Introduction to Topological Manifolds by John Lee (prereqs)

• Topology
• FallenApple
In summary: It took me about 4 months to read and work through the problems in the book. The material is definitely helpful for understanding physics that uses manifolds, as the book is very well-written and provides a lot of motivation for the theory.
FallenApple
I'm interested in this subject. This is a graduate text and I believe the prereqs are mostly a math degree, which I somewhat have(B.S in Applied Math from a few years back). The thing is, I forgot details about things. For example, I know how to do an epsilon delta proof and can read one when presented, but I don't remember the particular theorems/definitions( uniform convergence, ratiotests etc), I remember the philosophy behind a vector space/subspace of linear algebra, but I don't remember all the details of factorizations of matricies, all the theorems etc.

Is it possible to study this book while just looking up the needed details on the side as I go along?

Yes, it is definitely possible. I read much of this book and worked through a considerable portion of the problems before finishing my physics degree. It is pretty self contained, it even has an appendix with some prereq. results from set theory, metric spaces and group theory. The presentation of the material in the book is also really very good, it provides a lot of motivation for the theory.

FallenApple
FallenApple said:
I'm interested in this subject. This is a graduate text and I believe the prereqs are mostly a math degree, which I somewhat have(B.S in Applied Math from a few years back). The thing is, I forgot details about things. For example, I know how to do an epsilon delta proof and can read one when presented, but I don't remember the particular theorems/definitions( uniform convergence, ratiotests etc), I remember the philosophy behind a vector space/subspace of linear algebra, but I don't remember all the details of factorizations of matricies, all the theorems etc.

Is it possible to study this book while just looking up the needed details on the side as I go along?

Don't worry, you're fine. The book is extremely good, and the best thing is that it requires very very little prereqs. I have guided multiple people through this excellent book, most of which never had any formal proof-based math courses. Everything you need is detailed in the appendix which you should read first and should be pretty well-known to you. Don't make a mistake though, the book is not easy, it just has very little prereqs.

FallenApple
micromass said:
Don't worry, you're fine. The book is extremely good, and the best thing is that it requires very very little prereqs. I have guided multiple people through this excellent book, most of which never had any formal proof-based math courses. Everything you need is detailed in the appendix which you should read first and should be pretty well-known to you. Don't make a mistake though, the book is not easy, it just has very little prereqs.

Thanks. Do you have any suggestions on how I should pace myself? This is mostly for casual self study.

I don't want to put so much time that wouldn't be able to work/study other subjects. But at the same time, putting in very little daily time really wouldn't make it stick.

How hard is this compared to say, real analysis on the level of Ross?

Cruz Martinez said:
Yes, it is definitely possible. I read much of this book and worked through a considerable portion of the problems before finishing my physics degree. It is pretty self contained, it even has an appendix with some prereq. results from set theory, metric spaces and group theory. The presentation of the material in the book is also really very good, it provides a lot of motivation for the theory.

Thanks for the input. How much time did take you to get through the book? Was it helpful for understanding physics that uses manifolds?

1. What are the prerequisites for studying "Introduction to Topological Manifolds" by John Lee?

The prerequisites for studying "Introduction to Topological Manifolds" are a solid understanding of undergraduate-level topology, including concepts such as continuity, compactness, and connectedness. It is also helpful to have some familiarity with basic algebraic structures such as groups, rings, and fields.

2. Do I need to have prior knowledge of differential geometry to understand this book?

No, you do not need to have prior knowledge of differential geometry to understand "Introduction to Topological Manifolds". However, some familiarity with basic concepts of multivariable calculus, such as partial derivatives and the chain rule, may be helpful.

3. Is this book suitable for self-study, or is it better to use it alongside a course?

This book can be used for self-study, as it includes detailed explanations and examples. However, it may be beneficial to use it alongside a course or with the guidance of a teacher or tutor to fully grasp the material.

4. Are there any additional resources that can supplement my understanding of this book?

Yes, John Lee has also written a companion text titled "Introduction to Smooth Manifolds" that covers the more advanced topic of smooth manifolds. This can be a helpful resource for further understanding the concepts in "Introduction to Topological Manifolds". Additionally, there are many online resources and lecture notes available that can provide additional explanations and examples.

5. What kind of problems and exercises can I expect to find in this book?

The book includes a variety of problems and exercises, including both theoretical and computational problems. Some exercises are designed to test your understanding of the material, while others encourage you to think creatively and explore the concepts further. Solutions to selected exercises are provided in the back of the book.

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