11.6 Stiefel-Whitney classes

The last example of the characteristic classes is the Stiefel-Whitney class. In contrast to the rest of the characteristic classes, the Stiefel-Whitney class cannot be expressed in terms of the curvature of the bundle. The Stiefel-Whitney class is important in physics since it tells us whether a manifold admits a spin or not. Let us start with a brief review of a spin bundle.

11.6.1 Spin bundles

Let $TM\stackrel{\pi }{\to }M$ be a tangent bundle with $\dim M=m$ . The bundle $TM$ is assumed to have a fibre metric and the structure group $G$ is taken to be $\mathrm{O}(m)$ . If, furthermore, $M$ is orientable, $G$ can be reduced down to $\mathrm{SO}(m)$ . Let $LM$ be the frame bundle associated with $TM$ . Let $t_{ij}$ be the transition function of $LM$ which satisfies the consistency condition (9.6)

$$t_{ij}{t}_{jk}{t}_{ki}=I{t}_{ii}=I.$$

A spin structure on $M$ is defined by the transition function ${\tilde{t}}_{ij}\in \mathrm{SPIN}(m)$ such that

$$\phi \left({\tilde{t}}_{ij}\right)=t_{ij}{\tilde{t}}_{ij}{\tilde{t}}_{jk}{\tilde{t}}_{ki}=I{\tilde{t}}_{ii}=I\text{\hspace{1em}(11.129)}$$

where $\phi$ is the double covering $\mathrm{SPIN}(m)\to \mathrm{SO}(m)$ . The set of ${\tilde{t}}_{ij}$ defines a spin bundle $PS(M)$ over $M$ and $M$ is said to admit a spin structure (of course, $M$ may admit many spin structures depending on the choice of ${\tilde{t}}_{ij}$ ).

It is interesting to note that not all manifolds admit spin structures. Non-admittance of spin structures is measured by the second Stiefel-Whitney class which takes values in the Čech cohomology group $H^2(M;{\mathbb{Z}}_2)$ .

11.6.2 Čech cohomology groups

Let ${\mathbb{Z}}_2$ be the multiplicative group $\{-1,+1\}$ . A Čech $r$ -cochain is a function $f(i_0,i_1,\dots ,i_r)\in {\mathbb{Z}}_2$ , defined on $U_{i_0}\cap U_{i_1}\cap \cdots \cap U_{i_r}\ne \varnothing$ , which is totally symmetric under an arbitrary permutation $P$ ,

$$f(i_{P(0)},\dots ,i_{P(r)})=f(i_0,\dots ,i_r).$$

Let $C^r(M,{\mathbb{Z}}_2)$ be the multiplicative group of Čech $r$ -cochains. We define the coboundary operator $\delta :C^r(M;{\mathbb{Z}}_2)\to C^{r+1}(M;{\mathbb{Z}}_2)$ by

$$\left(\delta f\right)\left(i_0,\dots ,i_{r+1}\right)=\prod _{j=0}^{r+1}f\left(i_0,\dots ,{\hat{i}}_j,\dots ,i_{r+1}\right)\text{\hspace{1em}(11.130)}$$

where the variable below the $\hat{\mu }$ is omitted. For example,

$$\begin{array}{rl}\left(\delta f_0\right)\left(i_0,i_1\right)& =f_0\left(i_1\right)f_0\left(i_0\right)f_0\in C^0(M;{\mathbb{Z}}_2)\\ & \left(\delta f_1\right)\left(i_0,i_1,i_2\right)=f_1\left(i_1,i_2\right)f_1\left(i_0,i_2\right)f_1\left(i_0,i_1\right)f_1\in C^1(M;{\mathbb{Z}}_2).\end{array}$$

Since we employ the multiplicative notation, the unit element of $C^r(M;{\mathbb{Z}}_2)$ is denoted by 1. We verify that $\delta$ is nilpotent:

$$({\delta }^{2}f)(i_0,\dots ,i_{r+2})=\prod _{j,k=1}^{r+1}f(i_0,\dots ,{\hat{i}}_j,\dots ,{\hat{i}}_k,\dots ,i_{r+2})=1$$

since $-1$ always appears an even number of times in the middle expression (for example if $f(i_0,\dots ,{\hat{i}}_j,\dots ,{\hat{i}}_k,\dots ,i_{r+2})=-1$ , we have $f(i_0,\dots ,{\hat{i}}_k,\dots ,{\hat{i}}_j,\dots ,i_{r+2})=-1$ from the symmetry of $f$ ). Thus, we have proved, for any Čech $r$ -cochain $f$ , that

$${\delta }^{2}f=1.\text{\hspace{1em}(11.131)}$$

The cocycle group $Z^r(M;{\mathbb{Z}}_2)$ and the coboundary group $B^r(M;{\mathbb{Z}}_2)$ are defined by

$$Z^r(M;{\mathbb{Z}}_2)=\{f\in C^r(M;{\mathbb{Z}}_2)|\delta f=1\}\text{\hspace{1em}(11.132)}$$

$$B^r(M;{\mathbb{Z}}_2)=\{f\in C^r(M;{\mathbb{Z}}_2)|f=\delta f^{\prime },f^{\prime }\in C^{r-1}(M;{\mathbb{Z}}_2).\text{\hspace{1em}(11.133)}$$

Now the $r$ th Čech cohomology group $H^r(M;{\mathbb{Z}}_2)$ is defined by

$$H^r(M;{\mathbb{Z}}_2)=\ker {\delta }_r/\mathrm{im}{\delta }_{r-1}=Z^r(M;{\mathbb{Z}}_2)/B^r(M;{\mathbb{Z}}_2).\text{\hspace{1em}(11.134)}$$

11.6.3 Stiefel-Whitney classes

The Stiefel-Whitney class $w_r$ is a characteristic class which takes its values in $H^r(M;{\mathbb{Z}}_2)$ . Let $TM\stackrel{\pi }{\to }M$ be a tangent bundle with a Riemannian metric. The structure group is $\mathrm{O}(m)$ , $m=\dim M$ . We assume $\{U_i\}$ is a simple open covering of $M$ , which means that the intersection of any number of charts is either empty or contractible. Let $\{e_{i\alpha }\}$ ( $1\le \alpha \le m$ ) be a local orthonormal frame of $TM$ over $U_i$ . We have $e_{i\alpha }=t_{ij}{e}_{j\alpha }$ where $t_{ij}:U_i\cap U_j\to \mathrm{O}(m)$ is the transition function. Define the Čech 1-cochain $f(i,j)$ by

$$f(i,j)\equiv \det \left(t_{ij}\right)=\pm 1.\text{\hspace{1em}(11.135)}$$

This is, indeed, an element of $C^1(M;{\mathbb{Z}}_2)$ since $f(i,j)=f(j,i)$ . From the cocycle condition $t_{ij}{t}_{jk}{t}_{ki}=I$ , we verify that

$$\begin{array}{rl}\delta f(i,j,k)& =\det \left(t_{ij}\right)\det \left(t_{jk}\right)\det \left(t_{ki}\right)\\ & =\det (t_{ij}{t}_{jk}{t}_{ki})=1.\end{array}\text{\hspace{1em}(11.136)}$$

Hence, $f\in Z^1(M,{\mathbb{Z}}_2)$ and it defines an element $[f]$ of $H^1(M;{\mathbb{Z}}_2)$ . Now we show that this element is independent of the local frame chosen. Let $\{{\bar{e}}_{i\alpha }\}$ be another frame over $U_i$ such that ${\bar{e}}_{i\alpha }=h_i{e}_{i\alpha }$ , $h_i\in \mathrm{O}(m)$ . From ${\bar{e}}_{i\alpha }={\bar{t}}_{ij}{\bar{e}}_{j\alpha }$ , we find ${\bar{t}}_{ij}=h_i{t}_{ij}{h}_j^{-1}$ . If we define the 0-cochain $f_0$ by $f_0(i)\equiv \det h_i$ , we find that

$$\begin{array}{rl}\tilde{f}(i,j)& =\det \left(h_i{t}_{ij}{h}_j^{-1}\right)=\det \left(h_i\right)\det \left(h_j\right)\det \left(t_{ij}\right)\\ & =\delta f_0(i,j)f(i,j)\end{array}$$

where use has been made of the identity $\det h_j^{-1}=\det h_j$ for $h_j\in \mathrm{O}(m)$ . Thus, $f$ changes by an exact amount and still defines the same cohomology class $[f]$ .¹

This special element $w_1(M)\equiv [f]\in H^1(M;{\mathbb{Z}}_2)$ is called the first Stiefel-Whitney class.

Theorem 11.6

Let $TM\stackrel{\pi }{\to }M$ be a tangent bundle with fibre metric. $M$ is orientable if and only if $w_1(M)$ is trivial.

Proof

If $M$ is orientable, the structure group may be reduced to $\mathrm{SO}(m)$ and $f(i,j)=\det (t_{ij})=1$ , and hence $w_1(M)=1$ , the unit element of ${\mathbb{Z}}_2$ . Conversely, if $w_1(M)$ is trivial, $f$ is a coboundary; $f=\delta f_0$ . Since $f_0(i)=\pm 1$ , we can always choose $h_i\in \mathrm{O}(m)$ such that $\det (h_i)=f_0(i)$ for each $i$ . If we define the new frame ${\bar{e}}_{i\alpha }=h_i{e}_{i\alpha }$ , we have transition functions ${\tilde{t}}_{ij}$ such that $\det ({\tilde{t}}_{ij})=1$ for any overlapping pair $(i,j)$ and $M$ is orientable. [Suppose $f(i,j)=\det t_{ij}=-1$ for some pair $(i,j)$ . Then we may take $f_0(i)=-1$ and $f_0(j)=+1$ , hence $\det {\tilde{t}}_{ij}=-\det t_{ij}=+1$ .]

Theorem 11.6 shows that the first Stiefel-Whitney class is an obstruction to the orientability. Next we define the second Stiefel-Whitney class. Suppose $M$ is an $m$ -dimensional orientable manifold and $TM$ is its tangent bundle. For the transition function $t_{ij}\in \mathrm{SO}(m)$ , we consider a ‘lifting’, ${\tilde{t}}_{ij}\in \mathrm{SPIN}(m)$ such that

$$\phi \left({\tilde{t}}_{ij}\right)=t_{ij}{\tilde{t}}_{ji}={\tilde{t}}_{ij}^{-1}\text{\hspace{1em}(11.137)}$$

where $\phi :\mathrm{SPIN}(m)\to \mathrm{SO}(m)$ is the $2:1$ homomorphism (note that we have an option $t_{ij}\leftrightarrow {\tilde{t}}_{ij}$ or $-{\tilde{t}}_{ij}$ ). This lifting always exists locally. Since

$$\phi \left({\tilde{t}}_{ij}{\tilde{t}}_{jk}{\tilde{t}}_{ki}\right)=t_{ij}{t}_{jk}{t}_{ki}=I$$

we have ${\tilde{t}}_{ij}{\tilde{t}}_{jk}{\tilde{t}}_{ki}\in \ker \phi =\{\pm I\}$ . For ${\tilde{t}}_{ij}$ to define a spin bundle over $M$ , they must satisfy the cocycle condition,

$${\tilde{t}}_{ij}{\tilde{t}}_{jk}{\tilde{t}}_{ki}=I.\text{\hspace{1em}(11.138)}$$

Define the Čech 2-cochain $f:U_i\cap U_j\cap U_k\to {\mathbb{Z}}_2$ by

$${\tilde{t}}_{ij}{\tilde{t}}_{jk}{\tilde{t}}_{ki}=f(i,j,k)I.\text{\hspace{1em}(11.139)}$$

It is easy to see that $f$ is symmetric and closed. Thus, $f$ defines an element $w_2(M)\in H^2(M,{\mathbb{Z}}_2)$ called the second Stiefel-Whitney class. It can be shown that $w_2(M)$ is independent of the local frame chosen.

Exercise 11.5

Suppose we take another lift $-{\tilde{t}}_{ij}$ of $t_{ij}$ . Show that $f$ changes by an exact amount under this change. Accordingly, $[f]$ is independent of the lift. [Hint: Show that $f(i,j,k)\to f(i,j.k)\delta f_1(i,j,k)$ where $f_1(i,j)$ denotes the sign of $\pm {\tilde{t}}_{ij}$ .]

Theorem 11.7

Let $TM$ be the tangent bundle over an orientable manifold $M$ . There exists a spin bundle over $M$ if and only if $w_2(M)$ is trivial.

Proof

Suppose there exists a spin bundle over $M$ . Then we define a set of transition functions ${\tilde{t}}_{ij}$ such that ${\tilde{t}}_{ij}{\tilde{t}}_{jk}{\tilde{t}}_{ki}=I$ over any overlapping charts $U_i,U_j$ and $U_k$ , hence $w_2(M)$ is trivial. Conversely, suppose $w_2(M)$ is trivial, namely

$$f(i,j,k)=\delta f_1(i,j,k)=f_1(j,k)f_1(i,k)f_1(k,i)$$

$f_1$ being a 1-cochain. We consider the 1-cochain $f_1(i,j)$ defined in exercise 11.5. If we choose new transition functions ${\tilde{t}}_{ij}^{\prime }\equiv {\tilde{t}}_{ij}{f}_1(i,j)$ , we have

$${\tilde{t}}_{ij}^{\prime }{\tilde{t}}_{jk}^{\prime }{\tilde{t}}_{ki}^{\prime }=[\delta f_1(i,j,k){]}^2=I$$

and, hence, $\{{\vec{i}}_{ij}^{\prime }\}$ defines a spin bundle over $M$ .

We outline some useful results:

(a)

$$w_1\left(\mathbb{C}{P}^m\right)=1w_2\left(\mathbb{C}{P}^m\right)=\{\begin{array}{ll}1& m\text{o d d}\\ x& m\text{e v e n}\end{array}\text{\hspace{1em}(11.140)}$$

$x$ being the generator of $H^2({\mathbb{CP}}^m;{\mathbb{Z}}_2)$

(b)

$$w_1\left(S^m\right)=w_2\left(S^m\right)=1\text{\hspace{1em}(11.141)}$$

(c)

$$w_1\left({\Sigma }_g\right)=w_2\left({\Sigma }_g\right)=1\text{\hspace{1em}(11.142)}$$

${\Sigma }_g$ being the Riemann surface of genus $g$ .

Endnotes

来源与版本

正文：英文 · OCR 机器稿 · 待校对

核对状态：OCR 机器稿 · 待校对

原书 PDF

来源版本：second-edition-local

来源 PDF SHA-256：fcd24c3371d092a8c699ccf5cf0765462924b6c48412804231076e8084b1940d

OCR 来源 SHA-256：0b6324f01172a3548cd34382e82fd8bfae574602b6b2cfaa357ba96bc2c35478

OCR 产物 SHA-256：d4cce2cdc5a164ab8716b3c0bcbd854d55d606924ba80e4a8be0104bba0d42e3

Footnotes

1. Note that the multiplicative notation is being used. ↩