Math 221C: Differential topology

Syllabus

Lectures: MW 12:30 – 1:45 in GIRV 2115.

Office Hours: MTW 2:00 – 2:50 outside the Coral Tree Café.

Text: Differential topology by Guillemin and Pollack.

Homework: Homework will be on gradescope.

Topics: The goal is to cover most of the first three chapters of the textbook. This includes: smooth manifolds, transversality, tangent bundles, Borsuk-Ulam theorem, orientation and intersection number, Lefschetz fixed point theorem, and vector fields.

Tentative lecture plan

• Week 1: Chapter 1 sections 1, 2.
• Week 2: Chapter 1 sections 3, 4.
• Week 3: Chapter 1 sections 5, 6.
• Week 4: Chapter 1 sections 7 (skim Morse theory), 8, Chapter 2 section 1.
• Week 5: Chapter 2 sections 2, 3.
• Week 6: Chapter 2 section 4, and skim sections 5, 6.
• Week 7: Chapter 3 sections 1, 2, 3.
• Week 8: Chapter 3 sections 4, 5.
• Week 9: Chapter 3 skim sections 6, 7.
• Week 10: Peak at Chapter 4, and review.

Imaginary prerequisite "Math 221c-minus"

Inverse Function Theorem

Suppose $U$ is an open neighborhood of 0 in $\mathbb{R}^n$, $f \colon U \to \mathbb{R}^n$ is smooth, $f(0) = 0$, and $df_0$ is invertible. Then there exists an open neighborhood $V$ of $0$ in $U$, and an open neighborhood $W$ of $0$ in $\mathbb{R}^n$, such that $f \colon V \to W$ is a diffeomorphism.

Sard's Theorem

Suppose $U$ is an open set in $\mathbb{R}^n$, and $f \colon U \to \mathbb{R}^m$ is smooth. Let $C$ be the set of $x \in U$ such that $df_x$ is not surjective. Then $f(C)$ has measure zero in $\mathbb{R}^m$.

Partitions of Unity

This is a technical result that has a lot of slightly different versions. Here is a very simple version for just two open sets.

Suppose $X$ is a manifold, and $U$ and $V$ are open sets in $X$ with $X = U \cup V$. Then there exists smooth functions $f,g \colon X \to [0,1]$ such that

• $f(x) = 1$ and $g(x) = 0$ for all $x \in U \setminus V$.
• $f(x) = 0$ and $g(x) = 1$ for all $x \in V \setminus U$.
• $f(x) + g(x) = 1$ for all $x \in X$.

A stronger version uses countably infinitely many open sets and functions, and also demands that the functions have two bonus properties:

• Instead of just "zero outside its open set", each function can be "zero in an open neighborhood of the complement of its open set".
• Locally finite: each point has an open neighborhood where all but finitely many of the functions are zero.

One way to think of this is as a tool for defining "piecewise" functions. The functions it gives you are "piecewise" constant functions, except that they have to smoothly transition between 0 and 1. You can use this to "piece together" other functions. For example, the function that is $x^2$ for $x \le 0$ and $x^3$ for $x \ge 0$ is not smooth, but you can use a partition of unity to smooth it out near the origin.

Classification of compact 1-manifolds

Every compact connected one-dimensional manifold with boundary is diffeomorphic to $[0,1]$ or $S^1$.

A compact non-connected manifold is the disjoint union of finitely many connected components.

References

• Guillemin and Pollack, Differential topology.
• Guillemin and Pollack, errata.
• Spivak, A comprehensive introduction to differential topology.
• Stackexchange discussion of the meaning of proper embedding.
• Ryan Blair's lecture on the proof of Sard's theorem.
• Marston Moore's "delightful film" about Morse.
• Stackexchange discussion about regular projections of knots.
• A proof of the Jordan curve theorem for not-necessarily smooth curves.
• Terence Tao's introduction to forms.