Math 132: Differential Topology (Spring 2026)

See the Canvas course website for announcements, assignments, and other course materials.

Meeting time and location:

Tue Thu 9:00 AM - 10:15 AM at Science Center 310.
First day of class: Jan 27, 2026,   last day of class: Apr 28, 2026.

Instructor:

Sunghyuk Park
Email: sunghyukpark at math dot harvard dot edu
Office hours: Tue 10:15 AM - 12:00 PM at Science Center 231 (or by appointment)

Course assistants:

Course Assistant	Email	Office Hours
Benjamin Walter	bwalter at college dot harvard dot edu	Mon 8-10 PM at Leverett Dining Hall
Evan Tsingos	evantsingos at college dot harvard dot edu	Thu 7-9 PM at Math Lounge

Course goals:

This course is an introduction to the topology of smooth manifolds and smooth maps, from the perspective of oriented intersection theory.
Here are the topics I am planning to talk about:

• Smooth manifolds and smooth maps
• Transversality, mod 2 intersection theory, winding numbers, Jordan-Brouwer separation theorem, Borsuk-Ulam theorem
• Oriented intersection theory, Lefschetz fixed point theorem, vector fields, index and degree, Poincare-Hopf theorem, Euler characteristic
• Differential forms, integration, de Rham cohomology, Stokes theorem, Gauss-Bonnet theorem

Tentative schedule

(*Click on the topics below to see the lecture notes.)

Date	Topic	Date	Topic
1/27	1. Manifolds and Smooth Maps. smooth manifolds	1/29	derivatives and tangents
2/3	inverse function theorem and immersions	2/5	submsersions
2/10	transversality	2/12	homotopy and stability
2/17	Sard's theorem and Morse functions	2/19	Whitney embedding theorem
2/24	2. Transversality and Intersection. manifolds with boundary	2/26	1-manifolds and some consequences
3/3	transversality	3/5	mod 2 intersection theory
3/10	winding numbers and Jordan-Brouwer separation theorem	3/12	Borsuk-Ulam theorem
3/17	No class (Spring Recess)	3/19	No class (Spring Recess)
3/24	3. Oriented Intersection Theory. orientation	3/26	Midterm exam (either during the usual class time and location, or at FAS Testing Center; TBD)
3/31	oriented intersection number	4/2	Lefschetz fixed-point theoem
4/7	vector fields and Poincaré-Hopf theorem	4/9	Hopf degree theorem
4/14	Euler characteristics and triangulations	4/16	4. Integration on Manifolds. tensor and exterior algebras
4/21	differential forms	4/23	integration on manifolds
4/28	Stokes theorem

Textbook:

Guillemin & Pollack - Differential Topology [pdf]

Other useful references:

• Milnor - Topology from the Differentiable Viewpoint [pdf]
• Dan Freed's Lecture Notes

Prerequisites:

Mathematics 22a,b, 23a,b or 25a,b or 55a,b or 112; or an equivalent background in mathematics (vector calculus, introductory real analysis, and basic topology).

Grading:

There will be weekly* homework assignments (20%*) (to be submitted through Gradescope on our Canvas course website), as well as in-person midterm and final exams (80%*). (* subject to change)

Make-Up Exam Policy: There will be no make-up exams. If you miss an exam for a valid reason (e.g. documented illness or family emergency), your score on the other exam score will replace the missed exam.

AI policy:

While there is no specific restriction on the use of generative AI tools in this course, the in-person exams will carry much greater weight than the assignments to ensure a fair and accurate evaluation of your understanding.

Harvard College Honor Code:

"Members of the Harvard College community commit themselves to producing academic work of integrity – that is, work that adheres to the scholarly and intellectual standards of accurate attribution of sources, appropriate collection and use of data, and transparent acknowledgement of the contribution of others to their ideas, discoveries, interpretations, and conclusions. Cheating on exams or problem sets, plagiarizing or misrepresenting the ideas or language of someone else as one's own, falsifying data, or any other instance of academic dishonesty violates the standards of our community, as well as the standards of the wider world of learning and affairs."