 Seifert 矩阵、 Alexander 多项式、 Conway 多项式、 Jones 多项式等。最后把 各种不变量的关系叙述得非常清楚。
本书图文并茂，习题非常丰富。内容安排从浅入深，章节的衔接紧凑。对初学者容易忽视的要点讲得很清楚。大部分定理有严格的证明，但又不拘泥于一些繁琐的证明细节，容易使读者掌握要点。
由于所用的准备知识少，所有的证明几乎都基于平面上的 Reidemeister 变换，纽结的基本群只是简单介绍一下，代数拓扑的工具没有使用。从这点来看对于具有较深数学基础的读者可以较快浏览本书后再选择更加高深的纽结理论的书籍阅读。
下面两段国外的评论中第一篇是一个数学教授写的，第二篇是自学过这本书的一个研究生写的，颇有代表性。（杨劲根）

国外评论摘选
1) This book is an excellent introduction to knot theory for the serious, motivated undergraduate students, beginning graduate
students,mathematicains in other disciplines, or mathematically oriented scientists who want to learn some knot theory. Prequisites are a bare minimum: some linear algebra and a course in modern algebra should suffice, though a first geometrically oriented topology course (e. g., a course out of Armstrong, or Guillemin/Pollack) would be helpful.
Many different aspects of knot theory are touched on, including some of the polynomial invariants, knot groups, Alexander polynomial and related abelian invariants, as well as some of the more geometric invariants.
This book would serve as a nice complement to C. Adams Knot Book in that Livingston covers fewer topics, but goes into more mathematical detail. Livingston also includes many excellent exercises. Were an undergraduate to request that I do a reading course in knot theory with him/her, this would be one of the two books I'd use (Adam's book would be the other).
This book is intentionally written at a more elementary level than, say Kaufmann (On Knots), Rolfsen (Knots and Links), Lickorish (Introduction to Knot Theory) or Burde-Zieshcang (Knots), and would be a good stepping stone to these classics. 2) I really do enjoy this book - but picked it up as a means of teaching myself Knot Theory... as was the case with many of my text books in college, brevity (for the sake of publishing costs) makes some concepts more of a challenge to grasp. Overall, the illustrations are great, and if you do the exercizes, the material tends to flow more easliy. It seemed to me the book worked backwards a bit - first covering a subject, than introducing it comprehensively later on - not what I'm used to. Keep in mind, I'm not a Mathematician, merely a graduate student of mathematics, who is interested in learning about this subject on my own.

书名： Riemannian Geometry, 3rd ed.
作者： M.P. Do Carmo
出版商：Springer Verlag (2004) ISBN-13: 978-3540204930
页数： 322
适用范围：数学专业研究生
预备知识：微积分，线性代数
习题数量：较大
习题难度： 适中
推荐强度: 10
书评：本书是一本标准的黎曼几何教材和参考书，其作者是巴西著名几何学家 Do Carmo 教授。作者写作风格清晰明了，全书共十三章，前四章介绍了黎曼几何的基本概念， 如黎曼度量、黎曼联络、测地线和曲率等；第五章介绍了 Jacobi 场这个重要的工具， 阐明了测地线与曲率的关系；第六章对等距浸入介绍了第二基本形式及相关的基本公式。 该书从第七章开始， 主要介绍了整体问题，涉及曲率与拓扑和比较几何中的一些基本结果。该书自成体系， 是目前黎曼几何最好的入门书之一。 对于想从事研究整体微分几何及相关领域的读者，该书也适合自学。 （东瑜昕）

国外评论摘选
i) This book based on graduate course on Riemannian geometry covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results are treated in detail. contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced. For this third edition, some topics have been added and worked out in the same spirit. (L'ENSEIGNEMENT MATHEMATIQUE, Vol. 50, (3-4), 2004)
ii) This book is based on a graduate course on Riemannian geometry and analysis on manifolds that was held in Paris . Classical results on the relations between curvature and topology are treated in detail. The book is almost self-contained, assuming in general only basic calculus. It contains nontrivial exercises with full solutions at the end. Properties are always illustrated by many detailed examples. ( EMS Newsletter, December 2005)

书名： Foundations of Differential Geometry (in two volumes)
作者： Shoshichi Kobayashi & Katsumi Nomizu
出版商：John Wiley & Sons, Inc. (1996)
页数： Vol.I : 329 , Vol.II: 468
适用范围：数学专业研究生
预备知识：微积分，线性代数，微分流形、 Lie 群基础知识
习题数量：无
推荐强度：10
书评：
本书共两卷，旨在系统介绍微分几何的基础内容，其作者是著名的几何学家 S. Kobayashi 和 K. Nomizu 。第一卷首先概要地介绍了微分流形、李群和纤维丛的概念，然后主要介绍了主丛上的联络论、向量丛上的线性联络和仿射联络、黎曼流形上的黎曼联络，还涉及空间形式、仿射联络或黎曼度量的自同构群等。第二卷主要介绍了一些经典的专题 , 如子流形理论、Morse 指标理论 , 复流形、齐性空间和对称空间、示性类理论等。本书内容翔实、处理严谨，行文精练， 自二十世纪六十年代问世以来，一直被认为是经典的微分几何参考书。 1996 年John Wiley & Sons 出版社将其选入经典图书系列重印了其第三版，可见其影响。 对于想从事微分几何和相关领域研究的读者，这是一本很好的参考书。（东瑜昕）

国外评论摘选
1) The two-volume set by Kobayashi and Nomizu has remained the definitive reference for differential geometers since their appearance in 1963(volume 1) and 1969 (volume 2). Over the decades, many readers have developed a love/hate relationship with these difficult, challenging texts. For example, in a 2006 edition of a competing text, the author remarked that every differential geometer must have a copy of these tomes, but followed this judgment by observing that their effective usefulness had probably passed away, comparing them to the infamously difficult texts of Bourbaki.
As a practicing differential geometer, I would argue that Kobayashi and Nomizu remains an essential reference even today, for a number of reasons. Volume 1 still remains unrivalled for its concise, mathematically rigorous presentation of the theory of connections on a principal fibre bundle---material that is absolutely essential to the reader who desires to understand gauge theories in modern physics. The essential core of Volume 1 is the development of connections on a principal fibre bundle, linear and affine connections, and the special case of Riemannian connections, where a connection must be fitted to the geometry that results from a pre-existing metric tensor on the underlying manifold, M. Volume 2 offers thorough introductions to a number of classical topics, including submanifold theory, Morse index theory, homogeneous and symmetric spaces, characteristic classes, and complex manifolds.
The influence of the texts by Kobayashi and Nomizu can be seen in most of the subsequent differential geometry texts, both in organization and content, and especially in the adoption of notation. If there was a particularly fine point in your favorite introductory differential geometry text that you never completely understood, the odds are good that you will find the answer, fully developed and presented at an entirely different mathematical level, in Kobayashi and Nomizu. It is not an unreasonable analogy to say that learning differential geometry without having your own copy of Kobayashi/Nomizu is like studying literature in the complete ignorance of Shakespeare.
Let there be no mistake about the advanced level of these texts. The Preface to Volume 1 clearly states that the authors presume the reader to be familiar with differentiable manifolds, Lie groups, and fibre bundles, as developed in the (now classical) texts by Chevalley, Montgomery-Zippin, Pontrjagin, and Steenrod. Today's reader is far more likely to have studied these subject from more recent books like those by Boothby, Hall, and Husemoller, but whatever the source, a familiarity IS presumed. The lightning review provided in Chapter I of Volume 1 will be extremely tough going for the reader who is new to these topics. It should also be noted that in 329 pages of Volume 1 and 470 pages of Volume 2, not a single diagram or picture is to be found! Those drawn to geometry for its visual aspects will find Kobayashi/Nomizu totally lacking in visual aids.
As with so many classic references in mathematics, the hardbound edition of Kobayashi and Nomizu is no longer in print. Copies appear sporadically on the used book market at absolutely obscene prices. The Classics Library paperback edition is still available, but the serious student will find that the paperbacks simply do not fare well under serious, sustained use.

书名： Introduction to Lie groups and Lie algebras
作者： A.A.Sagle & R.E.Walde
出版商： Academic Press (1973)
页数： 361
适用范围：大学数学系研究生低年级教材
预备知识：抽象代数
习题数量：较小
习题难度： 中等
推荐强度： 9.5
书评： 本书详细介绍了李群和李代数的基本知识以及半单李代数的结构。本书的特点是起点很低 , 对欧氏空间中的微分、张量积、模及其表示以及微分流形和 Riemann 流形的基本内容都作了一定的介绍， 学生只需要有最基本的群论知识就可学习李群和李代数，而不需要事先掌握较多的几何基础 , 同时本书对于李群和李代数的介绍又是相当完全的。 （周子翔）