Stiefel-Whitney Classes

In this chapter, we are going to study the Stiefel-Whitney classes on vector bundle. Somehow, we will postpone the concrete construction of the characteristic classes. The whole process would be

1. Axioms of Stiefel-Whitney classes;
2. Some examples and simple calculation of Stiefel-Whitney classes;
3. Construction of Stiefel-Whitney classes.

The whole process is a lot more similar to the universal property in category theory. The construction can be (perhaps/maybe) forgotten. What we are going to keep in mind is the basic property of Stiefel-Whitney classes.

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Axioms of Stiefel-Whitney Classes

Given a vector bundle $\pi:E\rightarrow X$ of rank $k$.

[Axiom 1] The $i$-th Stiefel-Whitney class of $E$, denoted $w_i(E)$, is

$$\begin{gathered} w_i(E)\in H^{i}(X,\mathbb{Z}/2)\\ w_0(E)=1,\quad w_i(E)=0~\text{if~}i>k \end{gathered}$$

Followed by Stiefel-Whitney classes, Stiefel-Whitney number arises, as a element in cohomology ring.

[Def] (Stiefel-Whitney number)

$$w(E):=w_0(E)+w_1(E)+\cdots$$

[Axiom 2] If there is a pullback diagram of vector bundle

then

$$w_i(f^*E)=f^*w_i(E).$$

Here the pullback on the right-hand side is the pullback on cohomology.

[Axiom 3] (Whitney product formula) Suppose that $E,F\rightarrow X$. Then

$$w(E\oplus F)=w(E)w(F)$$

[Axiom 4] (Nontriviality) Let $\eta_1$ be the tautological bundle over $\mathbb{R}P^1$, then

$$w_1(\eta_1)\neq 0\in H^1(\mathbb{R}P^1,\mathbb{Z}/2)$$

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Property and First Examples

[Prop] Let $E\rightarrow X$ be a vector bundle of rank $k$.

1. $w_i(\underline{\mathbb{R}^k})=0$, $\forall i>0$;
2. $w(E\oplus\mathbb{R}^k)=w(E)w(\underline{\mathbb{R}}^k)=w(E)$;
3. If $E=F\oplus\underline{\mathbb{R}^l}$, $\mathrm{rank}(F)=l$, then

$$w_i(E)=w_i(F)=0, \quad \forall i>n-l.$$

4. $w:(\widetilde{KO}(X),\oplus)\rightarrow (H^*(X;\mathbb{Z}/2,\cup))$ preserves operation.

[e.g.]

1. $w(TS^n)=1$;

2. Consider the inclusion map $\mathbb{R}P^k\hookrightarrow\mathbb{R}P^n$​.

   

   Since $\boxed{H^*(\mathbb{R}P^n);\mathbb{Z}/2=(\mathbb{Z}/2)[a]/a^{n+1}}$ where $a\in H^1(\mathbb{R}P^n;\mathbb{Z}/2)$, then

   $$w(\eta_n)=a+1.$$

3. Notice that $\eta_n\subset\mathbb{R}P^n\times\mathbb{R}^{n+1}=\underline{\mathbb{R}^{n+1}}=\eta_n\oplus\eta_n^\perp$. Thus

$$\begin{gathered} w(\eta_n)w(\eta_n^\perp)=1, \\ w(\eta_n^\perp)=(1+a)^{-1}=1+a+\cdots \end{gathered}$$

4. (Tangent bundle of $\mathbb{R}P^n$)

   $$\begin{gathered} \pi:S^n\overset{2:1}{\rightarrow}\mathbb{R}P^n\\ \pi^*:T(\mathbb{R}P^n)\overset{\cong}{\rightarrow} TS^n \end{gathered}$$

   We can think of

   $$\begin{gathered} T_xS^n=\{v\in\mathbb{R}^{n+1}:v\cdot x=0\}\\ T_{-x}S^n=\{v\in\mathbb{R}^{n+1}:v\cdot (-x)=0\}\\ \end{gathered}$$

   and

   $$T_\alpha S^n=\{\{(x,v),(-x,-v)\}:x\in\alpha, v\cdot x=0\}.$$

   Therefore, for $\alpha=\{-x,x\}\in\mathbb{R}P^n$, there is an isomorphism

   $$\begin{aligned} T_\alpha\mathbb{R}P^n&\rightarrow\mathrm{Hom}(\eta_n|_\alpha,\eta_n^\perp|_\alpha)\\ \{(x,v),(-x,-v)\}&\mapsto (x\mapsto v) \end{aligned}$$

   With such correspondence, we are now in a position to calculate the Stiefel-Whitney class of $T(\mathbb{R}P^n)$.

   $$\begin{aligned} &\quad T\mathbb{R}P^n\oplus\underline{\mathbb{R}}\\ &\cong\mathrm{Hom}(\eta_n,\eta_n^\perp)\oplus\mathrm{Hom}(\eta_n,\eta_n)\\ &\cong\mathrm{Hom}(\eta_n,\eta_n\oplus\eta_n^\perp)\\ &\cong\mathrm{Hom}(\eta_n,\underline{\mathbb{R}^{n+1}})\\ &\cong(\eta_n^*)^{\oplus(n+1)}\\ &\cong(\eta_n)^{\oplus(n+1)}\\ \end{aligned}$$

   Thus,

   $$\begin{aligned} &w(T\mathbb{R}P^n)\\ =&w(\eta_n)^{n+1}\\ =&(1+a)^{n+1}\\ =&1+\binom{n+1}{1}a+\cdots+\binom{n+1}{n}a^n. \end{aligned}$$

   Notice here the coefficients are in $\mathbb{Z}/2$.

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Application

[e.g.] (Immersion Problem)

[e.g] (Obstruction of cobordism)

Let $M^n$ be a closed $C^\infty$-manifold. We might ask “does there exist $W^{n+1}$, such that $\partial W=M$?” . Recall that

$$w(T\mathbb{R}P^n)=(1+a)^n$$

In particular, set $n=2$, we have $w(T\mathbb{R}P^2)=1+a^2$. Consider the long exact sequence of $(W,M)$, using Poincare-Lefschetz duality,

Notice that we have smooth structure: denote $i:M\hookrightarrow N$.

$$\begin{aligned} TW|_M &=TM\oplus \nu(M)\\ &=TM\oplus\underline{\mathbb{R}} \end{aligned}$$

Let $[M]\in H_n(M)$ be the fundamental class of $M$. Then

$$\begin{aligned} &\langle w_{\alpha_1}(TM)\cdots w_{\alpha_k}(TM),[M]\rangle\\ =&\langle i^*w_{\alpha_{1}}(TW)\cdots i^*w_{\alpha_{k}}(TW),[M]\rangle\\ =&\langle w_{\alpha_{1}}(TW)\cdots w_{\alpha_k}(TW),i^*[M]\rangle\\ =&0. \end{aligned}$$

Therefore, $\mathbb{R}P^2$ cannot be the boundary of a three-manifold. Otherwise, $w_2(T\mathbb{R}P^2)=0$ and $w_1(T\mathbb{R}P^2)w_1(T\mathbb{R}P^2)=0$, contradicting to $w_2(T\mathbb{R}P^2)=a^2\neq 0$.