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|The embedded tangent functor
Let us fix a setting we will refer to all throughout the rest of this chapter.
- $M \subset \mathbb{R}^{h}$ is an embedded smooth manifold of dimension $m, p \in M$. $N \subset \mathbb{R}^{k}$ is an embedded smooth manifold of dimension $n, q \in N$.
- $f: M \rightarrow N$ is a smooth map, $f(p)=q$.
- $\phi: W \rightarrow U \subset \mathbb{R}^{m}$ is a chart of $M$ at $p, \phi(p)=a$, with inverse local parametrization $\psi: U \rightarrow W \subset M$.
- $f_{U, U^{\prime}}: U \rightarrow U^{\prime}$ is a representation of $f$ in local coordinates at $p$. Recall that this is obtained as follows: we take a local chart of $M$ at $p$, for simplicity still denoted $(W, \phi)$, and a local chart $\left(W^{\prime}, \phi^{\prime}\right)$ of $N$ at $q, \phi^{\prime}(q)=b$, such that $f(W) \subset W^{\prime}$; then
$$
f_{U, U^{\prime}}=\phi^{\prime} \circ f \circ \psi: U \rightarrow U^{\prime}
$$
(U and $U^{\prime}$ being open sets of $\mathbb{R}^{m}$ and $\mathbb{R}^{n}$, respectively). - Possibly by shrinking $W$, we can also assume that there is an open neighbourhood $\Omega$ of $p$ in $\mathbb{R}^{h}$ such that $\Omega \cap M=W$, a local smooth extension $\Phi: \Omega \rightarrow \mathbb{R}^{m}$ of $\phi$, and a local smooth extension $F: \Omega \rightarrow$ $\mathbb{R}^{k}$ of $f$.
We are going to define the tangent space $T_{p} M$ to $M \subset \mathbb{R}^{h}$ at each point $p \in M$ and for every smooth $f: M \rightarrow N, p \in M$, the differential $d_{p} f: T_{p} M \rightarrow T_{f(p)} N$. This is done in the following lemma. The facts collected in the lemma are easy consequences of the definitions and of the results of Chapter 1. The reader can justify them in detail as an exercise.
数学代写|拓扑学作业代写topology代考|Immersions,submersions,embeddings, Monge charts
The notions of immersion and submersion extend immediately to maps between embedded smooth manifolds: $f: M \rightarrow N$ is an immersion (resp. submersion) if for every $x \in M, d_{x} f$ is injective (surjective). We say that $f: M \rightarrow N$ is an embedding if $f$ is a diffeomorphism between $M$ and its image $f(M)$ (in particular, the inclusion $M \subset \mathbb{R}^{h}$ is an embedding). The proof of the following proposition is of a local nature and follows from Lemmas $2.4$ and $2.7$.
Proposition 2.12. (1) Let $f: M \rightarrow N$ be a surjective submersion; then for every $q \in N, Y=f^{-1}(q)$ is a submanifold of $M, \operatorname{dim} Y=\operatorname{dim} M-\operatorname{dim} N$.
(2) If $f: M \rightarrow N$ is an embedding, then $f(M)$ is a submanifold of $N$.
(3) $f: M \rightarrow N$ is an embedding if and only if $f$ is an immersion and $a$ homeomorphism to its image.
(4) If $f: M \rightarrow N$ is both an immersion and a submersion, then it is a local diffeomorphism.
We have seen in Example $2.9$ a distinguished local graph chart of $S^{n}$. Here we show that such a kind of chart exists for every embedded smooth $m$-manifold $M \subset \mathbb{R}^{h}$ at every point. For every multi-index $J=\left(j_{1}, \ldots, j_{m}\right)$, $|J|=m$, let $J^{\prime},\left|J^{\prime}\right|=h-m$, be its complementary multi-index. Denote by $\mathbb{R}{J}$ the subspace of $\mathbb{R}^{h}$ generated by $\left(e{j_{1}}, \ldots, e_{j_{m}}\right)$; hence we have the orthogonal direct sum decomposition $\mathbb{R}^{h}=\mathbb{R}{J} \oplus \mathbb{R}{J^{\prime}}$ and the orthogonal projection to $\mathbb{R}{J}, \pi{J}(x)=\left(x_{j_{1}}, \ldots, x_{j_{m}}\right)$. For every $p \in M$, denote by $\pi_{J, p}: \mathbb{R}^{h} \rightarrow \mathbb{R}{J}$ the composition of the translation $x \rightarrow x-p$, followed by $\pi{J}$. Denote by $\phi_{J, p}$ the restriction of $\pi_{J, p}$ to (any suitable subset of) $M$.
Proposition 2.13 (Monge charts). For every embedded smooth $m$-manifold $M \subset \mathbb{R}^{h}$, for every $p \in M$, there exists $J,|J|=m$, and an open neighbourhood $W$ of $p$ in $M$ such that $\left(W, \phi_{J, p}\right)$ is a chart of $M$ at $p$. The inverse local parametrization is of the form $\psi_{J, p}: U \rightarrow W, U \subset \mathbb{R}{J}, \psi{J, p}(y)=\left(y, f_{J, p}(y)\right)$ (by using the above decomposition $\mathbb{R}^{h}=\mathbb{R}{J} \oplus \mathbb{R}{J^{\prime}}$ ). Hence at every point $p$, $M$ is locally a graph of a smooth function defined on some $\mathbb{R}_{J}$.
Proof. By elementary linear algebra, there exists $J$ such that the restriction of $\pi_{J}$ to $T_{p} M$ is a linear isomorphism to $\mathbb{R}{J}$. As $d{p} \phi_{J, p}$ coincides with such a restriction, then $\phi_{J, p}$ is a local diffeomorphism.
拓扑学代写
数学代写|拓扑学作业代写TOPOLOGY代考|THE EMBEDDED TANGENT FUNCTOR
让我们修复一个我们将在本章其余部分中引用的设置。
- 米⊂RH是一个嵌入的光滑流形米,p∈米. ñ⊂Rķ是一个嵌入的光滑流形n,q∈ñ.
- F:米→ñ是一张光滑的地图，F(p)=q.
- φ:在→在⊂R米是一张图表米在p,φ(p)=一种, 具有逆局部参数化ψ:在→在⊂米.
- F在,在′:在→在′是代表F在当地坐标p. 回想一下，这是通过以下方式获得的：我们取一个局部图表米在p，为简单起见仍表示(在,φ), 和一个局部图表(在′,φ′)的ñ在q,φ′(q)=b, 这样F(在)⊂在′; 然后
F在,在′=φ′∘F∘ψ:在→在′
在一种nd$在′$b和一世nG这p和ns和吨s这F$R米$一种nd$Rn$,r和sp和C吨一世在和l是. - 可能通过收缩在，我们也可以假设有一个开放的邻域Ω的p在RH这样Ω∩米=在, 局部平滑扩展披:Ω→R米的φ, 和局部平滑扩展F:Ω→ Rķ的F.
我们将定义切线空间吨p米到米⊂RH在每个点p∈米并且对于每一个平滑F:米→ñ,p∈米, 微分dpF:吨p米→吨F(p)ñ. 这是在以下引理中完成的。引理中收集的事实是定义和第 1 章结果的简单推论。读者可以通过练习来详细证明它们的合理性。
数学代写|拓扑学作业代写TOPOLOGY代考|IMMERSIONS,SUBMERSIONS,EMBEDDINGS, MONGE CHARTS
浸没和浸没的概念立即扩展到嵌入式光滑流形之间的映射：F:米→ñ是一种沉浸r和sp.s在b米和rs一世这n如果对于每个X∈米,dXF是内射的s在rj和C吨一世在和. 我们说F:米→ñ是一个嵌入如果F是之间的微分同胚米及其图像F(米) 一世np一种r吨一世C在l一种r,吨H和一世nCl在s一世这n$米⊂RH$一世s一种n和米b和dd一世nG. 以下命题的证明是局部性质的，并且来自引理2.4和2.7.
提案 2.12。1让F:米→ñ是一个完全的淹没；那么对于每个q∈ñ,是=F−1(q)是一个子流形米,暗淡是=暗淡米−暗淡ñ.
2如果F:米→ñ是一个嵌入，那么F(米)是一个子流形ñ.
3 F:米→ñ是嵌入当且仅当F是一种沉浸和一种与其图像同胚。
4如果F:米→ñ既是浸入又是浸没，则它是局部微分同胚。
我们已经在示例中看到2.9一个杰出的局部图表小号n. 在这里我们展示了这样一种图表存在于每个嵌入的平滑米-歧管米⊂RH在每一点。对于每个多索引$S^{n}$. Here we show that such a kind of chart exists for every embedded smooth $m$-manifold $M \subset \mathbb{R}^{h}$ at every point. For every multi-index $J=\left(j_{1}, \ldots, j_{m}\right)$, $|J|=m$, let $J^{\prime},\left|J^{\prime}\right|=h-m$, be its complementary multi-index. Denote by $\mathbb{R}{J}$ the subspace of $\mathbb{R}^{h}$ generated by $\left(e{j_{1}}, \ldots, e_{j_{m}}\right)$; hence we have the orthogonal direct sum decomposition $\mathbb{R}^{h}=\mathbb{R}{J} \oplus \mathbb{R}{J^{\prime}}$ and the orthogonal projection to $\mathbb{R}{J}, \pi{J}(x)=\left(x_{j_{1}}, \ldots, x_{j_{m}}\right)$. For every $p \in M$, denote by $\pi_{J, p}: \mathbb{R}^{h} \rightarrow \mathbb{R}{J}$ the composition of the translation $x \rightarrow x-p$, followed by $\pi{J}$. Denote by $\phi_{J, p}$ the restriction of $\pi_{J, p}$ to (any suitable subset of) $M$.
Proposition 2.13 (Monge charts). For every embedded smooth $m$-manifold $M \subset \mathbb{R}^{h}$, for every $p \in M$, there exists $J,|J|=m$, and an open neighbourhood $W$ of $p$ in $M$ such that $\left(W, \phi_{J, p}\right)$ is a chart of $M$ at $p$. The inverse local parametrization is of the form $\psi_{J, p}: U \rightarrow W, U \subset \mathbb{R}{J}, \psi{J, p}(y)=\left(y, f_{J, p}(y)\right)$ (by using the above decomposition $\mathbb{R}^{h}=\mathbb{R}{J} \oplus \mathbb{R}{J^{\prime}}$ ). Hence at every point $p$, $M$ is locally a graph of a smooth function defined on some $\mathbb{R}_{J}$.
证明。由初等线性代数，有Ĵ这样的限制圆周率Ĵ到吨p米是 $\mathbb{R} {J}的线性同构.一种sd {p} \phi_{J, p}C这一世nC一世d和s在一世吨Hs在CH一种r和s吨r一世C吨一世这n,吨H和n\phi_{J, p}$ 是一个局部微分同胚。
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