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I am studying differential geometry and topology by myself. Not being a math major person and do not have rigorous background in analysis, manifolds, etc. I have background in intermediate linear algebra and multivariate calculus. To embark on the study, I delved into stackexchange past answers and other websites.

From these questions and their answers, I found that Milnor's Topology from a Differentiable Viewpoint, Lee's Introduction to Smooth Manifolds, Tu's An Introduction to Manifolds should work for self-study. I am not looking for a theorem and proof style book, but rather getting concepts such as topology, manifold, Lie groups, moving frames, etc.

When I start reading even the introductory chapters from books, I find that many books simply assume that the reader would already know concepts as homomorphism, isomorphism, wedge product, cotangent space, etc. This assumption is not true for many readers (like I). As a result, it is not possible to move ahead without knowing these stuff.

I further found that there is a large amount of literature devoted to these topics. I found, a branch of mathematics, abstract algebra, deals with homomorphism and other listed topics. Learning everything is a daunted task, in fact, only some portion might be needed for my purpose.

Differential geometry and topology have diverse applications and many people, who are from in different areas of sciences and who are not pure mathematicians, may need to learn these areas. Can someone suggest a 'self contained' introductory book that will sufficiently cover the subject-matter? If such book is not there, can someone mention references that will (quickly and with sufficient depth) cover the assumed prerequisites for learning topology and differential geometry (homomorphism, isomorphism, wedge product, cotangent space, etc.)? So that one does not have to entirely learn abstract algebra, which looks hard method.

Inputs are very much appreciated!

Edit: I believe that this is not a "personal advice" question as the links provided in the question are still valid questions and they belong to category "reference request."

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  • $\begingroup$ It is something along the lines of "impossible" to predict every mathematical notion that will show up when you're studying differential geometry. It would probably be wise to learn the content of an undergraduate course on abstract algebra and linear algebra, and then learn everything else that shows up piecemeal, as you come along the notions. Artin's "Algebra" is probably a good source. $\endgroup$ – user98602 Jan 2 '16 at 2:09
  • $\begingroup$ Edited the question $\endgroup$ – gyeox29ns Jan 2 '16 at 14:07
  • $\begingroup$ Differential Geometry - do Carmo, and Topological Manifolds - John M. Lee; both are expositary with selected exercises. $\endgroup$ – Beginner Jan 2 '16 at 14:09
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http://www.topologywithouttears.net/

This website and its contents should be useful.

If you want to learn some basic algebra, but nothing too in depth take a look at Fraleigh's Abstract Algebra.

For linear algebra , Axler's "Linear Algebra Done Right" is a good introduction.

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  • $\begingroup$ mentioned that already in the question - last list point. The book is good but does not cover everything what I am looking for. $\endgroup$ – gyeox29ns Jan 2 '16 at 2:06
  • $\begingroup$ How in depth of an introduction do you need? It might help to explain your job and what you would need topology for in the question. $\endgroup$ – Brandon Thomas Van Over Jan 2 '16 at 2:14
  • $\begingroup$ As far as prerequisites go, enough depth would enable a reader to read the main text on differential geometry and manifolds without any problems. $\endgroup$ – gyeox29ns Jan 2 '16 at 14:14

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