Machine-translated from Chinese.
与实向量丛的差异
复共轭丛
一个秩$2n$实向量丛的复结构指一个连续映射$J:E(\xi)\rightarrow E(\xi),$
限制在每个纤维上为$\mathbb{R}$线性映射, 且$J(J(v))=-v.$
可以将$J$理解为虚部.
注 1. 配备复结构的秩$2n$实向量丛等价于秩$n$复向量丛. 设$M^{2n}$为实流形, $\tau_M$的复结构称为$M$上的近复结构. 实流形$M^{2n}$配备近复结构且$J$满足一些微分方程(C-R方程?)当且仅当$M^n$为复流形.
定义 2. 给定复向量丛$\omega,$ 它的复共轭为复向量丛$\overline\omega,$ 使得恒等映射$E(\omega)\xrightarrow{f=\mathrm{id} }E(\overline\omega)$是复共轭线性的, 即$f(\lambda e)=\overline\lambda e.$ 因此$J_{\overline\omega}=-J_{\omega}.$
假设$\omega$有一个Hermitian度量, 即
$$
\left<{}\lambda v,w\right>=\lambda\left<{}v,w\right>,\quad \left<{}v,\lambda w\right>=\overline\lambda\left<{}v,w\right>,\quad \left<{}v,w\right>=\overline{\left<{}w,v\right>}.
$$
那么有$\overline{\omega}\xrightarrow{\cong_\mathbb{C} } \operatorname{Hom}_{\mathbb{C} }(\omega,\varepsilon)=:\omega^V,$
$v\mapsto(w\mapsto\left<{}w,v\right>).$
一般地, 作为复向量丛, $\omega\not\cong \overline\omega.$
这和实版本很不一样.
一个例子是考虑$\gamma^1:\mathbb{C}\rightarrow E\rightarrow \mathbb{C}\mathrm{P}^\infty,$
$$
\gamma^1\otimes \overline{\gamma}^1\cong \gamma^1\otimes \operatorname{Hom}_\mathbb{C}(\gamma^1,\varepsilon)\cong \varepsilon.
$$
因此$c_1(\gamma^1)+c_1(\overline\gamma^1)=c_1(\varepsilon)=0,$
从而$c_1(\overline\gamma^1)=-c_1(\gamma^1)=\tau\in H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z}).$
这就说明了$\overline\gamma^1\not\cong \gamma^1.$
注 3. 在代数几何中, 记$\gamma^1$为$O(-1),$ $\overline\gamma^1=O(1).$ $\mathbb{C}\cong \mathbb{R}^2$上的复共轭改变定向, $(\overline\gamma^1)_\mathbb{R}=\overline{(\gamma^1)}_\mathbb{R}.$
SW类与Chern示性类
定理 4. 设$\xi$为复向量丛, 忘掉它的复结构, 那么$w_{2i+1}(\xi_{\mathbb{R} })=0,$ $w_{2i}(\xi_\mathbb{R})=c_i(\xi)\mod 2.$ 即$H^{2i}(B;\mathbb{Z})\rightarrow H^{2i}(B;\mathbb{Z}_2),$ $c_i(\xi)\mapsto w_{2i}(\xi_\mathbb{R}).$ 特别地, $w_{2n}(\xi_\mathbb{R})=c_n(\xi)=e(\xi_\mathbb{R})\mod 2.$
注意到, 我们有纤维丛
$$
\mathbb{R}\mathrm{P}^1\rightarrow \mathbb{R}\mathrm{P}^{2n-1}\rightarrow \mathbb{C}\mathrm{P}^{n-1}, \quad \ell\mapsto \ell\otimes_\mathbb{R}\mathbb{C}.
$$
考虑$\widehat f:E(\xi)\rightarrow \mathbb{C}^\infty,$
是$\mathbb{C}$线性的且依纤维单射, 有
$$
\mathbb{C}^n\hookrightarrow E(\xi)\xrightarrow{\widehat f}\mathbb{C}^\infty.
$$
考虑实投影丛,
$$
\mathbb{R}\mathrm{P}^{2n-1}\rightarrow \mathbb{R}\mathrm{P}(E(\xi)_\mathbb{R})\rightarrow \mathbb{R}\mathrm{P}^\infty;
$$
考虑复投影丛,
$$
\mathbb{C}\mathrm{P}^{n-1}\rightarrow \mathbb{C}\mathrm{P}(E(\xi))\rightarrow \mathbb{C}\mathrm{P}^\infty.
$$
这两行中每一列又都是纤维为$\mathbb{R}\mathrm{P}^1$的纤维丛.
这样我们可以得到$H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z})\rightarrow H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z}_2)\rightarrow H^2(\mathbb{R}\mathrm{P}^\infty;\mathbb{Z}_2),$
$\beta\mapsto \overline\beta\mapsto \alpha^2,$ 可由Leray-Hirsch证明.
对于右侧第一个方块,
$H^2(\mathbb{R}\mathrm{P}(E);\mathbb{Z}_2)\leftarrow H^2(\mathbb{R}\mathrm{P}^\infty;\mathbb{Z}_2),$
$x^2(\xi_\mathbb{R})\leftarrow \alpha^2;$
$H^2(\mathbb{C}\mathrm{P}(E);\mathbb{Z}_2)\leftarrow H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z}_2),$
$\overline x_\mathbb{C}(\xi)\leftarrow \overline\beta.$
那么模$2$的Chern类依定义,
$$
\overline x_\mathbb{C}(\xi)^n+\overline c_1(\xi)\overline x_\mathbb{C}(\xi)^{n-1}+\cdots+\overline c_n(\xi)=0\in H^{2n}(\mathbb{C}\mathrm{P}(\xi);\mathbb{Z}_2),
$$
打到了
$$
x(\xi_{\mathbb{R} })^{2n}+\overline c_1(\xi)[x(\xi_\mathbb{R})^2]^{n-1}+\cdots +\overline c_n(\xi)=0\in H^{2n}(\mathbb{R}\mathrm{P}(\xi_\mathbb{R});\mathbb{Z}_2).
$$
示性类与阻碍性理论
阻碍性理论
何时纤维丛$F\rightarrow E\rightarrow B$有一个截面?
想法是若$B$为一个CW复形, 考虑$B^0\subset B^1\subset B^2\subset \cdots,$
尝试按顺序定义骨架上的截面.
定理 5. 假设$F\rightarrow E\rightarrow B$为一个纤维丛, $B$为CW复形, $F$是$n$-simple的, 即$\pi_1(F,x_0)$作用在$\pi_n(F,x_0)$上是平凡的. 令$s$为$B^{n-1}$截面, 可被延拓为$B^n$上截面. 那么存在障碍类$ob(s)\in H^{n+1}(B;\pi_n(F))$(局部系数), 使得$ob(s)=0$当且仅当它可以被延拓到$B^{n+1}$上.
注 6. 找到一个截面的同伦等于找到$F\rightarrow E\times I\rightarrow B\times I\hookleftarrow B\times\partial I$延拓两个截面$s,s'.$ 阻碍在$H^{n+1}(B\times I,B\times\partial I;\pi_n(F))\cong \pi^n(B;\pi_n(F))$中.
注 7. 假设$F$($n-1$)连通, 截面的障碍在$H^i(B;\pi_{i-1}(F))=0$中, $i\le n;$ 截面同伦的障碍在$H^i(B;\pi_i(F))=0$中, $i\le n-1.$ 因此存在$B^n$上的截面, 并且$s|_{B^{n-1} }$在同伦意义下唯一. $ob(s)\in H^{n+1}(B;\pi_n(F))$与截面选取无关. 这称为第一障碍类.
文章最后更新于 2022-11-09 17:19:15
Difference from real vector bundle
complex conjugate bundle
The complex structure of a real vector bundle of rank $2n$ refers to a continuous map $J:E(\xi)\rightarrow E(\xi),$
Restricted to a linear map of $\mathbb{R}$ on each fiber, and $J(J(v))=-v.$
$J$ can be understood as the imaginary part.
Note 1. A real vector bundle of rank $2n$ equipped with a complex structure is equivalent to a complex vector bundle of rank $n$. Assume $M^{2n}$ is a real manifold, and the complex structure of $\tau_M$ is called a near-complex structure on $M$. The real manifold $M^{2n}$ is equipped with a near-complex structure and $J$ satisfies some differential equations (C-R equations?) if and only if $M^n$ is a complex manifold.
Definition 2. Given a complex vector bundle $\omega,$, its complex conjugate is the complex vector bundle $\overline\omega,$, so that the identity map $E(\omega)\xrightarrow{f=\mathrm{id} }E(\overline\omega)$ is complex conjugate linear, that is, $f(\lambda e)=\overline\lambda e.$ and therefore $J_{\overline\omega}=-J_{\omega}.$
Suppose $\omega$ has a Hermitian metric, that is
$$
\left<{}\lambda v,w\right>=\lambda\left<{}v,w\right>,\quad \left<{}v,\lambda w\right>=\overline\lambda\left<{}v,w\right>,\quad \left<{}v,w\right>=\overline{\left<{}w,v\right>}.
$$
Then there is $\overline{\omega}\xrightarrow{\cong_\mathbb{C} } \operatorname{Hom}_{\mathbb{C} }(\omega,\varepsilon)=:\omega^V,$
$v\mapsto(w\mapsto\left<{}w,v\right>).$
In general, as a complex vector bundle, $\omega\not\cong \overline\omega.$
This is very different from the real version.
An example is to consider $\gamma^1:\mathbb{C}\rightarrow E\rightarrow \mathbb{C}\mathrm{P}^\infty,$
$$
\gamma^1\otimes \overline{\gamma}^1\cong \gamma^1\otimes \operatorname{Hom}_\mathbb{C}(\gamma^1,\varepsilon)\cong \varepsilon.
$$
Therefore $c_1(\gamma^1)+c_1(\overline\gamma^1)=c_1(\varepsilon)=0,$
Thus $c_1(\overline\gamma^1)=-c_1(\gamma^1)=\tau\in H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z}).$
This explains $\overline\gamma^1\not\cong \gamma^1.$
Note 3. In algebraic geometry, denote $\gamma^1$ as the complex conjugate change orientation on $O(-1),$ $\overline\gamma^1=O(1).$ $\mathbb{C}\cong \mathbb{R}^2$, $(\overline\gamma^1)_\mathbb{R}=\overline{(\gamma^1)}_\mathbb{R}.$
SW type and Chern type
Theorem 4. Let $\xi$ be a complex vector bundle, forget its complex structure, then $w_{2i+1}(\xi_{\mathbb{R} })=0,$ $w_{2i}(\xi_\mathbb{R})=c_i(\xi)\mod 2.$ is $H^{2i}(B;\mathbb{Z})\rightarrow H^{2i}(B;\mathbb{Z}_2),$ $c_i(\xi)\mapsto w_{2i}(\xi_\mathbb{R}).$ In particular, $w_{2n}(\xi_\mathbb{R})=c_n(\xi)=e(\xi_\mathbb{R})\mod 2.$
Notice that we have fibrous plexuses
$$
\mathbb{R}\mathrm{P}^1\rightarrow \mathbb{R}\mathrm{P}^{2n-1}\rightarrow \mathbb{C}\mathrm{P}^{n-1}, \quad \ell\mapsto \ell\otimes_\mathbb{R}\mathbb{C}.
$$
Consider $\widehat f:E(\xi)\rightarrow \mathbb{C}^\infty,$
is $\mathbb{C}$ linear and injective according to fibers, we have
$$
\mathbb{C}^n\hookrightarrow E(\xi)\xrightarrow{\widehat f}\mathbb{C}^\infty.
$$
Consider the real projection bundle,
$$
\mathbb{R}\mathrm{P}^{2n-1}\rightarrow \mathbb{R}\mathrm{P}(E(\xi)_\mathbb{R})\rightarrow \mathbb{R}\mathrm{P}^\infty;
$$
Consider the complex projection bundle,
$$
\mathbb{C}\mathrm{P}^{n-1}\rightarrow \mathbb{C}\mathrm{P}(E(\xi))\rightarrow \mathbb{C}\mathrm{P}^\infty.
$$
Each column in these two rows is a fiber bundle with fiber $\mathbb{R}\mathrm{P}^1$.
This way we can get $H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z})\rightarrow H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z}_2)\rightarrow H^2(\mathbb{R}\mathrm{P}^\infty;\mathbb{Z}_2),$
$\beta\mapsto \overline\beta\mapsto \alpha^2,$ can be proved by Leray-Hirsch.
For the first square on the right,
$H^2(\mathbb{R}\mathrm{P}(E);\mathbb{Z}_2)\leftarrow H^2(\mathbb{R}\mathrm{P}^\infty;\mathbb{Z}_2),$
$x^2(\xi_\mathbb{R})\leftarrow \alpha^2;$
$H^2(\mathbb{C}\mathrm{P}(E);\mathbb{Z}_2)\leftarrow H^2(\mathbb{C}\mathrm{P}^\infty;\mathbb{Z}_2),$
$\overline x_\mathbb{C}(\xi)\leftarrow \overline\beta.$
Then the Chern class modulo $2$ is defined as,
$$
\overline x_\mathbb{C}(\xi)^n+\overline c_1(\xi)\overline x_\mathbb{C}(\xi)^{n-1}+\cdots+\overline c_n(\xi)=0\in H^{2n}(\mathbb{C}\mathrm{P}(\xi);\mathbb{Z}_2),
$$
Hit
$$
x(\xi_{\mathbb{R} })^{2n}+\overline c_1(\xi)[x(\xi_\mathbb{R})^2]^{n-1}+\cdots +\overline c_n(\xi)=0\in H^{2n}(\mathbb{R}\mathrm{P}(\xi_\mathbb{R});\mathbb{Z}_2).
$$
Characteristic classes and hindrance theory
hindrance theory
When does a fiber bundle $F\rightarrow E\rightarrow B$ have a cross section?
The idea is that if $B$ is a CW complex, consider $B^0\subset B^1\subset B^2\subset \cdots,$
Try to define the sections on the skeleton sequentially.
Theorem 5. Suppose $F\rightarrow E\rightarrow B$ is a fiber bundle, $B$ is a CW complex, $F$ is $n$-simple, that is, the action of $\pi_1(F,x_0)$ on $\pi_n(F,x_0)$ is trivial. Let $s$ be the $B^{n-1}$ section, which can be extended to the upper section of $B^n$. Then there is an obstacle class $ob(s)\in H^{n+1}(B;\pi_n(F))$ (local coefficient), such that $ob(s)=0$ can be extended to $B^{n+1}$ if and only if it can be extended.
Note 6. Finding the homotopy of a section is equivalent to finding that $F\rightarrow E\times I\rightarrow B\times I\hookleftarrow B\times\partial I$ extends the two sections $s,s'.$ with the obstruction in $H^{n+1}(B\times I,B\times\partial I;\pi_n(F))\cong \pi^n(B;\pi_n(F))$.
Note 7. Assume that $F$($n-1$) are connected, the obstacles of the section are in $H^i(B;\pi_{i-1}(F))=0$, $i\le n;$ and the obstacles of the homotopy section are in $H^i(B;\pi_i(F))=0$, $i\le n-1.$ therefore there is a section on $B^n$, and $s|_{B^{n-1} }$ is unique in the sense of homotopy. $ob(s)\in H^{n+1}(B;\pi_n(F))$ has nothing to do with the selection of the section. This is called the first obstacle type.
The article was last updated on 2022-11-09 17:19:15
