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I want to learn about smooth manifolds, I have never studied them before, but I have a good background in Algebra. Can any one recomend some good introductory books?

Thanks

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Check out Introduction to Smooth Manifolds by John Lee, Differentiable Manifolds and Riemannian Geometry by William Boothby and Differentiable Manifolds by Lawrence Conlon.

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    Lee's book seems very good.2012-11-06
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    Thanks for the compliments. I just want to point out that the PDF that @amWhy linked to is a very early (and very buggy!) draft of the first edition of my _Introduction to Smooth Manifolds_, which somehow managed to escape into the wild and has now been posted on a number of sites in violation of copyright. Download it if you want, but _caveat emptor_. The published first edition is more reliable, and the second edition is much better. Just so you know.2012-11-07
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    @JackLee Thanks for your caveat! Good to know! I will, at once, delete my comment with the link - impressive as it was, even if buggy, as you say. I did not know that there was a violation of the copyright!2012-11-07
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    Here's a link to [Lee's book](http://books.google.com/books/about/Introduction_to_Smooth_Manifolds.html?id=eqfgZtjQceYC), for anyone interested...2012-11-07
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    [Springer link](http://link.springer.com/book/10.1007/978-1-4419-9982-5/page/1). Perhaps useful to someone2012-12-14