Differential Topology Lecture 2 Notes

From last lecture, we know topological manifolds are Hausdorff, second-countable, and locally euclidean, or $\forall x \in M$, $\exists U = N(x)$ such that $\phi : U \to \tilde{U} \subseteq \mathbb{R}^n$ is a homeomorphism.

Definition A pair $(U, \phi)$ is called a chart.

Definition A topological manifold with boundary is Hausdorff and second countable such that $\forall x \in M$, $\exists U = N(x)$ such that $U$ is homeomorphic to $\tilde{U} \subseteq \mathbb{H}^n$ (half space, $\mathbb{H} = \{ x \in \mathbb{R}\, |\, x \ge 0 \}$).

Smooth Manifolds

If we have an $n$-dimensional topological manifold, $X$, when is $f: X \to \mathbb{R}^n$ smooth?

We have that there is a chart $(U, \phi)$, so we want a function $f \circ \phi^{-1} \colon \tilde{U} \to \mathbb{R}^n$ because $\tilde{U} \subseteq \mathbb{R}^n$ or something. $f$ will be smooth if $\phi^{-1}$ is smooth.