Differential forms in algebraic topology

Summer 2020 Tutorial

Dates: July 6-August 13

Meetings: M-Th, 3pm-4:30pm

Textbook: Bott and Tu

Instructor: Joshua Wang

Email: jxwang@math.harvard.edu

Office hours: T, 8pm-9pm and Th, 10am-11am

Zoom Meeting ID: 978 4988 2048


August 10: The soft deadline for the final paper is August 24 while the hard deadline is August 31.

July 5: Our first week of meetings will be Tuesday (July 7) through Friday (July 10) instead of Monday through Thursday. This is to accommdate a conflicting Zoom meeting which includes updates on plans for the coming academic year. We will meet Monday (July 13) through Thursday (July 16) as planned the following week.


Week 1

July 7: overview, basics of differential forms

July 8: the de Rham complex on Euclidean space (§1)

July 9: the de Rham complex on smooth manifolds I (first subsection of §2)

July 10: the de Rham complex on smooth manifolds II

Week 2

July 13: the Mayer-Vietoris sequence I (last two subsections of §2)

July 14: the Mayer-Vietoris sequence II

July 15: orientation and integration (§3)

July 16: Stokes' theorem

Week 3

July 20: the Poincaré lemma and homotopy invariance I (first two subsections of §4)

July 21: the Poincaré lemma and homotopy invariance II

July 22: Poincaré duality I (first three subsections of §5)

July 23: Poincaré duality II

Week 4

July 27: the Poincaré dual of an oriented submanifold I (last subsection of §5)

July 28: the Poincaré dual of an oriented submanifold II

July 29: the Künneth formula and fiber bundles (fourth subsection of §5)

July 30: vector bundles I (first two subsections of §6)

Week 5

August 3: vector bundles II

August 4: the Thom isomorphism I (third and fourth subsections of §6)

August 5: the Thom isomorphism II

August 6: the Euler class I (fifth subsection of §6)

Week 6

August 10: the Euler class II (last subsection of §11)

August 11: the Čech-de Rham complex I (§8)

August 12: the Čech-de Rham complex II (first subsection of §9)

August 13: the spectral sequence of a double complex (third and fourth subsections of §14)

Course information

In this tutorial, we will explore algebraic topology through the lens of differential forms. We will begin with differential forms on Euclidean space, and eventually reach Poincaré duality, characteristic classes, and spectral sequences all in a concrete hands-on way. Armed with these concepts, we will continue into homotopy theory and compute some homotopy groups of spheres.

The purpose of this course is not just to understand algebraic topology more concretely; it also serves as a gateway into geometric topology, more advanced topics in algebraic topology, and even topics in analysis like Hodge theory and index theory.


First courses in analysis, algebra, and topology are required. Some familiarity with smooth manifolds is expected. Those who have encountered algebraic topology already will benefit from seeing this material from a different perspective.


The course will be taught over Zoom, and the meetings will be recorded. For each meeting, students are expected either to attend class with Zoom video on, or to watch the recording and submit solutions to at least one of the exercises from the class, collected at the end of the week.

When I was an undergraduate, a professor recommended Bott and Tu to me about a dozen times. "Josh, have I recommended this book to you before? You should read it; it's a beautiful book."