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对于纤维丛
$$
F\rightarrow E\xrightarrow{\pi} B,\quad \pi_1B=0.
$$
$$
E^2_{p,q}=H_p(B;H_q(F))\Rightarrow E^\infty_{p,q}=G_pH_{p+q}(E).
$$
考虑一些例子. 对于纤维丛
$$
S^1\rightarrow S^\infty\rightarrow \mathbb{C}\mathrm{P}^\infty,
$$
$$
E^2_{p,q}=H_p(\mathbb{C}\mathrm{P}^\infty)\otimes H_q(S^1),
$$
由$S^\infty$的同调,
$E^2_{p,q}$上的$\partial_2:E^2_{2k,0}\cong E^2_{2k-2,1}$必须是同构,
$k\ge 1.$
对于Hopf纤维化
$$
S^1\rightarrow S^3\rightarrow S^2=\mathbb{C}\mathrm{P}^1,
$$
也可以通过类似的方法得到谱序列的结构. 计算$\partial_2$很少直接计算,
第一种常见的计算方式就是上面的倒推法, 另一种是利用函子性质.
注意到第二个例子中的纤维丛是第一个例子的拉回,
由映射$\mathbb{C}\mathrm{P}^1\rightarrow \mathbb{C}\mathrm{P}^\infty.$
那么由于Leray-Serre谱序列有函子性质, 链映射与微分可交换.
这样就可以通过某个谱序列的微分得到另一个谱序列的微分.
局部系数
对拓扑空间$B,$ 记其万有覆盖空间为$\widetilde{B},$ $\pi=\pi_1(B),$
$M$是一个(左)$\mathbb{Z}[\pi]$模($\pi$作用在$M$上). 定义
$$
C_n(B;M):=C_n(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]} M,
$$
那么可以如下行定义$\pi$作用在$\widetilde{B}$上,
诱导$\pi$在$C_n(\widetilde{B})$上的作用,
从而$C_n(\widetilde{B})$是一个(左)$\mathbb{Z}[\pi]$模.
$\,\forall\,c\in C_n(\widetilde{B}),$ $m\in M,$ $r\in \pi,$
$$
(r^{-1}c)\otimes m=c\otimes (rm), \text{或} (rc)\otimes (rm)=c\otimes m.
$$
$-\bigotimes_{\mathbb{Z}_\pi}M$是一个函子, $C_n(B;M)$是链复形.
它的同调$H_n(B,M)$是局部系数在$M$中的$B$的同调.
若$\pi_1B$在$M$上作用平凡, $B,\widetilde{B}$为CW复形. 那么
$$
C_n^{CW}(B;M)=C_n^{CW}(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]}M=C_n^{CW}(B;\mathbb{Z})\bigotimes_{\mathbb{Z} }M,
$$
$H_n(B;M)$就是$B$的系数在$M$中的同调.
若$M=\mathbb{Z}[\pi],$
$$
C_n(B;\mathbb{Z}[\pi])=C_n(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]}\mathbb{Z}[\pi]=C_n(\widetilde{B}),
$$
因此$H_n(B;\mathbb{Z}[\pi])=H_n(\widetilde{B}).$
若$M:=\mathbb{Z}[\pi/\pi'],$ $\pi'\subset \pi.$ 那么
$$
C_n(B;\mathbb{Z}[\pi/\pi'])=C_n(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]}\mathbb{Z}[\pi/\pi']=C_n(B'),\quad B'=\widetilde{B}/\pi'.
$$
$$
H_n(B;\mathbb{Z}[\pi/\pi'])=H_n(B';\mathbb{Z}).
$$
类似地, 定义
$$
C^n(B;M)=\operatorname{Hom}_{\mathbb{Z}[\pi]}(C_n(\widetilde{B}),M),
$$
为一个上链复形, 上同调记为$H^n(B;M).$
若$B$为有限复形,
$$
H^n(B;\mathbb{Z}[\pi])\cong H_c^n(\widetilde{B};\mathbb{Z})
$$
对于不可定向流形上的Poincaré对偶, 记$N$为闭的不可定向$n$维流形,
$\pi_1(N,x)\xrightarrow{\rho} \{\pm 1\}$作用在$\mathbb{Z}\cong H(N,N\setminus \{x\})$上.
记$H_k(N;\mathbb{Z})$为整系数同调,
$H_k(N;\widetilde{\mathbb{Z} })$为关于$\rho$中局部系数同调.
定理 1. 对$n$维闭流形$N,$ $H^k(N;\mathbb{Z})\cong H_{n-k}(N;\widetilde{\mathbb{Z} }),$ $H^k(N;\widetilde{\mathbb{Z} })\cong H_{n-k}(N;\mathbb{Z}).$ 同构由和基本类$[N]\in H_n(N;\widetilde{\mathbb{Z} })$的卡积给出.
回到谱序列
考虑$\pi_1B$不一定非零的纤维化
$$
F\rightarrow E\rightarrow B.
$$
回忆每个$\gamma:[0,1]\rightarrow B$给出了同伦$L_\gamma:F_{\gamma(0)}\rightarrow F_{\gamma(1)}.$
若$B$道路连通,
$$
\pi_1(B,b)\rightarrow h\operatorname{Aut}(F_b)/\sim \rightarrow \operatorname{Aut}(H_q(F)),
$$
$$
\gamma\mapsto L_\gamma\mapsto L_{\gamma^\ast }.
$$
$\pi_1(B,b)$作用在$H_q(F_b)$上, 称为与纤维化关联的单值作用(monodromy
action).
对于纤维丛
$$
S^1\rightarrow U(2)\rightarrow \mathbb{R}\mathrm{P}^3,
$$
构造通过$U(1)\hookrightarrow U(2),$
$\frac{U(2)}{U(1)}=\frac{SU(2)\approx S^3}{\left\{\pm 1\right\} }=\mathbb{R}\mathrm{P}^3.$
事实上, $\pi_1(\mathbb{R}\mathrm{P}^3)$平凡地作用在$H_i(S^1)$上,
因为它的作用是个180度旋转, 这不改变$S^1$($H_1(S^1)$生成元)的定向.
文章最后更新于 2022-09-27 23:15:45
review
for fiber bundles
$$
F\rightarrow E\xrightarrow{\pi} B,\quad \pi_1B=0.
$$
$$
E^2_{p,q}=H_p(B;H_q(F))\Rightarrow E^\infty_{p,q}=G_pH_{p+q}(E).
$$
Consider some examples. For fiber bundles
$$
S^1\rightarrow S^\infty\rightarrow \mathbb{C}\mathrm{P}^\infty,
$$
$$
E^2_{p,q}=H_p(\mathbb{C}\mathrm{P}^\infty)\otimes H_q(S^1),
$$
From the homology of $S^\infty$,
$\partial_2:E^2_{2k,0}\cong E^2_{2k-2,1}$ on $E^2_{p,q}$ must be isomorphic,
$k\ge 1.$
For Hopf fibrosis
$$
S^1\rightarrow S^3\rightarrow S^2=\mathbb{C}\mathrm{P}^1,
$$
The structure of the spectral sequence can also be obtained by a similar method. Calculating $\partial_2$ is rarely calculated directly,
The first common calculation method is the backward calculation method above, and the other is to use the functor properties.
Notice that the fiber bundle in the second example is a pullback of the first example,
By mapping$\mathbb{C}\mathrm{P}^1\rightarrow \mathbb{C}\mathrm{P}^\infty.$
Then, since the Leray-Serre spectral sequence has functor properties, chain mapping and differential are commutative.
In this way, the differential of one spectral sequence can be obtained by the differential of another spectral sequence.
local coefficient
For topological space $B,$, record its universal coverage space as $\widetilde{B},$ $\pi=\pi_1(B),$
$M$ is a (left) $\mathbb{Z}[\pi]$ module ($\pi$ acts on $M$). Definition
$$
C_n(B;M):=C_n(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]} M,
$$
Then you can define $\pi$ to act on $\widetilde{B}$ as follows:
Inducing the effect of $\pi$ on $C_n(\widetilde{B})$,
Thus $C_n(\widetilde{B})$ is a (left) $\mathbb{Z}[\pi]$ module.
$\,\forall\,c\in C_n(\widetilde{B}),$ $m\in M,$ $r\in \pi,$
$$
(r^{-1}c)\otimes m=c\otimes (rm), \text{或} (rc)\otimes (rm)=c\otimes m.
$$
$-\bigotimes_{\mathbb{Z}_\pi}M$ is a functor, $C_n(B;M)$ is a chain complex.
Its homology $H_n(B,M)$ is the homology of $B$ with local coefficients in $M$.
If $\pi_1B$ acts trivially on $M$, $B,\widetilde{B}$ is a CW complex. Then
$$
C_n^{CW}(B;M)=C_n^{CW}(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]}M=C_n^{CW}(B;\mathbb{Z})\bigotimes_{\mathbb{Z} }M,
$$
$H_n(B;M)$ is the homology of the coefficients of $B$ in $M$.
If $M=\mathbb{Z}[\pi],$
$$
C_n(B;\mathbb{Z}[\pi])=C_n(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]}\mathbb{Z}[\pi]=C_n(\widetilde{B}),
$$
Therefore $H_n(B;\mathbb{Z}[\pi])=H_n(\widetilde{B}).$
If $M:=\mathbb{Z}[\pi/\pi'],$ $\pi'\subset \pi.$ then
$$
C_n(B;\mathbb{Z}[\pi/\pi'])=C_n(\widetilde{B})\bigotimes_{\mathbb{Z}[\pi]}\mathbb{Z}[\pi/\pi']=C_n(B'),\quad B'=\widetilde{B}/\pi'.
$$
$$
H_n(B;\mathbb{Z}[\pi/\pi'])=H_n(B';\mathbb{Z}).
$$
Similarly, define
$$
C^n(B;M)=\operatorname{Hom}_{\mathbb{Z}[\pi]}(C_n(\widetilde{B}),M),
$$
is a cochain complex, and cohomology is denoted as $H^n(B;M).$
If $B$ is a finite complex,
$$
H^n(B;\mathbb{Z}[\pi])\cong H_c^n(\widetilde{B};\mathbb{Z})
$$
For the Poincaré dual on a non-orientable manifold, let $N$ be a closed non-orientable $n$-dimensional manifold,
$\pi_1(N,x)\xrightarrow{\rho} \{\pm 1\}$ acts on $\mathbb{Z}\cong H(N,N\setminus \{x\})$.
Denote $H_k(N;\mathbb{Z})$ as integer coefficient homology,
$H_k(N;\widetilde{\mathbb{Z} })$ is homologous with respect to the local coefficients in $\rho$.
Theorem 1. The isomorphism for $n$-dimensional closed manifold $N,$ $H^k(N;\mathbb{Z})\cong H_{n-k}(N;\widetilde{\mathbb{Z} }),$ $H^k(N;\widetilde{\mathbb{Z} })\cong H_{n-k}(N;\mathbb{Z}).$ is given by the card product with the basic class $[N]\in H_n(N;\widetilde{\mathbb{Z} })$.
Return to spectral sequence
Consider fibrosis where $\pi_1B$ is not necessarily non-zero
$$
F\rightarrow E\rightarrow B.
$$
Recall that each $\gamma:[0,1]\rightarrow B$ gives the homotopy $L_\gamma:F_{\gamma(0)}\rightarrow F_{\gamma(1)}.$
If $B$ road is connected,
$$
\pi_1(B,b)\rightarrow h\operatorname{Aut}(F_b)/\sim \rightarrow \operatorname{Aut}(H_q(F)),
$$
$$
\gamma\mapsto L_\gamma\mapsto L_{\gamma^\ast }.
$$
$\pi_1(B,b)$ acts on $H_q(F_b)$ and is called fibrosis-related single value effect (monodromy
action).
for fiber bundles
$$
S^1\rightarrow U(2)\rightarrow \mathbb{R}\mathrm{P}^3,
$$
Construct passed $U(1)\hookrightarrow U(2),$
$\frac{U(2)}{U(1)}=\frac{SU(2)\approx S^3}{\left\{\pm 1\right\} }=\mathbb{R}\mathrm{P}^3.$
In fact, $\pi_1(\mathbb{R}\mathrm{P}^3)$ trivially acts on $H_i(S^1)$,
Because its effect is a 180-degree rotation, this does not change the orientation of $S^1$ ($H_1(S^1)$ generator).
The article was last updated on 2022-09-27 23:15:45
