# Introduction to Vector Bundle

Stiefel-Whitney classes are defined on vector bundle. First we consider some properties of vector bundle. There is no gap between **Vector Bundle ABC** and **Vector Bundle DEF**, as they are designed for you to take a break.

### Vector Bundle ABC

Fix a topological space $B$, called *base space*.

**[Def]** (Real vector bundle) A **real vector bundle** $\xi$ over $B$ consists of the following data:

- Total space: $E(\xi)$, a topological space;
- Projection map: $\pi:E\rightarrow B$ a continuous map;
- $\forall b\in B$, the $\mathbb{R}$-vector space structure on $\pi^{-1}(b)$,

such that: $\forall p\in B$, $\exists U\subset B$ an open neighborhood of $p$ and an integer $k\geq 0$, and a continuous map

$h:U\times \mathbb{R}^n\rightarrow \pi^{-1}(U)$

such that $\boxed{h(b.\cdot):\mathbb{R}^n\rightarrow\pi^{-1}(b)}$ defines an $\mathbb{R}$-isomorphism of vector spaces for each $b\in U$. Here $\pi^{-1}(b)$ is called the fiber over $b$, denoted by $F_b$ or $F_b(\xi)$. Clearly, $F_b\neq\emptyset$.

**[Remark]** $\dim(F_b)$ is a locally constant function of $b$.

**[Remark]** $\mathbb{R}^n$-bundle is the special case for fiber bundle, with fiber $\mathbb{R}^n$ and structural group $\mathrm{GL}(n,\mathbb{R})$.

**[Def]** (Isomorphic vector bundle) If there exists $f:E(\xi)\rightarrow E(\eta)$, a homomorphism between total spaces, that restricted to isomorphisms

$f|_{F_b(\xi)}:F_b(\xi)\rightarrow F_{f(b)}(\eta)$

on each fiber over each point $b\in B$, then $\xi$ is **isomorphic** to $\eta$, denoted $\xi\simeq\eta$.

**[e.g.]** (Trivial bundle) Define the trivial bundle over $B$ by setting total space $E=B\times\mathbb{R}^n$ and projection map

$\begin{aligned} \pi:E&\rightarrow B\\ (b,v)&\mapsto b \end{aligned}$

**[e.g.]** (Tangent bundle) $E=TM$ and

$\begin{aligned} \pi:TM&\rightarrow M\\ (x,v)&\mapsto x \end{aligned}$

The fiber over $x$ is $\pi^{-1}(x)=\{(x,v):v\in T_xM\}$ with a canonical structure. The local triviality can also be verified. Such bundle is denoted by $\tau_M$.

**[Def]** (Parametrizable) If a smooth manifold $M$ has a trivial tangent bundle $\tau_M$, then $M$ is said to be **parametrizable**.

**[e.g.]** (2-dimensional unit ball) $S^2\subset\mathbb{R}^3$ is not parametrizable. In general, parametrizable manifolds must have Euler characteristic $0$, namely, $\chi(S)=0$. However, $\chi(S^2)=2$.

**[e.g.]** (Normal bundle) Let $M\subset\mathbb{R}^n$ be a smooth submanifold. The normal bundle of $M$, denoted $\nu_M$, is defined as follows:

- $E=\{(x,v):v\in\mathbb{R}^n,v\perp T_xM\}$;
- $\pi(x,v)=x$

It can be verified that $\pi^{-1}(x)$ has a $\mathbb{R}$-vector space structure. It suffices to verify local triviality.

**[e.g.]** (Canonical line bundle over $\mathbb{P}^n$) Define base space

$\mathbb{P}^n:=\{x\in S^n\subset\mathbb{R}^{n+1}\}/\{x,-x\}$

with total space

$E(\gamma_n^1)=\left\{\left(\{\pm x\},v\right)\in\mathbb{P}^n\times\mathbb{R}^{n+1}:v=\lambda x, \lambda\neq 0\right\}$

and projection map

$\begin{aligned} \pi:E&\rightarrow B\\ (\{\pm x,v\})&\mapsto \{\pm x\} \end{aligned}$

Here $\pi^{-1}(x)$ can be viewed as “line through $x$ and $-x$ in $\mathbb{R}^{n+1}$”, and thus with a natural $\mathbb{R}^1$-structure. Such line bundle is called the canonical line bundle over $\mathbb{P}^n$, denoted $\gamma_n^1$.

**[Theorem 1]** $\gamma_n^1$ is not a trivial bundle over $\mathbb{P}^n$, for $n\geq 1$.

The proof of the above theorem requires the notion of *cross section*, a generalization of vector field, to describe the condition of trivial bundle.

**[Def]** (Cross section) A **cross section** of a vector field $\xi$ with base space $B$ is a map

$s:B\rightarrow E(\xi)$

such that

$s(b)\in F_b(\xi), \quad \forall b\in B$

**Proof of Theorem 1.** Notice that there exists a nowhere zero cross section over trivial $\mathbb{R}$-bundle. It suffices to show that: there does not exist “nowhere zero cross section”. Let $s:\mathbb{P}^n\rightarrow E(\gamma_n^1)$ be a cross section. Then

$\begin{aligned} S^n\rightarrow&\mathbb{P}^n\overset{s}{\rightarrow}E(\gamma_n^1)\\ x\mapsto&\{\pm x\}\mapsto (\{\pm x\},\tilde{s}(x)) \end{aligned}$

where $\tilde{s}(x)\neq 0$ has the same direction of $x$, *i.e.* $\exists t:S^n\rightarrow\mathbb{R}-\{0\}$, *s.t.*

$\tilde{s}(x)=t(x)x$

with the property that $t(-x)=-t(x)$. Since $t\in C^0$, then by intermediate value theorem, $\exists x_0\in S^n$, *s.t.* $t(x_0)=0$. Contradiction! $\square$

**[Prop]** Let $E$ be a vector bundle over $B$ with rank $k$, $U\subset B$ be a open subset of $B$. Then $E_U\cong U\times\mathbb{R}^k$ iff $\exists s_1,\ldots,s_k\in\Gamma(E_U)$, *s.t.* $\forall x\in U$, $\{s_1(x),\ldots,s_k(x)\}$ are linearly independent.

### Vector Bundle DEF

**[Def]** (Quotient bundle) Let $F\hookrightarrow E$, define the fiber over $x\in B$ to be $(E/F)_x:=E_x/F_x$, total space $E/F=\bigsqcup E_x/F_x$, and $\pi$ as usual.

Suppose that $E$ and $F$ have rank $k$ and $l$ , respectively. The linear space structure on $(E/F)_x$ is given by the canonical quotient on linear spaces. We shall use the proposition to verify the local triviality of $E/F$. $\forall x\in X$, $U\subset B$ be a open neighborhood of $x$.

### More on Cross Section

Before we continue to further topics, we shall look into further theorems and propositions on cross section.

**[Prop]** Let $E\rightarrow X$ be a vector bundle, where $X$ is compact and Hausdorff. Let $Y\subset X$ be a closed subset of $X$, and $E|_Y$ be the restriction vector bundle. Suppose that $s\in\Gamma(E|_Y)$. Then $\exists\bar{s}\in\Gamma(E)$, *s.t.*

$\bar{s}|_Y=s.$

**[Remark]** In fact, we can only require $X$ to be paracompact. In a $T^4$ topological space, according to Tietz Theorem, any function can be extended to a global function.

**[Remark]** For other cases, when a space is compact + Hausdorff, we can think of method of *partition of unity*: first prove the theorem locally, and glue the local cases together with partition of unity.

**Proof of Prop.**

### Homotopy Invariance

In this part, we shall prove that “pullback has homotopy invariant”.

**[Theorem 2]** (Homotopy Invariance)

Let $X$ be a compact and Hausdorff space, $f_0,f_1:X\rightarrow Y$, and $F:f_0\simeq f_1$. Namely,

$\begin{gathered} F:X\times[0,1]\rightarrow Y\\ F|_{X\times\{0\}}=f_0,\quad F|_{X\times\{1\}}=f_1 \end{gathered}$

Let $E\rightarrow Y$ be a vector bundle over $Y$. Then $f_0^*E\cong f_1^*E$.

**Proof.** Define $\iota_t:X\rightarrow X\times [0,1]$ by $\iota_t(x)=(x,t)$. Then

$\begin{gathered} \iota_0^*(F^*E)=f_0^*E\\ \iota_1^*(F^*E)=f_1^*E\\ \end{gathered}$

We shall prove that $\tilde{E}:=F^*E$ has the “same” section on every restriction $\tilde{E}|_{X\times\{t\}}$. Define $E_0:=p^*f_0^*E$ be another vector bundle over $X\times [0,1]$.

Goal: Prove that $\tilde{E}|_{X\times\{t\}}\cong E_0|_{X\times\{t\}}$ . Consider the $\mathrm{Hom}$-bundle

$\mathrm{Hom}(E',E_0)\rightarrow X.$

If $t\in[0,1]$ satisfies that $\tilde{E}|_{X\times\{t\}}\cong E_0|_{X\times\{t\}}$, then by the correspondence, $\exists s\in\Gamma(\mathrm{Hom}{(E',E_0)}|_{X\times\{t\}})$, such that $s$ is an isomorphism on each fiber over $X\times\{t\}$. By **Prop** in cross section, since $X\times\{t\}$ is a compact set in $X$, then $s$ can be extended to a cross section $\bar{s}\in\Gamma(\mathrm{Hom}(\tilde{E},E_0))$.

Consider

$W:=\left\{(x,t)\in X\times[0,1]:\bar{s}|_{\mathrm{Hom}(\tilde{E},E_0)_{(x,t)}}\text{~isomorphic}\right\}$

Openness: since $\bar{s}$ is a cross section, then $\bar{s}$ depends on $(x,t)$ continuously, which implies that $\det(\bar{s})\neq 0$ is an open condition. Thus $W$ is an open set.

Since $X\times\{t\}\subset W$ is compact, then by Tubular Neighborhood Theorem, $\exists \varepsilon>0$, *s.t.*

$X\times (t-\varepsilon,t+\varepsilon)\subset W.$

Therefore, using isomorphism $\bar{s}$, we know that

$J:=\{t\in[0,1]:\tilde{E}|_{X\times\{t\}}\cong E_0|_{X\times\{0\}}\}\subset [0,1]$

is open and closed, which means that the restriction map is locally constant into $\mathrm{Vect}_\mathbb{R}(X)$. Thus $J=[0,1]$, and we prove that $\tilde{E}|_{X\times\{t\}}$ does not depend on $t$, which implies that

$f_0^*E=E'|_{X\times\{0\}}\cong E'|_{X\times\{1\}}=f_1^*E.$

This completes the proof. $\square$