MAT 548 Geometry and Topology I, Fall 2026.

  • Course description: Together with MAT 549, this course forms the first year topology/geometry sequence for PhD students.

    The course will cover important basic notions in smooth manifolds and algebraic topology, and discuss relations between geometry and topology. Topics on the smooth manifolds side will include the basics of smooth manifolds and smooth maps, Sard's theorem and applications, submersions, immersions and embeddings, the notion of transversality. Topics on the algebraic topology side will include CW-complexes and cellular maps, the notion of homotopy, homotopy extension property, the fundamental group and its computations and applications, the notion of higher homotopy groups, coverings, fiber bundles and fibrations, path lifting and homotopy lifting, topological classification of 1- and 2-manifolds. We will try to emphasize the connections between the smooth tools and algebraic topology: given a smooth map, how can you tell if it defines a covering or a fiber bundle? What are the topological consequences? How can you define and compute the degree of a map in different ways?

    A more detailed list of topics will be posted with the week-by-week schedule as the course progresses. We may need to make adjustments to the scope of the course as time permits.

    Strong foundation in point-set topology will be assumed as a prerequisite, athough some finer point-set bits will be discussed as necessary (for example, to establish properties of CW-complexes). Analysis in R^n is another prerequisite (including the inverse and implicit function theorems). It will also be expected that the students have seen smooth manifolds and the fundamental group before, so this material is not entirely new, but we will develop the foundations in detail and in greater depths than more basic courses do.

  • Instructor: Olga Plamenevskaya, office 2-112 Math, e-mail: olga@math.stonybrook.edu

  • Office hours: Monday 2-3pm; Tuesday 11:00-1pm.

  • Grader: Artem Aleshin. Artem will be happy to help with homework or other course-related questions if you come to his office hours.

  • Class meetings: TR 9:30am-10:50pm, Physics P-130.

  • Course webpage: http://www.math.stonybrook.edu/~olga/mat548-fall26/.

    All course information will be posted on the course webpage.

  • Homework: weekly homework will be posted on this page.

    Homework submission: you can drop off paper homework at Artem's office (Math 2-109), anytime before the deadline, or submit the homework online through Gradescope.

    Gradescope instructions:

    1. Login to the Gradescope via your Stony Brook credentials.
    2. Click “Enroll in Course” and enter the course code 5NV6JX.
    3. Find the corresponding homework assignment and upload your PDF.
    4. If you have any problems logging in, email the grader (see above for email)

    Week 1 (Tuesday 8/25, Thursday 8/27): Chapters 1 and 2 in Lee's book. Definitions and basic examples and properties of smooth manifolds and smooth maps. (We didn't talk about partitions of unity yet, but we will in Week 3.)

    Homework 1 due Friday, Sept 4, at 5pm. Homework can be dropped off at Artem's office (Math 2-109) or submitted online (see instructions above).

    The homework in this course is intended to be non-trivial. It is a good idea to talk to other students and/or come to the office hours (mine or Artem's).

    Week 2 (Tuesday 9/1, Thursday 9/3): Chapter 3, Lee's book (except the tangent bundle, which will be discussed next week). We followed Lee pretty closely but added some motivation/discussion. Some missing details (esp. tangent space to manifold with boundary) will be covered next week.

    We had a diagnostic point-set topology (+ questions on basic multivariable calc/analysis in R^n) on 9/1. The quiz does *not* count toward your grade but shows if you are ready for MAT 548 and helps identify your weaker areas that you should review.

    Homework 2 due Friday, Sept 11, at 5pm.

    Week 3 (Tuesday 9/8, Thursday 9/10): The remaining technical bits from Chapters 2 and 3 (partition of unity, more details on manifolds with boundary). The tangent bundle and the notion of a vector bundle in general. Orientable vector bundles, orientation of a smooth manifold (this is not in Chapter 3). Then move on to Chapters 4 and 5: local diffomorphisms, submersions, immersions, examples and properties. We proved the Rank Theorem for immersions only (submersions are similar but a bit more technical; Lee gives the proof in the general case). We defined smooth submanifolds and proved Theorem 5.8. (We work with embedded submanifolds only; we won't need immersed submanifolds. We only worked with manifolds without boundary for Ch 4 and 5 material so far.)

    Homework 3 due Friday, Sept 18, at 5pm.

    Week 4 (Tuesday 9/15, Thursday 9/17): (With big thanks to Artem for the Tuesday lecture.) We finished discussing the key material from Chapters 4 and 5. These chapters contain too many details, so we tried to streamline and focus on the most important parts. In particular, "submanifold" means "embedded submanifold" for us (not "immersed", even if sometimes we'll be interested in images of smooth immersions). We mostly only considered manifolds without boundary (although boundary will sometimes show up in the homework and class discussion later). We started Chapter 6 (sets of measure 0, moving towards Sard's theorem).

    Here's a summary of what you need to know in Chapters 4-5:
    -- The rank theorem (know the statement, have some idea of what the proof involves)
    -- Definitions of smooth submersion, immersion, embedding, local diffeomorphism; basic properties such as Thm 4.26, Prop 4.28
    -- Embeddings require a topological condition so you need to be careful (examples 4.19, 4.20). However in nice situations you can get away with an injective smooth immersion (Prop. 4.22, especially parts (b)(c)).
    -- Two ways to define a smooth submanifold: (a) via slice charts: if you have slice charts on a subset S, that shows that S is a topological manifold and allows to put a smooth structure on S such that the inclusion map is a smooth embedding; (b) if a subset S is a topological manifold and the inclusion is a smooth embedding, then this gives an equivalent definition (existence of slice charts follows from the rank theorem). The second definition applies in exactly the same way if the ambient manifold is allowed to have a boundary (a submanifold with boundary requires more care, but we didn't discuss that).
    -- The tangent space T_p (S) to a submanifold S of M can be identified with a subspace of T_p (M) via equivalence classes of curves or via derivations, and has a nice form in slice charts
    -- The preimage of a regular value of a smooth map F: M → N is a smooth submanifold of M. This follows from the rank theorem. The tangent space T_p(S) of this submanifold S can be identified with the kernel of d_p F in T_p(M). The codimension of S in M equals the dimension of N.
    -- Restricting smooth maps to submanifolds (Thm 5.27, Thm 5.29/Cor 5.30): these facts often help establish smoothness in basic examples and are quicker to use than working in charts
    -- Notions of an orientable vector bundle and an orientable manifold. (This is a bit of material from Chapters 10 and 15.) By definition, M is orientable if its tangent bundle TM is orientable (as a vector bundle over M). The preimage of a regular value of a smooth function F: R^n → R is orientable.

    Homework 4 due Friday, Sept 25, at 5pm.

    Week 5 (Tuesday 9/22, Thursday 9/24) Chapter 6: Sard's theorem. Whitney's embedding theorem (we focused on the compact case for the proof, although the theorem is true for non-compact manifolds as well, see Lee for details). Transversality (end of Chapter 6 + read about transversality in Guillemin--Pollack for pictures and intuition). One key point here is Thm 6.30, it generalizes the Regular Preimage theorem (Cor. 5.14) and gives a very useful tool to produce smooth manifolds in many situations. The second key point is based on Sard's theorem: transversality is "generic" and can be achieved by small perturbations. Thm 6.35 and its corollaries makes this idea precise. We'll prove Thm 6.35 and finish the discussion next week.

    Homework 5 due Friday, Oct 2, at 5pm.

    Week 6 (Tuesday 9/29, Thursday 10/1) Lee, Chapter 6, Guillemin--Pollack, Chapter 2, sections 1-3. We talked about manifolds with boundary in more detail and generalized the transverse preimage theorem to the case when the domain has non-empty boundary. As an application of Sard's theorem in this context, proved that a compact smooth manifold with nonempty boundary does not admit a (smooth) retraction onto its boundary. This material is in Guillemin--Pollack. We discussed Parametric Transversality and some examples. Will continue this topic with more applications next week. Parametric transverality (constructing a family of maps where the total map is transverse to a given submanifold) is easy when the target manifold is the Euclidean space. To deal with the general case, we embedded the target manifold M into R^n and sketched the proof that M admits a "tubular neighborhood" that is diffeomorphic to a neighborhood of the 0-section of the normal bundle. (In the compact case that we focused on, you can take an epsilon-neighborhood for a sufficiently small epsilon.) Parametric transversality and normal bundles/tubular neighborhoods are in Chapter 6 of Lee as well as Guillemin-Pollack. In my opinion, Guillemin--Pollack's exposition is easier to read (but remember that they only work with smooth manifolds that are submanifolds in R^n, so you need to adapt some arguments slightly to our definitions/setup).

    Homework 6 due Friday, Oct 9, at 5pm.

    The midterm exam will be on Thursday, Oct 15, 9:00am-10:50am, in our usual classroom.

    What's on the test: all of the key points of Chapters 1-6 in Lee (with the exception of the bits that we didn't cover), see the week-by-week description of topics above. This includes the basics of smooth manifolds, smooth maps, tangent bundle (different equivalent definitions, properties); diffeomorphisms, immersions, submersions, embeddings, local diffeomorphisms (basic properties of each class of maps, being able to detect them, applications of the rank theorem); smooth submanifolds, definitions and properties (all submanifolds are assumed embedded, we never work with immersed submanifolds); regular preimage theorem; partition of unity, bump functions, and being able to use them to create smooth functions on manifolds; manifolds with boundary, definition and their basic properties; transversality: basic definitions, visual intuition, working with maps and tangent spaces; transverse preimage theorem (including the case where the domain has non-empty boundary); Sard's theorem and applications, including parametric transversality and using it in examples; Whitney's embedding theorem.

    Many of the facts we discussed in class are basic corollaries of the definitions. You should be able to reproduce proofs of these basic results. You should also know all definitions and important statements and properties and theorems, and be able to use them.

    What's not on the test: vector bundles and orientations, since this is more advanced material that we are just beginning to learn. You do not need to know the more technical proofs (Sard's theorem, the details of rank theorem, proof of partition of unity in the non-compact case, etc) but you need to know the statements. I won't give you tricky questions about manifolds with boundary (but their basic properties are required for the test). This week's material (normal bundle, tubular neighborhoods, smooth approximations, degree of a map) is not on the test.

    How to prepare: First, make sure that you understand all the basic notions well. We discussed a number of examples and basic facts, and there are many basic facts that we mentioned quickly but didn't prove, or didn't even mention. Ask yourself some questions about some basics: why is X x {y} a smooth submanifold in X x Y? Why is T(X x Y)= TX x TY? Why is the quotient map S^2 to RP^2 a smooth submersion? How can we prove that a rotation of the sphere is a smooth map (and a diffeomorphism)? You should be able to ask a lot of similar questions and answer them easily. When you are comfortable with this, try some harder questions. Go through the past homeworks. For extra practice, Lee and Guillemin-Pollack have plenty of exercises you can try. Work on transversality and using Sard's theorem, these topics will definitely be on the test.

    The level of the exam questions will be similar to the easier homework questions, and there may be short quiz-type questions on definitions and basic concepts.

    Week 7 (Tuesday 9/29, Thursday 10/1) Lee, Chapter 6 (Smooth Approximation of Continuous Maps), Guillemin--Pollack Ch 2 Section 4 (Degree of a Map). Guillemin--Pollack discuss more general intersection theory but we only focused on the notion of degree (repeat the proof of Theorem in section 4 to prove homotopy invariance of degrees). We used smooth approximation (Lee Ch 6), applied to continuous maps and homotopies, to define degree for continuous maps and prove homotopy invariance. Proved Boundary Theorem for degree (if X = dW and F: X -> Y extends to W, then deg F = 0). Note that Guillemin--Pollack only work with degree mod 2; we defined the integer-valued degree for the oriented case by counting preimages of a regular value with sign. Only sketched the proof of homotopy invariance for the integer-valued degree (it follows the same strategy of examining the boundary of a 1-dimensional manifold, but one needs to carefully sort out all the relevant orientations). We won't have time to go through this in class, but Guillemin--Pollack Chapter 3 is accessible at this point, and works out all the details of oriented intersection theory. It would be a very useful (optional) read if you feel motivated.

    Week 8 (Fall Break, no class Tuesday 10/13; Thursday 10/15) Midterm was on 10/15.

    Homework 7 due Friday, Oct 23, at 5pm.

    Late homework policy: late homework is accepted until midnight on Friday, and we will allow up to 3 late homework submissions for each student during the semester.

    If you notice a typo or something looks weird, please let me know. I do make mistakes sometimes (often). Any questions, feel free to email and ask!

    Important: For each homework problem, please give a proof or detailed explanation as appropriate (unless otherwise stated). Please write up your solutions neatly, be sure to put your name on the first page and staple all pages. Illegible homework will not be graded. Although you are welcome to work with others to understand how to solve the problems, you have to write all the solutions on your own, in your own words. You should not look for solutions online or use AI to produce solutions. (See Academic Integrity Statement and AI Policy at the bottom of the page.)

  • Textbooks:

  • Exams: there will be a midterm (10/15, in class) and a final exam (December 11, 11:15 - 1:45 pm).

  • Grading Policy: The course grade will be determined by homeworks and class participation, one midterm, and the final exam.

    No make-up exams will be given for the midterm. If a student misses a midterm exam for a well-documented medical reason or other similar circumstances beyond the student's control, the student may be excused from the exam, with the final grade determined from the other exams, homework, and class participation. For the final exam, make-ups will be given ONLY in cases of properly documented medical reasons or other similar circumstances, at the instructor's discretion.

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