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|Extension of isotopies to diffeotopies
We recall a few notions. Let $N$ be a smooth boundaryless $n$-manifold. Let $M$ be a smooth $m$-manifold and
$$
F: M \times[0,1] \rightarrow N
$$
a smooth map such that $f_{t}$ is an embedding for every $t \in[0,1]$; then $F$ is an isotopy connecting $f_{0}$ and $f_{1}$.
A diffeotopy of $N$ (also called an ambient isotopy) is a smooth map
$$
G: N \times[0,1] \rightarrow N
$$
such that $g_{t}$ is a diffeomorphism for every $t \in[0,1]$. We will also assume that $g_{0}=\mathrm{id}_{N}$. Hence diffeotopies are special isotopies.
Definition 7.1. We say that an isotopy $F$ as above extends to an ambient isotopy if there is a diffeotopy $G$ of $N$ such that $f_{t}=g_{t} \circ f_{0}$ for every $t \in[0,1]$. Note that $\left{V_{t}=f_{t}(M)\right}$ is a 1-parameter family of submanifolds of $N$ (each diffeomorphic to $M$ ) and that $V_{t}=g_{t}\left(V_{0}\right)$, for every $t$.
数学代写|拓扑学作业代写topology代考|Gluing manifolds together along boundary components
Let $M_{1}$ and $M_{2}$ be $m$-compact manifolds with boundary, $V_{1}$ and $V_{2}$ unions of connected components of $\partial M_{1}$ and $\partial M_{2}$, respectively, and let $\rho: V_{1} \rightarrow V_{2}$ be a diffeomorphism. Consider the compact topological quotient space
$$
M_{1} \mathrm{I}{\rho} M{2}
$$
by the equivalence relation on the disjoint union $M_{1}$ II $M_{2}$ which identifies every $x \in V_{1}$ with $\rho(x) \in V_{2} ; \rho$ is called the gluing map. Denote the projection to the quotient space by
$$
q: M_{1} \mathrm{U} \mathrm{} M_{2} \rightarrow M_{1} \mathrm{H}{\rho} M{2}
$$
and the inclusion by
$$
i_{s}: M_{s} \rightarrow M_{1} \mathrm{II} M_{2}
$$
for $s=1,2$; finally, set
$$
j_{s}=q \circ i_{s} .
$$
It is clear that $j_{s}$ is a homeomorphism to its image.
数学代写|拓扑学作业代写TOPOLOGY代考|On corner smoothing
Here, we re-examine manifolds with corners, already covered in Section 4.10. Using tubular neighbourhoods and collars as in the previous section, it is not hard to see that every compact smooth $m$-manifold with corner $M$ verifies the following properties:
- $M$ is a topological $m$-manifold and contains a boundaryless compact smooth $(m-2)$-manifold $L$ (the corner locus) such that $M \backslash L$ is a smooth $m$-manifold with boundary.
- There is an open neighbourhood $U$ of $L$ in $M$ and a homeomorphism
$$
\phi: U \rightarrow L \times[0,1) \times[0,1)
$$
such that for every $x \in L, \phi(x)=(x, 0,0)$ and the restriction of $\phi$ to $U \backslash L$ is a diffeomorphism to $L \times[0,1) \times[0,1) \backslash L \times{(0,0)}$.
There is a natural corner smoothing procedure that gives a smooth structure on $M$ which is compatible with the given smooth structures on $L$ and $M \backslash L$.
To this end, let us fix a homeomorphism $\tau:[0,1) \times[0,1) \rightarrow B^{2}(0,1) \cap \mathbf{H}^{2}$ which is a diffeomorphism outside $(0,0)$ (for instance, do it by using polar coordinates). Then set
$$
\tau^{\prime}: L \times[0,1) \times[0,1) \rightarrow L \times\left(B^{2}(0,1) \cap \mathbf{H}^{2}\right), \tau^{\prime}(x, y, z)=(x, \tau(y, z))
$$
and take the composition $\tau^{\prime} \circ \phi: U \rightarrow L \times\left(B^{2}(0,1) \cap \mathbf{H}^{2}\right)$. Take on $U$ the differential structure such that $\tau^{\prime} \circ \phi$ is tautologically a diffeomorphism. A smooth atlas of this structure together with a smooth atlas of $M \backslash L$ makes a smooth atlas on $M$ which, by construction, is compatible with the given smooth structures. The induced smooth structure on $\partial M$ coincides, up to diffeomorphism, with the one obtained by gluing the closure of the components of $\partial M \backslash L$ along the common boundary. Arguing similarly to Proposition $7.6$, the corner smoothing produces a unique smooth structure up to diffeomorphism.
拓扑学代写
数学代写|拓扑学作业代写TOPOLOGY代考|EXTENSION OF ISOTOPIES TO DIFFEOTOPIES
我们回忆了几个概念。让ñ做一个光滑的无边界n-歧管。让米做一个光滑的米-歧管和
F:米×[0,1]→ñ
一个光滑的地图，使得F吨是每个的嵌入吨∈[0,1]; 然后F是同位素连接F0和F1.
一个微分拓扑ñ 一种ls这C一种ll和d一种n一种米b一世和n吨一世s这吨这p是是一张光滑的地图
G:ñ×[0,1]→ñ
这样G吨是每一个微分同胚吨∈[0,1]. 我们还将假设G0=一世dñ. 因此，diffeotopies 是特殊的同位素。
定义 7.1。我们说同位素F如果存在diffeotopy，如上所述延伸到环境同位素G的ñ这样F吨=G吨∘F0对于每个吨∈[0,1]. 注意\left{V_{t}=f_{t}(M)\right}\left{V_{t}=f_{t}(M)\right}是 1 参数的子流形族ñ 和一种CHd一世FF和这米这rpH一世C吨这$米$然后在吨=G吨(在0), 对于每个吨.
数学代写|拓扑学作业代写TOPOLOGY代考|GLUING MANIFOLDS TOGETHER ALONG BOUNDARY COMPONENTS
让米1和米2是米- 带边界的紧流形，在1和在2的连通分量的联合∂米1和∂米2, 分别让ρ:在1→在2是微分同胚。考虑紧拓扑商空间
$$
M_{1} \mathrm{I}{\rho} M{2}
$$
by the equivalence relation on the disjoint union $M_{1}$ II $M_{2}$ which identifies every $x \in V_{1}$ with $\rho(x) \in V_{2} ; \rho$ is called the gluing map. Denote the projection to the quotient space by
$$
q: M_{1} \mathrm{U} \mathrm{} M_{2} \rightarrow M_{1} \mathrm{H}{\rho} M{2}
$$
并由
一世s:米s→米1一世一世米2
为了s=1,2; 最后，设置
js=q∘一世s.
很清楚js是其图像的同胚。
数学代写|拓扑学作业代写TOPOLOGY代考|ON CORNER SMOOTHING
在这里，我们重新检查了带角的流形，已经在 4.10 节中介绍过。使用上一节中的管状邻域和项圈，不难看出每个紧凑平滑米-带角的歧管米验证以下属性：
- 米是一个拓扑米-manifold 并包含一个无边界紧致平滑(米−2)-歧管大号 吨H和C这rn和rl这C在s这样米∖大号是光滑的米-带边界的歧管。
- 有一个开放的社区在的大号在米和同胚
φ:在→大号×[0,1)×[0,1)
这样对于每个X∈大号,φ(X)=(X,0,0)和限制φ到在∖大号是微分同胚大号×[0,1)×[0,1)∖大号×(0,0).
有一个自然的拐角平滑程序，可以在米这与给定的平滑结构兼容大号和米∖大号.
为此，让我们修复一个同胚τ:[0,1)×[0,1)→乙2(0,1)∩H2这是外面的微分同胚(0,0) F这r一世ns吨一种nC和,d这一世吨b是在s一世nGp这l一种rC这这rd一世n一种吨和s. 然后设置
τ′:大号×[0,1)×[0,1)→大号×(乙2(0,1)∩H2),τ′(X,是,和)=(X,τ(是,和))
并取作文τ′∘φ:在→大号×(乙2(0,1)∩H2). 承担在差分结构使得τ′∘φ在重言式上是微分同胚。此结构的平滑图集与米∖大号制作光滑的地图集米通过构造，它与给定的光滑结构兼容。诱导光滑结构∂米一致，直到微分同胚，与通过粘合组件的闭合获得的∂米∖大号沿着共同的边界。与命题类似地争论7.6，角平滑产生了一个独特的平滑结构，直到微分同胚。
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