Möbius Bundle Construction

Lee Smooth Manifolds

Chapter 10 involves a lot of very demanding constructions. We recreate them fully here, using the following lemma as a given:

(Smooth Manifold Chart Lemma) Let $M$ be a set, and suppose we are given a collection $\{U_{\alpha }\}$ of subsets of $M$ together with maps ${\phi }_{\alpha }:U_{\alpha }\to {\mathbb{R}}^n$, such that the following properties are satisfied: (i) For each $\alpha$, ${\phi }_{\alpha }$ is a bijection between $U_{\alpha }$ and an open subset ${\phi }_{\alpha }(U_{\alpha })\subseteq {\mathbb{R}}^n$. (ii) For each $\alpha$ and $\beta$, the sets ${\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })$ and ${\phi }_{\beta }(U_{\alpha }\cap U_{\beta })$ are open in ${\mathbb{R}}^n$. (iii) Whenever $U_{\alpha }\cap U_{\beta }\ne \varnothing$, the map

$${\phi }_{\beta }\circ {\phi }_{\alpha }^{-1}:{\phi }_{\alpha }(U_{\alpha }\cap U_{\beta })\to {\phi }_{\beta }(U_{\alpha }\cap U_{\beta })$$

is smooth. (iv) Countably many of the sets $U_{\alpha }$ cover $M$. (v) Whenever $p,q$ are distinct points in $M$, either there exists some $U_{\alpha }$ containing both $p$ and $q$, or there exist disjoint sets $U_{\alpha },U_{\beta }$ with $p\in U_{\alpha }$ and $q\in U_{\beta }$. Then $M$ has a unique smooth manifold structure such that each $(U_{\alpha },{\phi }_{\alpha })$ is a smooth chart.

(10.3, Möbius Bundle Construction) Define $\sim$ on ${\mathbb{R}}^2$ as $(x,y)\sim (x^{\prime },y^{\prime })$ iff $(x^{\prime },y^{\prime })=(x+n,(-1{)}^{n}y)$ for some $n\in \mathbb{Z}$; let $E={\mathbb{R}}^2/\sim$ be the quotient space with quotient map $q$. One can visualize $E$ as a Möbius band made with $[0,1]\times \mathbb{R}$.

Consider the commutative diagram where ${\pi }_1$ is projection onto the first coordinate and $\epsilon$ is the smooth covering map $\epsilon (x)=e^{2\pi ix}$. $\epsilon \circ {\pi }_1$ is constant on the fibers of $q$ and thus descends to a continuous quotient map $\pi$. Intuitively $\pi$ is the projection of the Möbius band onto ${\mathbb{S}}^1$; we formalize this intuition by proving there is a unique smooth manifold structure on $E$ such that $q$ is a smooth covering map and $\pi :E\to {\mathbb{S}}^1$ is a smooth line bundle over ${\mathbb{S}}^1$.

Define smooth charts on $E$ as follows: for any $p\in E$, consider an evenly covered neighborhood $U$ of $\pi (p)$ and let $\tilde{U}$ be a sheet of $U$ in $\mathbb{R}$; one can quickly verify that $q$ restricts to a homeomorphism between ${\pi }_1^{-1}(\tilde{U})$ and ${\pi }^{-1}(U)$. By declaring these homeomorphisms as diffeomorphisms, we obtain a smooth structure on the topology of $E$. It is trivial that, under this smooth structure, $q$ is a smooth covering map. The smooth trivializations of $\pi$ also naturally fall out of the construction; using the same $U$ one can see ${\pi }^{-1}(U)\cong {\pi }_1^{-1}(\tilde{U})\cong U\times \mathbb{R}$. Such functions clearly satisfy the remaining requisite properties of tangent bundles. Uniqueness follows from the fact that, when the topology of the base of a smooth covering map is fixed, there is no other smooth structure such that the map continues to be a smooth covering.