《代数拓扑与微分形式》笔记(6-1)-向量丛基础 "Algebraic Topology and Differential Forms" Notes (6-1) — Fundamentals of Vector Bundles

Machine-translated from Chinese.

主要为了讨论紧垂直上同调与Thom类.

向量丛的局部平凡化

延续上节记号.

$E_x=\pi^{-1}(x)$是向量空间. 若$\,\exists\,\{U_\alpha\}$为$M$的开覆盖, 其上有微分同胚$\phi_\alpha:E_{U_\alpha}=\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times \mathbb{R}^n$, 将$E_x\mapsto \{x\}\times \mathbb{R}^n$, 且该映射为向量空间的同构, 则称$\pi$为秩$n$的光滑(实)向量丛, $\{(U_\alpha,\phi_\alpha)\}$为局部平凡化, $E$与$M$称为全空间与底流形.

向量丛为纤维为$\mathbb{R}^n$, 结构群为$\operatorname{GL}(n,\mathbb{R})$的纤维丛. 若将$\mathbb{R}$都换为$\mathbb{C}$, 即为复向量丛. 我们默认讨论光滑实向量丛.

光滑映射$s:U\rightarrow E$称为截面, 若$\pi\circ s=id$, 即要求$s(x)\in E_x$. $E$在$U$上所有截面全体记为$\Gamma(U,E)$. 零截面指$s: M\rightarrow E$, $s(x)=0_x\in E_x$. $U$上的$n$个截面$s_1,...,s_n$称为标架, 若对于每点$x\in U$, $s_1(x),...,s_n(x)$为$E_x$的一组基.

局部平凡化上的自然标架为$\varepsilon_1^\alpha, ...,\varepsilon_n^\alpha$, 由 $\varepsilon_i^\alpha(x)=\phi_\alpha^{-1}(x,e_i)$定义. 其中$e_1,...,e_n$为标准单位基. 反过来, 标架$s_1,...,s_n$也给出$E|_U$的平凡化: $\phi(\sum v_is_i(x))=(x,(v_1,...,v_n))$, 即将$E|_x$上的坐标直接打为$\mathbb{R}^n$上的坐标.

标架与局部平凡化之间的相互转化是研究向量丛的基本方法.

向量丛本质: 转移函数. $\,\forall\,x\in U_{\alpha\beta}$, $\phi_{\alpha}\circ(\phi_\beta)^{-1}:\{x\}\times \mathbb{R}^n\rightarrow \{x\}\times \mathbb{R}^n$为自同构. 存在$g_{\alpha\beta}(x)\in \operatorname{GL}(n,\mathbb{R})$使得, $\,\forall\,v\in \mathbb{R}^n$, $\phi_\alpha\phi_\beta^{-1}(x,v)=(x,g_{\alpha\beta}(x)v)$.

其满足cocycle条件: $g_{\alpha\beta}=g_{\beta\alpha}^{-1}\:\:\text{on}\:\:U_{\alpha\beta};$ $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma} \:\:\text{on}\:\:U_{\alpha\beta\gamma}.$

引理 1.1. 若有另一平凡化$\{(U_\alpha,\phi_\alpha')\}$诱导转移函数$g_{\alpha\beta}'$, 则存在光滑矩阵映射$\lambda_\alpha,\lambda_\beta$, 使得$g_{\alpha\beta}(x)=\lambda_\alpha(x)g'_{\alpha\beta}(x)\lambda^{-1}_\beta(x)$.

证: 注意到$\phi_\alpha\circ{\phi'_\alpha}^{-1}$为自同构, 可定义$\lambda_\alpha$, 使得$\phi_\alpha{\phi'_\alpha}^{-1}(x,v)=(x,\lambda_\alpha(x)v).$ 类似地, 定义$\lambda_\beta.$ 接下来, 在$\phi_\alpha\phi_\beta^{-1}$中合适地插入$\phi_\alpha',\phi_\beta'$ 即可.

定义两个转移函数等价即为存在这样的光滑矩阵映射使得他们相抵, 注意这里转移函数允许源于两个不同的向量丛. 所以同一向量丛两个不同平凡化, 对应的转移函数等价.

光滑映射$f:E\rightarrow E'$称为丛映射,H 若其将每根纤维映到纤维, 且是线性的. 也成为同态, 记其全体为$\operatorname{Hom}(E,E').$ 当限制在纤维上是同构时, 也称$f$为同构, $E$和$E'$同构.

结构群的约化

若$\pi:E\rightarrow M$存在局部平凡化, 转移函数只取值于$GL(n,\mathbb{R})$的子群$H$, 那么称向量丛$E$的结构群可约化到$H$.

也就是对于一般的转移函数$g$, 我们希望找到等价的$g'$只取值于较好的$H$. 一般的, $H$为(特殊)正交群和行列式为正的$GL^+(n,\mathbb{R}).$ 若确实可约化到$GL^+(n,\mathbb{R})$, 则称向量丛$E$可定向. 称局部平凡化定向, 若$\det g_{\alpha\beta}>0$. 称两个定向平凡化等价, 若 $\phi_\alpha{\phi_\alpha'}^{-1}$Jacobian为正.

这样若$M$连通, $E$的所有定向平凡化分为两个等价类, 每个等价类为一个定向. 给定定向的可定向向量丛称为定向向量丛.

命题 1.1. 秩$n$的向量丛$E$结构群总能约化到$O(n)$. 能约化到$SO(n)$当且仅当$E$可定向.

证: 通过自然标架诱导局部内积, 再利用单位分解合成整体内积. 之后做Gram-Schimit正交化定义新的正交单位标架, 给一个局部平凡化即可.

向量丛的计算

设$E$为秩$n$向量丛, $\{(U_\alpha,\phi_\alpha)\}$为局部平凡化, 转移函数$g_{\alpha\beta}$; $E'$为秩$m$向量丛, $\{(U_\alpha,\phi'_\alpha)\}$为局部平凡化, 转移函数$g'_{\alpha\beta}$. 可以假设两个开覆盖一样是因为$\{(U_\alpha,\phi_\alpha)\}$和$\{(U_{\alpha \beta},\phi_{\alpha})\}$是等价的, 开覆盖$\{U_{\alpha \beta}\}$为开覆盖$\{U_\alpha\}$和$\{U_\beta\}$交织而成.

直和: $E\oplus E'$为一个秩$n+m$向量丛, 局部平凡化为$\{(U_\alpha, \phi_\alpha\oplus\phi_\alpha')\}$, 转移函数为$\begin{pmatrix} g_{\alpha\beta}&0\\ 0&g_{\alpha\beta}' \end{pmatrix}$

张量积: $E\otimes E'$为一个秩$nm$向量丛, 局部平凡化为$\{(U_\alpha,\phi_\alpha\otimes\phi_\alpha')\}$, 转移函数为$g_{\alpha}\otimes g_{\alpha}'.$

$\operatorname{Hom}(E,E')$为$E$到$E'$丛映射全体, 同构于$E^\ast \otimes E'.$

对偶丛: 回忆泛函分析中, $f:V\rightarrow W$为有限维空间上的线性泛函, $f^\ast :W^\ast \rightarrow V^\ast$为对偶空间上的线性泛函, 它的表示矩阵为$f$表示矩阵的转置, 因此我们也以$f^t$记$f^\ast$.

$E^\ast$即为$E$的对偶空间, 称为对偶丛. 局部平凡化为$\{(U_\alpha,{(\phi^t_\alpha)}^{-1})\}$. 转移函数$(g_{\alpha\beta}^t)^{-1}$.

拉回丛: 设有$f:N\rightarrow M$为光滑映射, 则可构造向量丛: $\pi:f^{-1}E\rightarrow N$, 称为$E$由$f$的拉回丛.

$f^{-1}E=\{(y,e)|f(y)=\pi(e)\}\subset N\times E.$ 即对每点$y\in N$, 将$f(y)\in M$的纤维$E|_{f(y)}$移植过来称为$f^{-1}E|_{y}$. 它是$N\times E$的唯一最大子集, 使得$\pi \circ (f\times \mathrm{id}) =f\circ \operatorname{pr}_1$.

乘积丛的拉回丛为乘积丛, 因此$f^{-1}E$可局部平凡化, 转移函数为$f^\ast g_{\alpha\beta}$.

向量丛的同伦性

定理 1.1 (向量丛的同伦性质). 设$Y$是紧流形, $f_0,f_1:Y\rightarrow X$同伦且$\pi:E\rightarrow X$是向量丛, 则$f_0^{-1}E\cong f_1^{-1}E$, 即同伦映射诱导同构丛.

证: 只需证$\,\forall\,t_0\in I$, $\,\exists\,\varepsilon>0$, $\,\forall\,t\in O(t_0,\varepsilon)$, $f_{t}^{-1}E\cong f_{t_0}^{-1}E$. 由$I$紧性则结论成立.

其中$f:Y\times I\rightarrow X$为$f_0,f_1$间的同伦映射, $f_t(y)=f(y,t)$. 取$\pi:Y\times I\rightarrow Y$为投射, 则$f^{-1}E|_{Y\times \{t_0\} }\cong\pi^{-1} f_{t_0}^{-1}E|_{Y\times \{t_0\} },$ 即将$f(y,t_0)$分为两步:$f_{t_0}(\pi(y,t_0)).$

回忆$\operatorname{Hom}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$也是一个向量丛, 每根纤维是所有线性映射全体; 取其子群$\operatorname{Iso}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$, 每根纤维是其中的同构全体. 因此我们只需要给出一个$(Y,t_0)$附近的一个截面即可.

我们取$\operatorname{Hom}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$的一个局部平凡化$\{(U_\alpha,\phi_\alpha)\}$, 由于$Y,I$紧, 由管状邻域引理(拓扑学), $\,\exists\,Y\times O(t_0,\varepsilon)$被有限个$U_\alpha$包含, 使得每个$(y,O(t_0,\varepsilon))$在某个$U_\alpha$内.

取$Y$的坐标邻域$\{V_k\}$覆盖$Y$, 使得$V_k\times O(t_0,\varepsilon)$包含在某个$U_\alpha$内. 截面$s(y,t_0)$已经在$Y$上整体定义. 在每个局部平凡化邻域内, $\operatorname{Hom}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$的$V_k\times O(t_0,\varepsilon)$上的纤维, 均可视为欧氏空间, 从而在其上任意将截面向外延拓. 即从$s(y,t_0)$出发, 延拓为$s(y,t)$, $(y,t)\in V_k\times O(t_0,\varepsilon).$ 进一步, 利用从属于$\{V_k\}$的单位分解将每个$V_k$上的延拓合并, 形成$Y\times O(t_0,\varepsilon)$上的整体延拓, 从而截面在条状区域内整体定义.

延拓时注意, 非同构线性映射全体为闭集, 因此我们可以在开集内进行延拓. 截面给出了, 从而同构在$t_0$附近成立. 由紧性, 在$I$上成立, 特别地, $f_0^{-1}E\cong f_1^{-1}E.$

推论 1.1. 可缩空间上的向量丛是平凡丛.

文章最后更新于 2021-09-23 14:45:08

Mainly to discuss tight vertical cohomology and Thom classes.

Local trivialization of vector bundles

Continuation of the previous section’s mark.

$E_x=\pi^{-1}(x)$ is a vector space. If $\,\exists\,\{U_\alpha\}$ is the open cover of $M$, There is a diffeomorphism $\phi_\alpha:E_{U_\alpha}=\pi^{-1}(U_\alpha)\rightarrow U_\alpha\times \mathbb{R}^n$ on it, Let $E_x\mapsto \{x\}\times \mathbb{R}^n$, and this mapping be the isomorphism of the vector space, Then $\pi$ is called smooth (real) of rank $n$ vector bundle, $\{(U_\alpha,\phi_\alpha)\}$ is local trivialization, $E$ and $M$ are called full space with bottom manifold.

The vector bundle is the fiber $\mathbb{R}^n$, The structure group is $\operatorname{GL}(n,\mathbb{R})$ fiber plexus. If $\mathbb{R}$ is replaced with $\mathbb{C}$, it is a complex vector bundle. By default we discuss smooth real vector bundles.

The smooth map $s:U\rightarrow E$ is called Section, if $\pi\circ s=id$, That is, it is required that $s(x)\in E_x$. $E$ and all sections on $U$ are collectively recorded as $\Gamma(U,E)$. zero section Refers to $s: M\rightarrow E$, $s(x)=0_x\in E_x$. $n$ sections $s_1,...,s_n$ on $U$ are called frame, if for each point $x\in U$, $s_1(x),...,s_n(x)$ is a set of bases of $E_x$.

The natural frame on local trivialization is $\varepsilon_1^\alpha, ...,\varepsilon_n^\alpha$, Defined by $\varepsilon_i^\alpha(x)=\phi_\alpha^{-1}(x,e_i)$. Where $e_1,...,e_n$ is the standard unit base. In turn, Frame $s_1,...,s_n$ also gives the trivialization of $E|_U$: $\phi(\sum v_is_i(x))=(x,(v_1,...,v_n))$, That is, the coordinates on $E|_x$ are directly converted to the coordinates on $\mathbb{R}^n$.

The mutual transformation between the frame and local trivialization is the basic method for studying vector bundles.

The essence of vector bundles: transfer functions. $\,\forall\,x\in U_{\alpha\beta}$, $\phi_{\alpha}\circ(\phi_\beta)^{-1}:\{x\}\times \mathbb{R}^n\rightarrow \{x\}\times \mathbb{R}^n$ is an automorphism. There exists $g_{\alpha\beta}(x)\in \operatorname{GL}(n,\mathbb{R})$ such that, $\,\forall\,v\in \mathbb{R}^n$, $\phi_\alpha\phi_\beta^{-1}(x,v)=(x,g_{\alpha\beta}(x)v)$.

its satisfaction cocycle conditions: $g_{\alpha\beta}=g_{\beta\alpha}^{-1}\:\:\text{on}\:\:U_{\alpha\beta};$ $g_{\alpha\beta}g_{\beta\gamma}=g_{\alpha\gamma} \:\:\text{on}\:\:U_{\alpha\beta\gamma}.$

Lemma 1.1. If there is another trivialization $\{(U_\alpha,\phi_\alpha')\}$ that induces the transfer function $g_{\alpha\beta}'$, then there is a smooth matrix mapping $\lambda_\alpha,\lambda_\beta$ such that $g_{\alpha\beta}(x)=\lambda_\alpha(x)g'_{\alpha\beta}(x)\lambda^{-1}_\beta(x)$.

Certificate: Note that $\phi_\alpha\circ{\phi'_\alpha}^{-1}$ is an automorphism, Definable $\lambda_\alpha$, Make$\phi_\alpha{\phi'_\alpha}^{-1}(x,v)=(x,\lambda_\alpha(x)v).$ Similarly, define $\lambda_\beta.$ next, Insert $\phi_\alpha',\phi_\beta'$ appropriately within $\phi_\alpha\phi_\beta^{-1}$ That’s it.

Defining the equivalence of two transfer functions means that there is such a smooth matrix mapping that makes them cancel out, Note that the transfer function here is allowed to originate from two different vector bundles. So two different trivializations of the same vector bundle, The corresponding transfer functions are equivalent.

The smooth map $f:E\rightarrow E'$ is called a bundle map, H if it maps each fiber to fiber, And it is linear. It also becomes a homomorphism, and its entire body is written as $\operatorname{Hom}(E,E').$ When the isomorphism is restricted to the fibers, $f$ is also called isomorphism, and $E$ and $E'$ are isomorphisms.

Reduction of structural groups

If $\pi:E\rightarrow M$ has local trivialization, The transfer function only takes the value of the subgroup $H$ of $GL(n,\mathbb{R})$, Then the structural group of vector bundle $E$ can be called reduce to $H$.

That is, for the general transfer function $g$, we hope to find the equivalent $g'$ that only takes the value of the better $H$. Generally, $H$ is a (special) orthogonal group and $GL^+(n,\mathbb{R}).$ whose determinant is positive If it can indeed be reduced to $GL^+(n,\mathbb{R})$, it is called a vector bundle $E$ Orientable. local trivialization Orientation, if $\det g_{\alpha\beta}>0$. two directional trivializations Equivalent, if $\phi_\alpha{\phi_\alpha'}^{-1}$Jacobian is positive.

In this way, if $M$ is connected, all orientations of $E$ are trivially divided into two equivalence classes, and each equivalence class is an orientation. A directional vector bundle with a given orientation is called a directional vector bundle.

Proposition 1.1. The vector bundle $E$ structure group of rank $n$ can always be reduced to $O(n)$. It can be reduced to $SO(n)$ if and only if $E$ can be oriented.

Certificate: The local inner product is induced through the natural frame, and then the partition of unity is used to synthesize the global inner product. Then do Gram-Schimit orthogonalization to define a new orthogonal unit frame and give it a local trivialization.

Calculation of vector bundles

Let $E$ be a vector bundle of rank $n$, $\{(U_\alpha,\phi_\alpha)\}$ be a local trivialization, Transfer function $g_{\alpha\beta}$; $E'$ is the rank $m$ vector bundle, $\{(U_\alpha,\phi'_\alpha)\}$ is local trivialization, transfer function $g'_{\alpha\beta}$. It can be assumed that the two open covers are the same because $\{(U_\alpha,\phi_\alpha)\}$ and $\{(U_{\alpha \beta},\phi_{\alpha})\}$ are equivalent, Open coverage $\{U_{\alpha \beta}\}$ is intertwined with open coverage $\{U_\alpha\}$ and $\{U_\beta\}$.

Naowa: $E\oplus E'$ is a vector bundle of rank $n+m$, Locally trivialized to $\{(U_\alpha, \phi_\alpha\oplus\phi_\alpha')\}$, The transfer function is $\begin{pmatrix} g_{\alpha\beta}&0\\ 0&g_{\alpha\beta}' \end{pmatrix}$

tensor product: $E\otimes E'$ is a vector bundle of rank $nm$, Locally trivialized to $\{(U_\alpha,\phi_\alpha\otimes\phi_\alpha')\}$, The transfer function is $g_{\alpha}\otimes g_{\alpha}'.$

$\operatorname{Hom}(E,E')$ is the entire cluster mapping from $E$ to $E'$, Isomorphic to $E^\ast \otimes E'.$

dual bundle: Recall that in functional analysis, $f:V\rightarrow W$ is a linear functional on a finite-dimensional space, $f^\ast :W^\ast \rightarrow V^\ast$ is the linear functional on the dual space, Its representation matrix is $f$, which represents the transpose of the matrix, so we also use $f^t$ to denote $f^\ast$.

$E^\ast$ is the dual space of $E$, which is called the dual bundle. Locally trivialized to $\{(U_\alpha,{(\phi^t_\alpha)}^{-1})\}$. Transfer function$(g_{\alpha\beta}^t)^{-1}$.

Pull back to the bush: Assuming that $f:N\rightarrow M$ is a smooth mapping, a vector bundle can be constructed: $\pi:f^{-1}E\rightarrow N$, is called the pullback bundle of $E$ from $f$.

$f^{-1}E=\{(y,e)|f(y)=\pi(e)\}\subset N\times E.$ That is, for each point $y\in N$, Transplanting the fibers $E|_{f(y)}$ from $f(y)\in M$ is called $f^{-1}E|_{y}$. It is the single largest subset of $N\times E$, Make$\pi \circ (f\times \mathrm{id}) =f\circ \operatorname{pr}_1$.

The pullback bundle of the product bundle is the product bundle, so $f^{-1}E$ can be locally trivialized, The transfer function is $f^\ast g_{\alpha\beta}$.

Homotopy of vector bundles

Theorem 1.1 (Homotopy property of vector bundles). Assume $Y$ is a compact manifold, $f_0,f_1:Y\rightarrow X$ is homotopy and $\pi:E\rightarrow X$ is a vector bundle, then $f_0^{-1}E\cong f_1^{-1}E$ is a homotopy map-induced isomorphism bundle.

Certificate: Just prove $\,\forall\,t_0\in I$, $\,\exists\,\varepsilon>0$, $\,\forall\,t\in O(t_0,\varepsilon)$, $f_{t}^{-1}E\cong f_{t_0}^{-1}E$. From the compactness of $I$, the conclusion holds.

Where $f:Y\times I\rightarrow X$ is the homotopy mapping between $f_0,f_1$, $f_t(y)=f(y,t)$. Take $\pi:Y\times I\rightarrow Y$ as the projection, Then $f^{-1}E|_{Y\times \{t_0\} }\cong\pi^{-1} f_{t_0}^{-1}E|_{Y\times \{t_0\} },$ Divide $f(y,t_0)$ into two steps: $f_{t_0}(\pi(y,t_0)).$

Recall that $\operatorname{Hom}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$ is also a vector bundle, Each fiber is the ensemble of all linear maps; Take its subgroup $\operatorname{Iso}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$, Each fiber is an isomorphic whole within it. Therefore we only need to give a section near $(Y,t_0)$.

We take a local trivialization $\{(U_\alpha,\phi_\alpha)\}$ of $\operatorname{Hom}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$, Since $Y,I$ is compact, by the tubular neighborhood lemma (topology), $\,\exists\,Y\times O(t_0,\varepsilon)$ is contained by a limited number of $U_\alpha$, Make every $(y,O(t_0,\varepsilon))$ within a certain $U_\alpha$.

Take the coordinate neighborhood $\{V_k\}$ of $Y$ to cover $Y$, Make $V_k\times O(t_0,\varepsilon)$ contained in a certain $U_\alpha$. Section $s(y,t_0)$ has been defined globally on $Y$. Within each locally trivialized neighborhood, Fiber on $\operatorname{Hom}(f^{-1}E,\pi^{-1}f^{-1}_{t_0}E)$’s $V_k\times O(t_0,\varepsilon)$, can be regarded as a Euclidean space, and the section can be extended to the extension arbitrarily on it. That is, starting from $s(y,t_0)$, The extension is $s(y,t)$, $(y,t)\in V_k\times O(t_0,\varepsilon).$ and further, Merge the extensions on each $V_k$ using the partition of unity belonging to $\{V_k\}$, Forming an overall continuation of $Y\times O(t_0,\varepsilon)$, The cross section is thus defined as a whole within the strip area.

When extending, please note that all non-isomorphic linear maps are closed sets, so we can extend them within open sets. The cross section is given, so the isomorphism holds near $t_0$. By compactness, it holds on $I$, in particular, $f_0^{-1}E\cong f_1^{-1}E.$

Corollary 1.1. A vector bundle on a shrinkable space is a trivial bundle.

The article was last updated on 2021-09-23 14:45:08