Topology — Lecture 2
Date: 1/14/26
Last time
Definition
A topological space $ M $ is a smooth $ n $-manifold if:
-
$ M $ is second countable and Hausdorff.
-
(Locally Euclidean)
For all $ p \in M $, there exists an open set $ U \subset M $ with
$ p \in U $ and a map
\(\phi : U \to \mathbb{R}^n\) which is a homeomorphism onto its image. -
(Smooth compatibility)
If $ (U,\phi) $, $ (V,\psi) $ are charts with $ U \cap V \neq \varnothing $, then
\(\phi \circ \psi^{-1}, \quad \psi \circ \phi^{-1}\) are $ C^\infty $ maps (as maps between open sets of $ \mathbb{R}^n $).
Remark
$ M $ is a topological $ n $-manifold if only conditions (1) and (2) hold.
Examples
-
$ \mathbb{R}^n $:
$ U = \mathbb{R}^n $, $ \phi = \mathrm{id} $. -
$ S^2 \subset \mathbb{R}^3 $ (sphere).
-
Non-example:
A space where removing a point disconnects neighborhoods in a way not possible in $ \mathbb{R}^n $.
Non-example (connectedness argument)
If there exists an open set $ U \ni p $ with a homeomorphism
\(\phi : U \to \mathbb{R}^2,\)
this is not possible when removing $ p $ disconnects $ U $, since open subsets of $ \mathbb{R}^2 $ do not become disconnected by removing a single point.
Hence such spaces fail to be manifolds.
Atlases
Definition
Let $ M $ be a smooth manifold.
A collection of charts
\(\{ (U_\alpha, \phi_\alpha) \}_{\alpha \in A}\)
is an atlas if:
\(M = \bigcup_{\alpha \in A} U_\alpha\)
and the charts satisfy the smooth compatibility condition.
Example: Atlas for $ S^1 \subset \mathbb{C} $
Let \(U_1 = \{ e^{i\theta} : \theta \in (0, 2\pi) \}, \quad \phi_1(e^{i\theta}) = \theta\)
\[U_2 = \{ e^{i\theta} : \theta \in (-\pi, \pi) \}, \quad \phi_2(e^{i\theta}) = \theta\]On the overlap: \(U_1 \cap U_2 = \{ e^{i\theta} : \theta \in (-\pi,0) \cup (0,\pi) \}\)
The transition map: \((\phi_2 \circ \phi_1^{-1})(\theta) = \begin{cases} \theta + 2\pi, & \theta \in (-\pi,0) \\ \theta, & \theta \in (0,\pi) \end{cases}\)
This map is smooth. Likewise $ \phi_1 \circ \phi_2^{-1} $ is smooth.
Compatibility Lemma
Lemma
Suppose $ \mathcal{U} $ is an atlas for $ M $.
If charts $ (U,\phi) $ and $ (V,\psi) $ are each compatible with $ \mathcal{U} $, then
$ (U,\phi) $ and $ (V,\psi) $ are compatible with each other.
Proof (sketch)
Let $ p \in U \cap V $.
Then there exists $ \alpha \in A $ such that $ p \in U_\alpha $.
We know: \(\phi \circ \phi_\alpha^{-1}, \quad \phi_\alpha \circ \psi^{-1}\) are smooth by assumption.
Then \(\phi \circ \psi^{-1} = (\phi \circ \phi_\alpha^{-1}) \circ (\phi_\alpha \circ \psi^{-1})\) is a composition of smooth maps, hence smooth.
Similarly for $ \psi \circ \phi^{-1} $.
Corollary
If $ \mathcal{U} $ is an atlas for $ M $, and
\(\{ (V_\beta, \psi_\beta) \}_{\beta \in B}\)
are charts each compatible with $ \mathcal{U} $, then
\(\mathcal{U} \cup \{ (V_\beta, \psi_\beta) \}_{\beta \in B}\)
is also an atlas.
Smooth Structures
Definition
A smooth structure on $ M $ is a maximal atlas $ \mathcal{U} $ such that:
- If $ \mathcal{V} $ is another atlas with $ \mathcal{U} \subset \mathcal{V} $, then
$ \mathcal{U} = \mathcal{V} $.
There are no additional compatible charts.
The maximal atlas is unique.
Proposition
Any atlas for a smooth manifold $ M $ is contained in a unique maximal atlas.
Proof sketch:
Take the union of all charts compatible with the given atlas and apply the corollary.
Remark (Kervaire, 1960s)
There exist topological manifolds that do not admit any smooth structure.
Idea:
- There exist invariants of topological manifolds that vanish for smooth manifolds.
Examples
-
If $ M $ is a smooth manifold and $ U \subset M $ is open, then
$ U $ is a smooth manifold (with the induced topology). -
$ \mathrm{GL}(n,\mathbb{R}) = { A \in \mathbb{R}^{n \times n} : \det A \neq 0 } $
Since $ \det : \mathbb{R}^{n \times n} \to \mathbb{R} $ is continuous and
$ \mathbb{R} \setminus {0} $ is open,
$ \mathrm{GL}(n,\mathbb{R}) $ is open in $ \mathbb{R}^{n \times n} $ and hence a smooth manifold.
Product Smooth Structures
Example
If $ M $ is a smooth $ n $-manifold with atlas $ {(U_\alpha,\phi_\alpha)} $
and $ N $ is a smooth $ m $-manifold with atlas $ {(V_\beta,\psi_\beta)} $,
then \(M \times N\) is a smooth $ (n+m) $-manifold with atlas: \(\{ (U_\alpha \times V_\beta,\ \phi_\alpha \times \psi_\beta) \}_{(\alpha,\beta)}\)
Example
The $ n $-torus: \(\mathbb{T}^n = S^1 \times \cdots \times S^1\)
Smooth Maps
Definition
Let $ M $ be a smooth manifold and $ f : M \to \mathbb{R} $.
We say $ f $ is smooth ($ C^\infty $) if for every chart $ (U_\alpha,\phi_\alpha) $, \(f \circ \phi_\alpha^{-1} : \phi_\alpha(U_\alpha) \subset \mathbb{R}^n \to \mathbb{R}\) is a $ C^\infty $ function.
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