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.

All course information will be posted on the course webpage.

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.)

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.

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.)

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.

Please review the inverse and implicit function theorems, we will be using them!

Lee, Introduction to smooth manifolds (GTM 218), 2nd edition. We'll cover chapters 1-6 and small bits of Chapters 8, 9, and 10.

Guillemin, Pollack, Differential Topology. This book makes some simplifying assumptions and is not as rigorous or detailed as Lee, but it is very good at explaining key ideas and providing intuition and pictures and good examples.

Fomenko, Fuchs, Homotopical Topology (GTM 273). We will cover part of Chapter 1 only. This book has the advantage of discussing some important homotopy theory early on.

Hatcher, Algebraic Topology. We will cover the material of Chapters 0, 1, and a part of Chapter 4. This book is available online for free download.

Milnor, Topology from Differentiable Viewpoint. We won't be following this text in class, but this is a classic, everyone should read it. You'll be glad you did, it's a very illuminating read. The book makes some simplyfing assumptions from the start (defining smooth manifolds as submanifolds in R^n) but gets to important theorems very quickly. It covers some theorems that we won't get to.

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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AI Policy Use of AI tools is not allowed to obtain solutions of homework problems, or for any help on any tests. All work you submit should be your own, without any AI assistance. If you use AI as a study tool to learn any course-related material (for example, to clarify a point from a lecture or a textbook), remember that any mistakes or errors are reflected in your grade, even if they were introduced by AI tools. The final content of your work is your responsibility. Use of AI must be disclosed in detail if it helped you learn any relevant information (please state what you asked AI and what you learned).

Academic Integrity Statement Each student must pursue his or her academic goals honestly and be personally accountable for all submitted work. Representing another person's work as your own is always wrong. Faculty is required to report any suspected instances of academic dishonesty to the Academic Judiciary. For more comprehensive information on academic integrity, including categories of academic dishonesty, please refer to the academic judiciary website at http://www.stonybrook.edu/commcms/academic_integrity/index.html.

Critical Incident Management Stony Brook University expects students to respect the rights, privileges, and property of other people. Faculty are required to report to the Office of Student Conduct and Community Standards any disruptive behavior that interrupts their ability to teach, compromises the safety of the learning environment, or inhibits students' ability to learn.

Student Accessibility Support Center Statement If you have a physical, psychological, medical, or learning disability that may impact your course work, please contact the Student Accessibility Support Center, Stony Brook Union Suite 107, (631) 632-6748, or at sasc@stonybrook.edu. They will determine with you what accommodations are necessary and appropriate. All information and documentation is confidential.