Math 214: Differential Topology

UC Berkeley, Fall 2025

Instructor

Michael Hutchings
hutching@math.berkeley.edu.
Office: 923 Evans.
Tentative office hours: Wednesday 9:00-12:00.

Textbooks

The official textbook for the course is John Lee, Introduction to smooth manifolds, second edition. (The first edition presents the material in a different order and omits some key topics such as Sard's theorem.) Legal free download here from the UCB computer network.

The following are some other books which I recommend, in order of increasing difficulty. These are classics; I read all of them (except Munkres) when I was a student and really enjoyed them.

Munkres, Topology, second edition. Clearly and gently explains point set topology, if you need to review this. (However we won't be going into details of point set topology very much in the course.) Also gives a nice introduction to the fundamental group and the classification of surfaces. (Familiarity with the fundamental group is useful but we will not use this much.)
Milnor, Topology from the differentiable viewpoint. A beautiful little book which introduces some of the most important ideas of the subject.
Guillemin and Pollack, Differential topology. Explains the basics of smooth manifolds (defining them as subsets of Euclidean space instead of giving the abstract definition). More elementary than Lee's book, but gives nice explanations of transversality and differential forms (which we wil be covering).
Spivak, A comprehensive introduction to differential geometry, vol. I, 3rd edition. I considered using this as the text for this course. Volumes 2-5 are also good (but go beyond this course).
Bott and Tu, Differential forms in algebraic topology. As the title suggests, it introduces various topics in algebraic topology using differential forms. We will not be doing much algebraic topology in this class, but you might still enjoy looking at this book while we are discussing differential forms. Legal free download here from the UCB computer network.

Syllabus

The basic plan is to cover most of the material in chapters 1-19 of Lee's book (adding a few interesting things which are not in the book, and some bits from chapters 20 and 21). My goal is for you to understand the basic concepts listed below and to be able to work with them. This is all pretty essential material for graduate level geometry and topology. In class I will try to introduce the main ideas, explain where they come from, and demonstrate how to use them. I will tend to leave technical lemmas for you to read in Lee's book (or not).

Basic definitions: topological manifolds, smooth manifolds, smooth maps, diffeomorphisms. (Lee, chapters 1-2; we will discuss manifolds with boundary later.) A bit about classification of manifolds (not in the book).
Tangent vectors, tangent space, differential of a smooth map, tangent bundle. Calculations in coordinates. Inverse function theorem, implicit function theorem, and regular level sets. (Lee, chapter 3 and Appendix C)
Immersions, embeddings, and submanifolds. Isotopy, regular homotopy. Submersions. (Lee, chapters 4 and 5)
Transversality and Sard's theorem. (Lee, chapter 6. For more about this see Guillemin and Pollack.)
Introduction to Lie groups. (Lee, chapter 7 and a bit of chapter 21)
Vector fields and Lie bracket. The Lie algebra of a Lie group. (Lee, chapter 8)
Flows, integral curves, Lie derivative. Exponential map on a Lie group. (Lie, chapter 9 and a bit of chapter 20) Introduction to the Poincare-Hopf index theorem. (Guillemin and Pollack)
Vector bundles. (Lee chapter 10)
1-forms. Exterior derivative and wedge product of same. (Lee, chapter 11)
Riemannian metrics. (Lee, chapter 13)
Differential forms on manifolds: wedge product, pullback, exterior derivative, Lie derivative. (Lee, chapter 14)
Orientations. Integration of differential forms on oriented manifolds. (Lee chapters 15 and 16)
Stokes' theorem. The volume form on a Riemannian manifold and the divergence theorem. (Lee chapters 15 and 16; for alternate treatments of Stokes' theorem see Guillemin and Pollack or Bott and Tu.)
de Rham cohomology. Computation using the Mayer-Vietoris sequence. (Lee chapter 17) Degree of a smooth map and applications. (Milnor)
Introduction to the de Rham theorem. (Lee chapter 18)
Distributions and foliations. (Lee chapter 19)
Introduction to Morse theory, if time permits. (not in Lee's book)

Homework

Homework assignments will be posted here every week or two. The reader will give feedback on selected questions. A homework assignment will receive full credit if you make a reasonable effort and answer at least most of the questions. You are encouraged to collaborate on homework. One homework assignment can be missed without penalty.

HW#1, due 9/16 in class.
HW#2, due 9/30 at the beginning of class.
HW#3, due 10/14 at the beginning of class.
HW#4, due 10/25 at the beginning of class.
HW#5, due 11/18 at the beginning of class.
HW#6, due 12/12 at 9:40 AM.

Exams and grades

Course grades will be determined as follows:

Homework: 50%
Midterm: 25%
Final exam: 50%
Lowest exam score: -25%

Incomplete grades can be given only if both (1) an unanticipated event such as illness prevents you from completing the course, and (2) you are otherwise passing the course with a grade of C or above.

Electronic devices and AI

Electronic devices should not be used in class except as needed for learning the class material and when this does not distract other students. Appropriate uses of devices include taking lecture notes electronically, and looking up relevant mathematics. However if you have a question, it might be better to simply ask the question out loud, as other students may be wondering the same thing.

Use of ChatGPT and similar tools for graded work is not allowed. You can use these tools for studying, although I generally discourage this, as these AI tools currently make many errors (some blatant and some subtle), and when they answer questions correctly they often spare you from doing the work that you need to do in order to learn. There are also various ethical concerns with their use (e.g. they may be stealing human work or have an excessive impact on the environment). However AI will probably play some nontrivial role in mathematical work in the future. For now it can be fun (after you have learned the material) to test AI on math questions and see how it does.

DSP accommodations

Students requiring DSP accommodations should have a letter sent from the DSP office to the instructor, and should contact the instructor and/or reader to make any necessary arrangements.

Academic honesty

On homework, all collaborators and external sources must be explicitly acknowledged. Exams are expected to be taken within the time limits, and without aid from other people, books or notes, or the internet, unless explicitly allowed by the rules of the exam. The code of student conduct may be found here.

Lecture summaries

After each lecture, brief summaries will be posted here.

(Thursday 8/28) Definitions of topological manifolds and smooth manifolds (Lee chapter 1). Overview of classification results (not in Lee's book).
(Tuesday 9/2) Basic examples of smooth manifolds. Smooth maps and diffeomorphisms (Lee chapter 2).
(Thursday 9/4) Tangent vectors, tangent space, derivative of a smooth map (Lee chapter 3).
(Tuesday 9/9) No class (I have a medical procedure)
(Thursday 9/11) The tangent bundle; see Lee chapter 3. The inverse function theorem and local diffeomorphisms; see Lee chapter 4 and appendix C.
(Tuesday 9/16) Immersions, embeddings, submanifolds. Started on implicit function theorem. (See Lee chapters 4 and 5. What I call a submanifold, Lee calls an "embedded submanifold".)
(Thursday 9/18) Implicit function theorem. Some of the proof of the Whitney embedding theorem. See Lee chapter 6.
(Tuesday 9/23) Statement of Sard's theorem, proof in an easy case. The rest of the proof of the Whitney embedding theorem. Started discussing transversality. See Lee chapter 6. (Transversality is only briefly discussed in Lee and we will say more about it. Guillemin-Pollack discusses transversality extensively.)
(Thursday 9/25) Orientations; see Lee chapter 15. The Mobius band is not orientable. A compact hypersurface in R^n is orientable. The tangent bundle of any manifold (orientable or not) is orientable. Started to discuss the signed intersection number of oriented submanifolds intersecting transversely. (This is not in Lee but is discussed in Guillemn-Pollack chapter 3.)
(Tuesday 9/30) Vector fields (see Lee chapter 8). Introduction to the Poincare-Hopf index theorem (not in Lee, discussed in Guillemin-Pollack).
(Thursday 10/2) Proof that one can perturb a map to obtain transversality to a submanifold (see Lee Theorem 6.36). Introduction to manifolds with boundary (see Lee chapter 1).
(Tuesday 10/7) Sketch of the proof that intersection number is homotopy invariant. Introduction to Lie groups. (See Lee chapter 7.) Brief review of the fundamental group and the universal cover.
(Thursday 10/9) Lie group actions on manifolds. Commutator of vector fields. The Lie algebra of a Lie group. (See Lee chapter 8.)
(Tuesday 10/14) More about the Lie algebra of a Lie group. Pushforwards of vector fields and "f-related" vector fields. A Lie group homomorphism induces a Lie algebra homomorphism. (See Lee chapter 8.) Started discussing the flow of a vector field. (See Lee chapter 9.)
(Thursday 10/16) The exponential map on a Lie group. (See Lee chapter 20.) The Lie derivative of a vector field. The commutator of two vector fields is zero if and only if their flows commute. (See Lee chapter 9.)
(Tuesday 10/21) Midterm, in class. Will cover the material through 10/14.
(Thursday 10/23) Vector bundles (see Lee chapter 10). Started discussing 1-forms (see Lee chapter 11).
(Tuesday 10/28) 1-forms, fundamental theorem of line integrals, criteria for 1-forms to be exact.
(Thursday 10/30) 2-forms, d of a 1-form.
(Tuesday 11/4 and Thursday 11/6) k-forms. Wedge product, exterior derivative, pullback, integration. Statement of Stokes theorem. (See Lee chapters 14 and 16.)
(Tuesday 11/11) Academic and Administrative Holiday, no class
(Thursday 11/13) Proof of Stokes theorem. Application to prove the divergence theorem on a Riemannian manifold with boundary.
(Tuesday 11/18) Started discussing de Rham cohomology. Computed cohomology of R^n, and partially explained the top cohomology of an oriented manifold.
(Thursday 11/20) Top degree cohomology of a compact connected manifold (oriented or not). Degree of a map between compact connected oriented manifolds of the same dimension. Application to the Gauss-Bonnet theorem for surfaces in R^3.
(Tuesday 11/25) TBA
(Thursday 11/27) Academic and Administrative Holiday, no class
(Tuesday 12/2) TBA
(Thursday 12/4) TBA
(Tuesday 12/9) Reading/Review/Recitation week, optional review session
(Thursday 12/11) Reading/Review/Recitation week, optional review session
(Tuesday 12/16, 3-6pm, in the usual classroom, 215 Dwinelle) Final exam. Will cover the entire class, with somewhat more emphasis on the second half.