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.
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.
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 $\mathbb{R}^m$.
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:
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.
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.