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Künneth公式
设$M$, $F$是流形, 且$M$的de Rham上同调的维数是有限的, 则
$$H^n(M\times F)=\sum_{p+q=n} H^p(M)\otimes_\mathbb{R}H^q(F).$$
证: 假定$M$有有限好覆盖. 取$M\times F$到$F$, $M$的投射为$\rho,\pi$,
则其诱导$\psi:\Omega^{\ast }(M)\otimes \Omega^\ast (F)\rightarrow \Omega^\ast (M\times F)$,
$\psi(\omega\otimes \phi)=\pi^\ast \omega\wedge \rho^\ast \phi.$
由于$\pi^\ast$, $\rho^\ast$与$d$可交换, $\psi$可约化到上同调间的映射.
若$M\approx \mathbb{R}^m$(好覆盖仅为一个开集), 则由Poincaré引理,
$H^n(\mathbb{R}^m\times F)=H^n(\mathbb{R}^{m-1}\times F)=\cdots=H^n(F)=H^0(\mathbb{R}^m)\otimes H^n(F)= \sum_p H^{p}(\mathbb{R}^m)\otimes H^{n-p}(F).$
设$U,V$为$M$的开集, 固定$n$, 做上同调的MV序列.
将每项与$H^{n-p}(F)$做张量积, 再把每项关于$p$做直和, 得到新的MV序列.
该MV序列又可通过链映射$\psi$打到$M\times F$对应的MV序列上,
由zig-zag引理该图可交换.
交换图有点大...不打了先.
从而借助MV序列, 若Künneth公式对$U,V,U\cap V$成立, 其对$U\cup V$也成立.
由MV方法, 归纳好覆盖开集个数即可.
Leray-Hirsch定理
$M\times F$可看作$M$上以$F$为纤维的平凡纤维丛,
因此Künneth公式可以推广到更一般的情形, 这就是Leray-Hirsch定理.
先介绍一些纤维丛的概念.
设$G$是拓扑群, $F$是拓扑空间, 设$G$对$F$有一个左作用,
即$\,\forall\,g\in G$定义了一个连续同胚$g:F\rightarrow F$, 满足
$g_1(g_2(v))=(g_1g_2)(v)$, $\,\forall\,g_1,g_2\in G$, $v\in F.$
称作用是有效作用, 若$g(y)=y$,
$\,\forall\,y\in F\Rightarrow g=1\in G.$
在此基础上, 再加入$E,B$为拓扑空间, $\pi:E\rightarrow B$是连续满射.
设$B$有开覆盖$\{U_\alpha\}_{\alpha\in I}$满足:
(i) 对每个$\alpha\in I$, 有保纤维的同胚:
$\phi_\alpha:E_{U_\alpha}:=\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times F$,
即$\phi_\alpha$是同胚且$\operatorname{pr}_1\circ\phi_\alpha=\pi.$
(ii) $\phi_\alpha,\phi_\beta$诱导了$g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G$,
$(\phi_\alpha\circ \phi_\beta^{-1})(x,v)=(x,g_{\alpha\beta}(x)(v)).$
则称$\pi:E\rightarrow B$是以$F$为纤维的纤维丛, 结构群为$G$,
底空间为$B$, 全空间为$E$.
$\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$为纤维丛的一个局部平凡化.
对$x\in B$, $\pi^{-1}(x)$即称为点$x$的纤维, 记作$E_x$.
在研究上同调时, 假定$E,B,F$是光滑流形,
$G$无特殊说明为$F$的微分同胚群$\operatorname{Diff}(F)$.
设$\pi:E\rightarrow M$是以$F$为纤维的光滑纤维丛.
假定$E$上有上同调类$e_1,...,e_r$,
限制在每条纤维$E_x$上形成了$H^\ast (E_x)$的一组基, 则可定义
$$\psi:H^\ast (M)\otimes \mathbb{R}\{e_1,...,e_r\}\rightarrow H^\ast (E).$$
定理 1.1 (Leray-Hirsch 定理). 在上述假定下, 若$M$有有限好覆盖, 则$\psi$是同构, 有$H^\ast (E)\cong H^\ast (M)\otimes \mathbb{R}\{e_1,...,e_r\} \cong H^\ast (M)\otimes H^\ast (F).$
证: 由于有限好覆盖中的$U_\alpha\approx \mathbb{R}^n$为可缩空间,
其上纤维丛平凡,
对$E|_{U_\alpha}$的Leray-Hirsch定理退化为$U_\alpha\times F$的Künneth公式.
用MV方法即可借助好覆盖, 将该论断归纳到$M$上.
文章最后更新于 2021-09-22 18:30:43
Künneth formula
Assume $M$, $F$ are manifolds, and the dimension of de Rham cohomology of $M$ is finite, then
$$H^n(M\times F)=\sum_{p+q=n} H^p(M)\otimes_\mathbb{R}H^q(F).$$
Certificate: Assume that $M$ has finite good coverage. Take $M\times F$ to $F$, and the projection of $M$ is $\rho,\pi$,
Then it induces $\psi:\Omega^{\ast }(M)\otimes \Omega^\ast (F)\rightarrow \Omega^\ast (M\times F)$,
$\psi(\omega\otimes \phi)=\pi^\ast \omega\wedge \rho^\ast \phi.$
Since $\pi^\ast$, $\rho^\ast$ and $d$ are commutative, $\psi$ can be reduced to the mapping between cohomologies.
If $M\approx \mathbb{R}^m$ (a good covering is only an open set), then according to Poincaré’s lemma,
$H^n(\mathbb{R}^m\times F)=H^n(\mathbb{R}^{m-1}\times F)=\cdots=H^n(F)=H^0(\mathbb{R}^m)\otimes H^n(F)= \sum_p H^{p}(\mathbb{R}^m)\otimes H^{n-p}(F).$
Let $U,V$ be the open set of $M$, fix $n$, and make a cohomological MV sequence.
Make a tensor product of each item with $H^{n-p}(F)$, and then make a direct sum of each item with respect to $p$ to get a new MV sequence.
The MV sequence can be mapped to the MV sequence corresponding to $M\times F$ through chain mapping $\psi$.
This graph is commutative by the zig-zag lemma.
The exchange diagram is a bit big... I won’t go into it first.
Therefore, with the help of the MV sequence, if Künneth's formula holds for $U,V,U\cap V$, it also holds for $U\cup V$.
Using the MV method, it is enough to summarize the number of open sets covered.
Leray-Hirsch theorem
$M\times F$ can be regarded as an ordinary fiber bundle on $M$ with $F$ as the fiber.
Therefore, Künneth's formula can be extended to a more general situation, which is Leray-Hirsch theorem.
Let’s first introduce some concepts of fiber bundles.
Let $G$ be a topological group, $F$ be a topological space, let $G$ have a pair of $F$ left acting,
That is, $\,\forall\,g\in G$ defines a continuous homeomorphism $g:F\rightarrow F$, satisfying
$g_1(g_2(v))=(g_1g_2)(v)$, $\,\forall\,g_1,g_2\in G$, $v\in F.$
The function is effective effect, if $g(y)=y$,
$\,\forall\,y\in F\Rightarrow g=1\in G.$
On this basis, $E,B$ is added as the topological space, and $\pi:E\rightarrow B$ is the continuous surjection.
Assume $B$ has open coverage $\{U_\alpha\}_{\alpha\in I}$ and satisfies:
(i) For each $\alpha\in I$, there is a fiber-preserving homeomorphism:
$\phi_\alpha:E_{U_\alpha}:=\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times F$,
That is, $\phi_\alpha$ is a homeomorphism and $\operatorname{pr}_1\circ\phi_\alpha=\pi.$
(ii) $\phi_\alpha,\phi_\beta$ induces $g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G$,
$(\phi_\alpha\circ \phi_\beta^{-1})(x,v)=(x,g_{\alpha\beta}(x)(v)).$
Then $\pi:E\rightarrow B$ is said to have $F$ as fiber fiber plexus, structural group is $G$,
base space is $B$, full space is $E$.
$\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ is one of the fiber bundles local trivialization.
For $x\in B$, $\pi^{-1}(x)$ is called the fiber at point $x$, recorded as $E_x$.
When studying cohomology, it is assumed that $E,B,F$ is a smooth manifold,
$G$ The diffeomorphism group $\operatorname{Diff}(F)$ not specified as $F$.
Let $\pi:E\rightarrow M$ be a smooth fiber bundle with $F$ as the fiber.
Assume that there is a cohomology class $e_1,...,e_r$ on $E$,
Restricting to a set of bases forming $H^\ast (E_x)$ on each fiber $E_x$, we can define
$$\psi:H^\ast (M)\otimes \mathbb{R}\{e_1,...,e_r\}\rightarrow H^\ast (E).$$
Theorem 1.1 (Leray-Hirsch theorem). Under the above assumption, if $M$ has finite good coverage, then $\psi$ is isomorphic, and there is $H^\ast (E)\cong H^\ast (M)\otimes \mathbb{R}\{e_1,...,e_r\} \cong H^\ast (M)\otimes H^\ast (F).$
Certificate: Since $U_\alpha\approx \mathbb{R}^n$ in finite good coverage is a shrinkable space,
The fibers on it are ordinary,
The Leray-Hirsch theorem for $E|_{U_\alpha}$ degenerates into the Künneth formula for $U_\alpha\times F$.
The MV method can be used to summarize this conclusion on $M$ with the help of good coverage.
The article was last updated on 2021-09-22 18:30:43
