Lecture Notes on Elementary Topology and Geometry (Undergraduate Texts in Mathematics) by I.M. Singer (PDF)

Description

At the present time, the average undergraduate mathematics major finds mathematics heavily compartmentalized. After the calculus, he takes a course in analysis and a course in algebra. Depending upon his interests (or those of his department), he takes courses in special topics. Ifhe is exposed to topology, it is usually straightforward point set topology; if he is exposed to geom­ etry, it is usually classical differential geometry. The exciting revelations that there is some unity in mathematics, that fields overlap, that techniques of one field have applications in another, are denied the undergraduate. He must wait until he is well into graduate work to see interconnections, presumably because earlier he doesn’t know enough. These notes are an attempt to break up this compartmentalization, at least in topology-geometry. What the student has learned in algebra and advanced calculus are used to prove some fairly deep results relating geometry, topol­ ogy, and group theory. (De Rham’s theorem, the Gauss-Bonnet theorem for surfaces, the functorial relation of fundamental group to covering space, and surfaces of constant curvature as homogeneous spaces are the most note­ worthy examples.) In the first two chapters the bare essentials of elementary point set topology are set forth with some hint ofthe subject’s application to functional analysis.

User’s Reviews

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐Although ostensibly for undergraduates, this monograph text requires considerable ability in abstract thinking. A well developed capability and sophistication with certain prerequisite math skills is useful. The text does, however, accomplish its claim, namely providing a shortcut pathway toward some of the more fun stuff in advanced mathematics, like deRham cohomology theory, without requiring a long chain of prerequisite courses, usually amounting to an undergraduate major in mathematics. For students who have a natural affinity for advanced mathematics, this approach could provide additional motivation to pursue those interests. For non-math majors with a particular interest in cohomology theory, this shortcut works amazingly well.

⭐As a reference, in particular first two chapters, this book is believed to be one of the best introductory textbook on the rigorous set and topology; indeed it seems to be the case.Since I am still reading and taking notes on this topic, I am not able to write any review on the contents of this book, but as far as I’ve read, their style are clear and follow the standard notions (a few remarks must be stated but they are so far epsilon).Hopefully within a reasonable time scale, I would like add more to say here.

⭐Libro clasificado como usado como nuevo y sin marcas. Notablemente deteriorado y con paginas llenas de marcas y notas al dorso.

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⭐good

⭐I believe that one of the authors, I. Singer is a Fields Medalist, so this is co-written by one of the masters in the subject. It is one of the few books I’m aware of that covers point-set, algebraic, and differential topology. However, this is not in any way an exhaustive text. It is very spare and to the point. This is a small book packed with information. No exercises are included. It is not particularly advance in any one area of topology, so keep in mind that this is just an overview of all the main areas. However, just because it is not particularly advanced, does not mean it is easy to read for someone completely unfamiliar with the material. I tried reading the early chapters as a first introduction to topology, and didn’t know what was going on. Reading of it got better when I read it side-by-side with the books by Munkres and Armstrong. This book is pretty abstract and dense, so it will read slowly. A lot of the finer details in the subjects are found in other texts.

⭐学生時代に読み始めて、やっと最近読み終わりました。日本語訳も出ていたんですね、今は品切れのようですが。基本の集合論からガウスボンネの定理まで、入門書と言うにはかなり難しいですが、やっぱり数学の中ではトポロジーが一番面白いと思います。連続であるが一様連続ではない、とか、連結であるが道連結ではない、とか、最初のうちは楽しいことばかりで、平和って素晴らしい、なんて冗談を言っていられるんですが、基本群、被覆空間、単体複体、多様体、と進むうちに、少しずつ消化しきれないものが積み重なって行き、いつの間にか迷路にはまって自分がどこにいるのかわからなくなってしまう、トポロジーなんてもう無理！だったのですが、ある日たまたま、根上生也「トポロジカル宇宙完全版―ポアンカレ予想解決への道」を図書館で見つけて、もう一度迷路にはまってみたくなりました。同じく入門書の入門書としては結城浩「数学ガール/ポアンカレ予想」もおすすめです。ウィキペディアで調べたら、著者のI.M.Singerは2004年のアーベル賞受賞者でした。日本語訳のKindle版を出してほしいと思います。

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⭐トポロジーと幾何学にまたがる領域にまたがるテーマを丁寧に扱っていて、気に入っています。

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