如果你也在 怎样代写拓扑学topology这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。拓扑学topology在数学中，拓扑学（来自希腊语中的τόπος，”地方、位置”，和λόγος，”研究”）关注的是几何对象在连续变形下保持的属性，如拉伸、扭曲、皱缩和弯曲；也就是说，在不关闭孔、打开孔、撕裂、粘连或穿过自身的情况下。
拓扑学topology拓扑空间是一个被赋予了结构的集合，称为拓扑，它允许定义子空间的连续变形，以及更广泛地定义所有种类的连续性。欧几里得空间，以及更一般的，公制空间都是拓扑空间的例子，因为任何距离或公制都定义了一个拓扑。拓扑学中所考虑的变形是同构和同形。在这种变形下不变的属性是一种拓扑属性。拓扑属性的基本例子有：维度，它可以区分线和面；紧凑性，它可以区分线和圆；连接性，它可以区分一个圆和两个不相交的圆。
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数学代写|拓扑学作业代写topology代考|Stiefel manifolds
We introduce first the Stiefel manifolds. There are two versions that we call linear and orthogonal, respectively. For every $n \in \mathbb{N}$ and every $0 \leq k \leq n$, the linear Stiefel manifold $L_{n, k}$, as a set, is the set of ordered $k$-tuples $\left(v_{1}, \ldots, v_{k}\right)$ of linearly independent vectors in $\mathbb{R}^{n}$. By arranging each of them in an $n \times k$ matrix $A$ (so that $v_{j}$ is the $j$-column of $\left.A\right), L_{n, k} \subset M(n, k, \mathbb{R})$. It is an open subset: consider the smooth function $\delta: M(n, k, \mathbb{R}) \rightarrow \mathbb{R}$ defined in the proof of Proposition $1.10$; then $L_{n, k}=M(n, k, \mathbb{R}) \backslash \delta^{-1}(0)$. This specifies how $L_{n, k}$ is an embedded smooth manifold. As a particular case, we have $\operatorname{GL}(n, \mathbb{R})=L_{n, n}$. For every $P \in \operatorname{GL}(n, \mathbb{R}), A \rightarrow P A$ defines a diffeomorphism $L_{n, k} \rightarrow L_{n, k}$. This action is transitive; in particular, for every $A \in L_{n, k}$, there exists $P \in \operatorname{GL}(n, \mathbb{R})$ such that $P I_{n, k}=A$, where $I_{n, k}$ is the matrix whose columns are $e_{1}, \ldots, e_{k}$, the first $k$ vectors of the canonical basis of $\mathbb{R}^{n}$.
Now, let $S_{n, k} \subset L_{n, k}$ be the closed subset defined as $f^{-1}\left(I_{k}\right)$ where
$$
f: L_{n, k} \rightarrow S(k, \mathbb{R})
$$
is the smooth map $f(A)=A^{t} A$ with values in the space $S(k, \mathbb{R})$ of $k \times k$ symmetric matrices, which can be identified with $\mathbb{R}^{\frac{k(k+1)}{2}}$. In other words, we require that the columns of any $A \in S_{n, k}$ form a system of orthonormal vectors. As particular cases, we have $S_{n, 1}=S^{n-1}$ and $S_{n, n}=O(n)$, the classical (real) orthogonal group. As $M(n, k, \mathbb{R})=\left(\mathbb{R}^{n}\right)^{k}$, we see immediately that $S_{n, k} \subset\left(S^{n-1}\right)^{k}$; hence $S_{n, k}$ is compact. The above action of $\operatorname{GL}(n, \mathbb{R})$ on $L_{n, k}$ restricts to a transitive action of $O(n)$ on $S_{n, k}$ : for every $A \in S_{n, k}$, there exists $P \in O(n)$ such that $P A=I_{n, k}$. It follows that to prove that $S_{n, k}$ is an embedded smooth manifold in $\left(\mathbb{R}^{n}\right)^{k}$, it is enough to prove that there is a chart $(W, \phi)$ of $S_{n, k}$ at $J:=I_{n, k}$. Hence, it is enough to prove that $d_{J} f$ is surjective and conclude by applying Corollary 1.6. Let us compute $d_{J} f$ by the definition of the differential. Then
$$
\begin{aligned}
&d f_{J}(B)=\lim {t \rightarrow 0} \frac{(J+t B)^{t}(J+t B)-I{k}}{t} \
&=\lim {t \rightarrow 0}\left(J^{t} B+B^{t} J+t B^{t} B\right)=J^{t} B+B^{t} J \end{aligned} $$ We have to prove that, for every symmetric matrix $C \in S(k, \mathbb{R})$, there exists $B \in M(n, k, \mathbb{R})$ such that $J^{t} B+B^{t} J=C$. Set $B=\frac{1}{2} J C$. Then $$ J^{t} B+B^{t} J=\frac{1}{2} J^{t} J C+\frac{1}{2} C^{t} J^{t} J=\frac{1}{2} C+\frac{1}{2} C^{t}=C $$ because $C=C^{t}$. Summarizing, $S{n, k}$ is a compact embedded smooth manifold in $L_{n, k} \subset M(n, k, \mathbb{R})=\left(\mathbb{R}^{n}\right)^{k}$, of dimension
$$
\operatorname{dim} S_{n, k}=n k-\frac{k(k+1)}{2} .
$$
$S_{n, k}$ is called an orthogonal Stiefel manifold. In particular, the orthogonal group $O(n)$ is a compact embedded smooth submanifold of $\left(S^{n-1}\right)^{n}$ of dimension
$$
\operatorname{dim} O(n)=n^{2}-\frac{n(n+1)}{2}
$$
数学代写|拓扑学作业代写topology代考|Fibrations of Stiefel manifolds by Stiefel manifolds
In Section $4.3 .1$ we will formalize the general notion of a locally trivializable fibre bundle. Here we are going to see some concrete and motivating examples of this notion.
For every $0 \leq h<k \leq n, L_{n, k}$ is a submanifold (an open set) in the product $L_{n, h} \times L_{n, k-h}$ and denote by
$$
l_{k, h}: L_{n, k} \rightarrow L_{n, h}
$$
the restriction of the natural projection onto the first factor. This map is equivariant for the above actions of $\mathrm{GL}(n, \mathbb{R})$ on both Stiefel manifolds (i.e., $\left.l_{k, h}(P A)=P l_{k, h}(A)\right)$, hence to study local properties such as the smoothness of the map, it is enough to study the restriction of $l_{k, h}$ on $l_{k, h}^{-1}(\Omega)$ where $\Omega$ is a neighbourhood of $I_{n, h}$. Clearly, $l_{k, h}\left(I_{n, k}\right)=I_{n, h}$. The fibre $F_{k, h}:=l_{k, h}^{-1}\left(I_{n, h}\right)$ over $I_{n, h}$ is made by the $2 \times 2$ block matrices of the form
$$
Y(S, D):=\left(\begin{array}{cc}
I_{h} & S \
0 & D
\end{array}\right)
$$
where $(S, D) \in M(h, k-h, \mathbb{R}) \times L_{n-h, k-h}$. If $P \in \operatorname{GL}(n, \mathbb{R})$ is such that $P I_{n, h}=A$, then $P\left(l_{k, h}^{-1}\left(I_{n, h}\right)\right)=l_{k, h}^{-1}(A)$ and all fibres are diffeomorphic to each other. Let $\Omega$ be the open neighbourhood of $I_{n, h}$ made by matrices of the form
$$
X=\left(\begin{array}{l}
B \
R
\end{array}\right)
$$
where $B \in \mathrm{GL}(h, \mathbb{R})$. We define the smooth map $X \rightarrow P(X) \in \mathrm{GL}(n, \mathbb{R})$ as
$$
P(X)=\left(\begin{array}{cc}
B & 0 \
R & I_{n-h}
\end{array}\right)
$$
数学代写|拓扑学作业代写TOPOLOGY代考|Grassmann manifolds
For every $(n, k)$ as above, we are going to define now the Grassmann manifold $\mathfrak{G}_{n, k}$.
Denote by $G_{n, k}$ the set of linear subspaces of $\mathbb{R}^{n}$ of dimension $k$. Let $\mathfrak{G}_{n, k}$ be the closed subset of $S(n, \mathbb{R})=\mathbb{R}^{\frac{n(n+1)}{2}}$ defined by the polynomial matrix equations
$$
A^{2}-A=0, \operatorname{trace}(A)=k
$$
If $A \in S(n, \mathbb{R})$ satisfies $A^{2}-A$, then its spectrum of eigenvalues is ${0,1}$ and, by the spectral theorem for real symmetric matrices, the respective eigenspaces provide an orthogonal direct sum decomposition of $\mathbb{R}^{n}$; the last condition on the trace is equivalent to the fact that the eigenspace for the eigenvalue $\lambda=1$ has dimension equal to $k$ and also to the fact that $A$ has rank equal to $k$.
We fix a bijection $V \rightarrow A_{V}$ from $G_{n, k}$ to $\mathfrak{G}{n, k}$ as follows. For every $V \in G{n, k}$ we have the orthogonal direct sum decomposition $\mathbb{R}^{n}=V \oplus V^{\perp}$ ( $V^{\perp}$ being the orthogonal space to $V$ for the standard Euclidean scalar product) and the linear map $A_{V} \in \mathcal{L}\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)=M(n, \mathbb{R})$ such that $A_{V}\left(v+v^{\prime}\right)=v$. We readily verify that $A_{V} \in \mathfrak{G}{n, k}$. The inverse map $A \rightarrow V{A}$ is defined by setting $V_{A}$ equal to the eigenspace of $A$ relative to the eigenvalue $\lambda=1$.
拓扑学代写
数学代写|拓扑学作业代写TOPOLOGY代考|STIEFEL MANIFOLDS
我们首先介绍 Stiefel 流形。有两个版本，我们分别称为线性和正交。对于每一个$n \in \mathbb{N}$ and every $0 \leq k \leq n$, the linear Stiefel manifold $L_{n, k}$, as a set, is the set of ordered $k$-tuples $\left(v_{1}, \ldots, v_{k}\right)$ of linearly independent vectors in $\mathbb{R}^{n}$. By arranging each of them in an $n \times k$ matrix $A$ (so that $v_{j}$ is the $j$-column of $\left.A\right), L_{n, k} \subset M(n, k, \mathbb{R})$. It is an open subset: consider the smooth function $\delta: M(n, k, \mathbb{R}) \rightarrow \mathbb{R}$ defined in the proof of Proposition $1.10$; then $L_{n, k}=M(n, k, \mathbb{R}) \backslash \delta^{-1}(0)$. This specifies how $L_{n, k}$ is an embedded smooth manifold. As a particular case, we have $\operatorname{GL}(n, \mathbb{R})=L_{n, n}$. For every $P \in \operatorname{GL}(n, \mathbb{R}), A \rightarrow P A$ defines a diffeomorphism $L_{n, k} \rightarrow L_{n, k}$. This action is transitive; in particular, for every $A \in L_{n, k}$, there exists $P \in \operatorname{GL}(n, \mathbb{R})$ such that $P I_{n, k}=A$, where $I_{n, k}$ is the matrix whose columns are $e_{1}, \ldots, e_{k}$, the first $k$ vectors of the canonical basis of $\mathbb{R}^{n}$.
Now, let $S_{n, k} \subset L_{n, k}$ be the closed subset defined as $f^{-1}\left(I_{k}\right)$ where
$$
f: L_{n, k} \rightarrow S(k, \mathbb{R})
$$
is the smooth map $f(A)=A^{t} A$ with values in the space $S(k, \mathbb{R})$ of $k \times k$ symmetric matrices, which can be identified with $\mathbb{R}^{\frac{k(k+1)}{2}}$. In other words, we require that the columns of any $A \in S_{n, k}$ form a system of orthonormal vectors. As particular cases, we have $S_{n, 1}=S^{n-1}$ and $S_{n, n}=O(n)$, the classical (real) orthogonal group. As $M(n, k, \mathbb{R})=\left(\mathbb{R}^{n}\right)^{k}$, we see immediately that $S_{n, k} \subset\left(S^{n-1}\right)^{k}$; hence $S_{n, k}$ is compact. The above action of $\operatorname{GL}(n, \mathbb{R})$ on $L_{n, k}$ restricts to a transitive action of $O(n)$ on $S_{n, k}$ : for every $A \in S_{n, k}$, there exists $P \in O(n)$ such that $P A=I_{n, k}$. It follows that to prove that $S_{n, k}$ is an embedded smooth manifold in $\left(\mathbb{R}^{n}\right)^{k}$, it is enough to prove that there is a chart $(W, \phi)$ of $S_{n, k}$ at $J:=I_{n, k}$. Hence, it is enough to prove that $d_{J} f$ is surjective and conclude by applying Corollary 1.6. Let us compute $d_{J} f$ by the definition of the differential. Then
$$
\begin{aligned}
&d f_{J}(B)=\lim {t \rightarrow 0} \frac{(J+t B)^{t}(J+t B)-I{k}}{t} \
&=\lim {t \rightarrow 0}\left(J^{t} B+B^{t} J+t B^{t} B\right)=J^{t} B+B^{t} J \end{aligned} $$ We have to prove that, for every symmetric matrix $C \in S(k, \mathbb{R})$, there exists $B \in M(n, k, \mathbb{R})$ such that $J^{t} B+B^{t} J=C$. Set $B=\frac{1}{2} J C$. Then $$ J^{t} B+B^{t} J=\frac{1}{2} J^{t} J C+\frac{1}{2} C^{t} J^{t} J=\frac{1}{2} C+\frac{1}{2} C^{t}=C $$ because $C=C^{t}$. Summarizing, $S{n, k}$ is a compact embedded smooth manifold in $L_{n, k} \subset M(n, k, \mathbb{R})=\left(\mathbb{R}^{n}\right)^{k}$, of dimension
$$
\operatorname{dim} S_{n, k}=n k-\frac{k(k+1)}{2} .
$$
$S_{n, k}$ is called an orthogonal Stiefel manifold. In particular, the orthogonal group $O(n)$ is a compact embedded smooth submanifold of $\left(S^{n-1}\right)^{n}$ of dimension
$$
\operatorname{dim} O(n)=n^{2}-\frac{n(n+1)}{2}
$$
数学代写|拓扑学作业代写TOPOLOGY代考|FIBRATIONS OF STIEFEL MANIFOLDS BY STIEFEL MANIFOLDS
在部分4.3.1我们将形式化局部可平凡化纤维束的一般概念。在这里，我们将看到这个概念的一些具体和激励性的例子。
对于每一个0≤H<ķ≤n,大号n,ķ是一个子流形一种n这p和ns和吨在产品中大号n,H×大号n,ķ−H并表示为
lķ,H:大号n,ķ→大号n,H
自然投影对第一个因素的限制。该映射对于上述动作是等变的G大号(n,R)在两个 Stiefel 歧管上即$0 \leq h<k \leq n, L_{n, k}$ is a submanifold (an open set) in the product $L_{n, h} \times L_{n, k-h}$ and denote by
$$
l_{k, h}: L_{n, k} \rightarrow L_{n, h}
$$
the restriction of the natural projection onto the first factor. This map is equivariant for the above actions of $\mathrm{GL}(n, \mathbb{R})$ on both Stiefel manifolds (i.e., $\left.l_{k, h}(P A)=P l_{k, h}(A)\right)$, hence to study local properties such as the smoothness of the map, it is enough to study the restriction of $l_{k, h}$ on $l_{k, h}^{-1}(\Omega)$ where $\Omega$ is a neighbourhood of $I_{n, h}$. Clearly, $l_{k, h}\left(I_{n, k}\right)=I_{n, h}$. The fibre $F_{k, h}:=l_{k, h}^{-1}\left(I_{n, h}\right)$ over $I_{n, h}$ is made by the $2 \times 2$ block matrices of the form
$$
Y(S, D):=\left(\begin{array}{cc}
I_{h} & S \
0 & D
\end{array}\right)
$$
where $(S, D) \in M(h, k-h, \mathbb{R}) \times L_{n-h, k-h}$. If $P \in \operatorname{GL}(n, \mathbb{R})$ is such that $P I_{n, h}=A$, then $P\left(l_{k, h}^{-1}\left(I_{n, h}\right)\right)=l_{k, h}^{-1}(A)$ and all fibres are diffeomorphic to each other. Let $\Omega$ be the open neighbourhood of $I_{n, h}$ made by matrices of the form
$$
X=\left(\begin{array}{l}
B \
R
\end{array}\right)
$$
where $B \in \mathrm{GL}(h, \mathbb{R})$. We define the smooth map $X \rightarrow P(X) \in \mathrm{GL}(n, \mathbb{R})$ as
$$
P(X)=\left(\begin{array}{cc}
B & 0 \
R & I_{n-h}
\end{array}\right)
$$
数学代写|拓扑学作业代写TOPOLOGY代考|GRASSMANN MANIFOLDS
对于每一个(n,ķ)如上所述，我们现在要定义格拉斯曼流形Gn,ķ.
表示为Gn,ķ的线性子空间集Rn维度的ķ. 让Gn,ķ是的闭子集小号(n,R)=Rn(n+1)2由多项式矩阵方程定义
一种2−一种=0,痕迹(一种)=ķ
如果一种∈小号(n,R)满足一种2−一种，则其特征值谱为0,1并且，根据实对称矩阵的谱定理，各个特征空间提供正交直接和分解Rn; 迹上的最后一个条件等价于特征值的特征空间λ=1尺寸等于ķ还有一个事实是一种排名等于ķ.
我们修复一个双射在$V \rightarrow A_{V}$ from $G_{n, k}$ to $\mathfrak{G}{n, k}$ as follows. For every $V \in G{n, k}$ we have the orthogonal direct sum decomposition $\mathbb{R}^{n}=V \oplus V^{\perp}$ ( $V^{\perp}$ being the orthogonal space to $V$ for the standard Euclidean scalar product) and the linear map $A_{V} \in \mathcal{L}\left(\mathbb{R}^{n}, \mathbb{R}^{n}\right)=M(n, \mathbb{R})$ such that $A_{V}\left(v+v^{\prime}\right)=v$. We readily verify that $A_{V} \in \mathfrak{G}{n, k}$. The inverse map $A \rightarrow V{A}$ is defined by setting $V_{A}$ equal to the eigenspace of $A$ relative to the eigenvalue $\lambda=1$.
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