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I'm teaching a graduate (Master's) introduction to geometry and topology (e.g. some basics on manifolds, vector bundles, algebraic topology). What textbooks have you found are best for teaching a course like that?

I'm less worried about the specific list of topics but just wanted to know if people found a particular textbook to be particular good at explaining these concepts for the first time to beginning graduate students - and/or particularly good for structuring a course from.

(But if you wish, some possible topics are: Topological manifolds. The fundamental group and covering spaces. Applications. Singular homology and applications. Smooth manifolds. Differential forms and Stokes’ theorem, definition of de-Rham cohomology.)

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  • $\begingroup$ Tristan Needham's book is excellent (in my opinion) but (a) it may not cover what you want, and (b) I haven't taught from it. Link here. $\endgroup$ Sep 20, 2022 at 14:59
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    $\begingroup$ That list of topics, probably leads you to use Geometry of Differential Forms (Translations of Mathematical Monographs, Vol. 201) by Shigeyuki Morita. I haven't taught from it. For undergrads and beginning grad students John Lee and Jeffree Lee's books are essential. Those books have everything that other books leave out. If your students seek details, get the Lee books. $\endgroup$ Sep 20, 2022 at 23:05

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Disclaimer: I haven't taught the kind of course you describe, so please take my recommendations below with a grain of salt. Nevertheless, I hope they're helpful.

John M. Lee's Introduction to Topological Manifolds is a great place to start for the basics of topology, topological manifolds, the fundamental group, covering spaces and just a touch of homology. It's written at an introductory graduate level and the author has taken pains to write clearly and in detail. For smooth manifolds, you can follow up with his Introduction to Smooth Manifolds. However, if the latter book is too long and detailed for your course, you could switch to Loring W. Tu's An Introduction to Manifolds, which is also well written.

If your students have already had some general topology, you could either skip or go lightly through the early chapters of Introduction to Topological Manifolds, giving you more time for smooth manifolds later.

A book that I'm less familiar with, but looks promising, is the recent A Short Course in Differential Topology by Bjørn I. Dundas. It introduces smooth manifolds, the tangent space, regular values, vector bundles, flows and a few other topics, with the assumed background on general topology given in an appendix. Admittedly, it doesn't treat several of the topics in your list, but you said you're less worried about the specific topics. The author has a PDF version available at https://folk.uib.no/nmabd/dt/dtcurrent.pdf. The PDF isn't identical to the published version but it's close enough to give you a good impression. Note that despite the term "differential topology" in the title, it covers much of the smooth manifolds basics you'd find in the early chapters of many books of manifolds, be they heading toward analysis on manifolds, differential geometry or differential topology.

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