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 BB, called base space.

[Def] (Real vector bundle) A real vector bundle ξ\xi over BB consists of the following data:

  1. Total space: E(ξ)E(\xi), a topological space;
  2. Projection map: π:EB\pi:E\rightarrow B a continuous map;
  3. bB\forall b\in B, the R\mathbb{R}-vector space structure on π1(b)\pi^{-1}(b),

such that: pB\forall p\in B, UB\exists U\subset B an open neighborhood of pp and an integer k0k\geq 0, and a continuous map

h:U×Rnπ1(U)h:U\times \mathbb{R}^n\rightarrow \pi^{-1}(U)

such that h(b.):Rnπ1(b)\boxed{h(b.\cdot):\mathbb{R}^n\rightarrow\pi^{-1}(b)} defines an R\mathbb{R}-isomorphism of vector spaces for each bUb\in U. Here π1(b)\pi^{-1}(b) is called the fiber over bb, denoted by FbF_b or Fb(ξ)F_b(\xi). Clearly, FbF_b\neq\emptyset.

[Remark] dim(Fb)\dim(F_b) is a locally constant function of bb​.

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

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

fFb(ξ):Fb(ξ)Ff(b)(η)f|_{F_b(\xi)}:F_b(\xi)\rightarrow F_{f(b)}(\eta)

on each fiber over each point bBb\in B, then ξ\xi is isomorphic to η\eta, denoted ξη\xi\simeq\eta.

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

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

[e.g.] (Tangent bundle) E=TME=TM​ and

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

The fiber over xx is π1(x)={(x,v):vTxM}\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 τM\tau_M.

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

[e.g.] (2-dimensional unit ball) S2R3S^2\subset\mathbb{R}^3 is not parametrizable. In general, parametrizable manifolds must have Euler characteristic 00, namely, χ(S)=0\chi(S)=0. However, χ(S2)=2\chi(S^2)=2.

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

  1. E={(x,v):vRn,vTxM}E=\{(x,v):v\in\mathbb{R}^n,v\perp T_xM\};
  2. π(x,v)=x\pi(x,v)=x

It can be verified that π1(x)\pi^{-1}(x) has a R\mathbb{R}-vector space structure. It suffices to verify local triviality.

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

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

with total space

E(γn1)={({±x},v)Pn×Rn+1:v=λx,λ0}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

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

Here π1(x)\pi^{-1}(x) can be viewed as “line through xx and x-x in Rn+1\mathbb{R}^{n+1}”, and thus with a natural R1\mathbb{R}^1-structure. Such line bundle is called the canonical line bundle over Pn\mathbb{P}^n, denoted γn1\gamma_n^1.

[Theorem 1] γn1\gamma_n^1 is not a trivial bundle over Pn\mathbb{P}^n, for n1n\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 ξ\xi with base space BB is a map

s:BE(ξ)s:B\rightarrow E(\xi)

such that

s(b)Fb(ξ),bBs(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 R\mathbb{R}-bundle. It suffices to show that: there does not exist “nowhere zero cross section”. Let s:PnE(γn1)s:\mathbb{P}^n\rightarrow E(\gamma_n^1) be a cross section. Then

SnPnsE(γn1)x{±x}({±x},s~(x))\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 s~(x)0\tilde{s}(x)\neq 0 has the same direction of xx, i.e. t:SnR{0}\exists t:S^n\rightarrow\mathbb{R}-\{0\}, s.t.


with the property that t(x)=t(x)t(-x)=-t(x). Since tC0t\in C^0, then by intermediate value theorem, x0Sn\exists x_0\in S^n, s.t. t(x0)=0t(x_0)=0. Contradiction! \square

[Prop] Let EE be a vector bundle over BB with rank kk, UBU\subset B be a open subset of BB. Then EUU×RkE_U\cong U\times\mathbb{R}^k iff s1,,skΓ(EU)\exists s_1,\ldots,s_k\in\Gamma(E_U), s.t. xU\forall x\in U, {s1(x),,sk(x)}\{s_1(x),\ldots,s_k(x)\} are linearly independent.

Vector Bundle DEF

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

Suppose that EE and FF have rank kk and ll , respectively. The linear space structure on (E/F)x(E/F)_x is given by the canonical quotient on linear spaces. We shall use the proposition to verify the local triviality of E/FE/F. xX\forall x\in X, UBU\subset B be a open neighborhood of xx.

More on Cross Section

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

[Prop] Let EXE\rightarrow X be a vector bundle, where XX is compact and Hausdorff. Let YXY\subset X be a closed subset of XX, and EYE|_Y be the restriction vector bundle. Suppose that sΓ(EY)s\in\Gamma(E|_Y). Then sˉΓ(E)\exists\bar{s}\in\Gamma(E), s.t.


[Remark] In fact, we can only require XX to be paracompact. In a T4T^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 XX be a compact and Hausdorff space, f0,f1:XYf_0,f_1:X\rightarrow Y, and F:f0f1F:f_0\simeq f_1. Namely,

F:X×[0,1]YFX×{0}=f0,FX×{1}=f1\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 EYE\rightarrow Y be a vector bundle over YY. Then f0Ef1Ef_0^*E\cong f_1^*E.

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

ι0(FE)=f0Eι1(FE)=f1E\begin{gathered} \iota_0^*(F^*E)=f_0^*E\\ \iota_1^*(F^*E)=f_1^*E\\ \end{gathered}

We shall prove that E~:=FE\tilde{E}:=F^*E has the “same” section on every restriction E~X×{t}\tilde{E}|_{X\times\{t\}}. Define E0:=pf0EE_0:=p^*f_0^*E be another vector bundle over X×[0,1]X\times [0,1].

Goal: Prove that E~X×{t}E0X×{t}\tilde{E}|_{X\times\{t\}}\cong E_0|_{X\times\{t\}} . Consider the Hom\mathrm{Hom}-bundle

Hom(E,E0)X.\mathrm{Hom}(E',E_0)\rightarrow X.

If t[0,1]t\in[0,1] satisfies that E~X×{t}E0X×{t}\tilde{E}|_{X\times\{t\}}\cong E_0|_{X\times\{t\}}, then by the correspondence, sΓ(Hom(E,E0)X×{t})\exists s\in\Gamma(\mathrm{Hom}{(E',E_0)}|_{X\times\{t\}}), such that ss is an isomorphism on each fiber over X×{t}X\times\{t\}. By Prop in cross section, since X×{t}X\times\{t\} is a compact set in XX, then ss can be extended to a cross section sˉΓ(Hom(E~,E0))\bar{s}\in\Gamma(\mathrm{Hom}(\tilde{E},E_0)).


W:={(x,t)X×[0,1]:sˉHom(E~,E0)(x,t) isomorphic}W:=\left\{(x,t)\in X\times[0,1]:\bar{s}|_{\mathrm{Hom}(\tilde{E},E_0)_{(x,t)}}\text{~isomorphic}\right\}

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

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

X×(tε,t+ε)W.X\times (t-\varepsilon,t+\varepsilon)\subset W.

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

J:={t[0,1]:E~X×{t}E0X×{0}}[0,1]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 VectR(X)\mathrm{Vect}_\mathbb{R}(X). Thus J=[0,1]J=[0,1], and we prove that E~X×{t}\tilde{E}|_{X\times\{t\}} does not depend on tt, which implies that

f0E=EX×{0}EX×{1}=f1E.f_0^*E=E'|_{X\times\{0\}}\cong E'|_{X\times\{1\}}=f_1^*E.

This completes the proof. \square