Textbooks for learning: Lie algebra, topology, and differential geometry?

Question

I'm a PhD student trying to learn the math required to understand Gauge theory. From some reading over the past few weeks, I think I need to learn the following:

• Lie algebra / Lie groups
• Algebraic and geometric Topology
• Differential geometry

I only need understand the material enough to be literate (my research hopes only to use some results from these fields and not to contribute to them). Does anyone have any good recommendations for textbooks that assume little mathematical knowledge beyond some undergrad. courses in linear algebra, vector calc., probability, and PDEs?

In particular, I'm hoping to find a textbook that assumes little prior knowledge while being somewhat brief its exposition (in my reading so far I'm finding myself getting lost in the volume of definitions).

Thanks for reading, I appreciate any suggestions!

edit: hardmath asked where I feel I've been getting lost in definitions. I've started reading "A Course in Modern Mathematical Physics" by Peter Szekeres (suggested by a friend). This seems like a great book and I've learnt a lot in the first few chapters but I'm struggling with the amount of terminology. For example, by chapter 3 my notes contained about 70 definitions.

Answer 1

Here are my picks for best intro books:

Lie Algebras: https://www.amazon.com/Introduction-Algebras-Springer-Undergraduate-Mathematics/dp/1846280400

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Differential Geomtry: https://www.amazon.com/First-Steps-Differential-Geometry-Undergraduate/dp/146147731X

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Geometric Topology:

https://www.amazon.com/Introduction-Theory-Translations-Mathematical-Monographs/dp/0821810227

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Algebraic Topology:

Fundamental group / covering spaces: https://www.amazon.com/Introduction-Topological-Manifolds-Graduate-Mathematics/dp/1441979395

Homology/Cohomlogy: https://www.amazon.com/Elements-Algebraic-Topology-James-Munkres/dp/0201627280/ref=sr_1_2?dchild=1&keywords=munkres+algebraic+topology&qid=1607026571&s=books&sr=1-2

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Answer 2

As I suggested in comments, "An Introduction to Manifolds" by Loring Tu is a good place to start. Points in favour:

1. From the preface the author wrote it with the rationale of providing "a readable but rigorous introduction that gets the reader quickly up to speed, to the point where for example he or she can compute the de Rham cohomology of simple spaces." He goes on to explain that there is a "self-imposed absence of point-set topology in the prerequisites... [His] solution is to make the first four sections of the book independent of point-set topology and to place the necessary point-set topology in an appendix." The intention is that the reader study the first four sections and Appendix A simultaneously.
2. The book is the first volume of a series, so if you need more than what is in this book then you might find it in another book by the same author. It is the prequel to the famous "Differential Forms in Algebraic Topology" by Bott and Tu, and there is also "Differential Geometry" by Tu (ostensibly the third book but written with only the contents of the "Introduction" as prerequisites).
3. It's a Springer Universitext book, which at the time of writing makes it pretty cheap as far as textbooks go.

Point against: a perhaps-more-standard set of references for similar material is the series of books by John M Lee. Lee's books make better references in part because they are written to be more encyclopaedic, which might mean that Tu's books are slightly more approachable to someone learning on their own.