Introduction to Smooth Manifolds

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1. tamndNov 2025
   I first learned about manifolds through Introduction to Smooth Manifolds by John M. Lee. The book is dense but beautifully structured, guiding you from basic topology to smooth maps and tangent spaces with clear logic. It demands focus, yet every definition builds toward a deeper picture of how geometry works beneath the surface. Highly recommended.
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2. tamndNov 2025
   That's a great suggestion. I actually started with Topological Manifolds before moving on to Introduction to Smooth Manifolds and it really helped build a solid foundation.

   I havent read Loring Tus books before but let me look at them since I have been wanting to revisit the topic with a clearer and more relaxed approach.

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3. kdavisJan 2023
   * 4-Manifolds and Kirby Calculus by Andras I. Stipsicz and Robert E. Gompf

   * Differential Manifolds by Antoni A. Kosinski

   * Introduction to Smooth Manifolds by John M. Lee

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4. mirceamApr 2019
   Not a mathematician, but have spent some time self-learning math. I have not gone through the following books entirely, however I've used them when needed to learn or clarify a concept:

   > Group Theory - I would look for a good abstract algebra book that also covers groups, such as A First Course in Abstract Algebra, Fraleigh or Basic Algebra, Knapp. Another popular recommendation is Abstract Algebra, Dummit and Foote.

   > Topology I think you should first look at a real analysis course and understand metric space topology for Euclidean spaces. If you have taken calculus, you can try and go through Baby Rudin. For topology specifically you can look into Munkres or Introduction to Topological Manifolds, J Lee.

   > Measure Theory T. Tao has a good book on this topic.

   > Differential Geometry Introduction to Smooth Manifolds by J Lee

   > Probability Probability Theory, Achim Klenke

   > Game Theory Game Theory by Maschler, Solan & Zamir

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5. ajudsonMar 2019
   +1 on munkres. Also recommend Algebraic Topology by Hatcher as a followup (free on his website) and Introduction to Smooth Manifolds by Lee.
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6. MrBraJan 2015
   "An overview of Inter-universal Teichmüller Theory and [...] how we can begin to understand this theory":

   http://www.quora.com/Joseph-Heavner/Posts/An-overview-of-Int...

   Also, from an anonymous post on 4chan @ http://boards.4chan.org/sci/thread/6931488/are-you-ready-for... :

   "How to understand IUTeich from the bottom up:

   - Algebra (Israel Gelfand)

   - The Method of Coordinates (Israel Gelfand)

   - How to Prove It: A Structured Approach (Daniel Velleman)

   - Kiselev's Geometry - Planimetry and Stereometry

   - Trigonometry (Israel Gelfand)

   - What Is Mathematics? An Elementary Approach to Ideas and Methods (Richard Courant)

   - A Course of Pure Mathematics (G.H. Hardy)

   - Linear Algebra (Kenneth Hoffman and Ray Kunze)

   - Elementary Differential Equations (William Boyce and Richard DiPrima)

   - Topology (James Munkres)

   - Calculus On Manifolds (Michael Spivak)

   - Principles of Mathematical Analysis (Walter Rudin)

   - Real and Complex Analysis (Walter Rudin)

   - Functional Analysis (Walter Rudin)

   - Partial Differential Equations (Lawrence Evans)

   - Analysis On Manifolds (James Munkres)

   - Abstract Algebra (David Dummit and Richard Foote)

   - Algebraic Topology (Allen Hatcher)

   - Introduction to Smooth Manifolds (John Lee)

   - Foundations of Differentiable Manifolds and Lie Groups - (Frank Warner)

   - Galois Theory (Harold Edwards)

   - Linear Representations of Finite Groups (Jean-Pierre Serre and Leonhard Scott)

   - A Classical Introduction to Modern Number Theory (Kenneth Ireland and Michael Rosen)

   - Introduction to Analytic Number Theory (Tom Apostol)

   - Modular Functions and Dirichlet Series in Number Theory (Tom Apostol)

   - Riemannian geometry (Peter Petersen)

   - The Theory of the Riemann Zeta-Function (Edward Charles Titchmarsh)

   - An Introduction to Teichmüller Spaces (Yoichi Imayoshi and Masahiko Taniguchi)

   - A Course in p-adic Analysis (Alain Robert)

   - Foundations of p-Adic Teichmuller Theory (Shinichi Mochizuki)

   - Mochizuki's papers at http://www.kurims.kyoto-u.ac.jp/~motizuki/papers-english.htm... "

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