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|Dissections carried by generic Morse functions
We fix a nice atlas with collars on the triad $\left(M, V_{0}, V_{1}\right)$ adapted to the given Morse function $f: M \rightarrow[0,1]$. This means the following facts hold:
- The collars are of the form $V_{\left[0, \epsilon_{0}\right]}, V_{\left[1-\epsilon_{0}, 1\right]}$, for some $\epsilon_{0}>0, \epsilon_{0}<$ $c_{0}=f\left(p_{0}\right), c_{s}=f\left(p_{s}\right)<1-\epsilon_{0} .$
- Every critical point $p_{r}$ of $f$ is contained in a unique internal normal chart $\left(W_{r}, \phi_{r}\right)$, in such a way that $B_{r} \cap B_{r^{\prime}}=\emptyset$ if $r \neq r^{\prime}$ (recall that $B_{r}=\phi_{r}^{-1}\left(B^{m}(0,1 / 3)\right)$.
- Every $\left(W_{r}, \phi_{r}\right)$ is such that $\left(f \circ \psi_{r}-c_{r}\right): B^{m}(0,1 / 3) \rightarrow \mathbb{R}$ is in normal form according to Morse’s lemma of Section $1.11$ at $0=$ $\phi_{r}\left(p_{r}\right)$.
Then, we take $\epsilon>0$ such that the following hold: - $\epsilon_{0}<c_{0}-\epsilon, c_{0}+\epsilon<c_{1}-\epsilon, \ldots, c_{s-1}+\epsilon<c_{s}-\epsilon, c_{s}+\epsilon<1-\epsilon_{0}$.
- For every $r=0, \ldots, s, V_{c_{r}-\epsilon} \cap B_{r} \neq \emptyset$ and $V_{c_{r}+\epsilon} \cap B_{r} \neq \emptyset$, so that $V_{\left[c_{r}-\epsilon, c_{r}+\epsilon\right]}$ is the union of $V_{\left[c_{r}-\epsilon, c_{r}+\epsilon\right]} \cap B_{r}$ and its complement.
So we have the dissection of the triad $\left(M, V_{0}, V_{1}\right)$ associated to the Morse function $f$ :
$V_{\left[0, c_{0}-\epsilon\right]} \cup V_{\left[c_{0}-\epsilon, c_{0}+\epsilon\right]} \cup V_{\left[c_{0}+\epsilon, c_{1}-\epsilon\right]} \cup V_{\left[c_{1}-\epsilon, c_{1}+\epsilon\right]} \cup \cdots \cup V_{\left[c_{s}-\epsilon, c_{s}+\epsilon\right]} \cup V_{\left[c_{s}+\epsilon, 1\right]}$.
By applying the cylinder and Thom’s lemmas, we have: - $V_{\left[0, c_{0}-\epsilon\right]} \sim V_{0} \times\left[0, c_{0}-\epsilon\right], V_{\left[c_{s}+\epsilon, 1\right]} \sim\left[c_{s}+\epsilon, 1\right] \times V_{1}$.
- For every $r=0, \ldots, s-1, V_{\left[c_{r}+\epsilon, c_{r+1}-\epsilon\right]} \sim V_{c_{r}+\epsilon} \times\left[c_{r}+\epsilon, c_{r+1}-\epsilon\right]$.
- $V_{\left[0, c_{r}+\epsilon\right]} \sim V_{\left[0, c_{r+1}-\epsilon\right]}$.
For every $r=0, \ldots, s-1,\left(V_{\left[c_{r}-\epsilon, c_{r}+\epsilon\right]}, V_{c_{r}-\epsilon}, V_{c_{r}+\epsilon}\right)$ is an elementary triad in the sense that it carries a Morse function (the restriction of $f$ ) with only one critical point $\left(p_{r}\right.$ of a certain index $\left.q_{r}\right)$.
数学代写|拓扑学作业代写topology代考|Handle decompositions
Let $\left(M, V_{0}, V_{1}\right)$ be a triad as before. By definition, a handle decomposition of the triad is a sequence of nested triads of the form
$$
\left(M_{0}, V_{0}, V_{1,0}\right) \subset\left(M_{1}, V_{0}, V_{1,1}\right) \subset\left(M_{2}, V_{0}, V_{1,2}\right) \subset \cdots \subset\left(M_{k}, V_{0}, V_{1, k}\right)
$$
such that the following hold:
- $V_{1, k}=V_{1}$, and $\left(M_{k}, V_{0}, V_{1}\right)$ is diffeomorphic to $\left(M, V_{0}, V_{1}\right)$ via a diffeomorphism which is the identity in a neighbourhood of $V_{0} \mathrm{I} V_{1}$.
- For every $r=0, \ldots, k-1,\left(M_{r+1}, V_{0}, V_{1, r+1}\right)$ is obtained by attaching a $q$-handle (of dimension $m$ ) to $\left(M_{r}, V_{0}, V_{1, r}\right)$ at $V_{1, r}$ (for some $q$ ).
Handle decompositions are diffeomorphic if they are related by a diffeomorphism which is the identity near the boundary and respects the sequences of nested triads. We can also normalize the form of a given handle decomposition by stipulating that it starts with a “right” collar $C_{0}$ of $V_{0}$ and ends with a “left” collar $C_{1}$ of $V_{1}$.
As an immediate corollary of Proposition $9.3$, we have the existence of handle decompositions for every triad.
数学代写|拓扑学作业代写TOPOLOGY代考|Moves on handle decompositions
There are two basic ways to modify a given handle decomposition of a triad $\left(M, V_{0}, V_{1}\right)$ (up to equivalence of triads).
Handle sliding. This is synonymous with modifying the attaching map of a handle in the decomposition, say $H_{r}$, staying in the same isotopy class. We have already noticed in Chapter 7 that, up to diffeomorphism, this does not modify $M_{r}$; then we can continue the decomposition by composing the subsequent attaching maps with such a diffeomorphism. Finally, we obtain a decomposition diffeomorphic to the given one (possibly by attaching a final collar of $V_{1}$ to normalize the form).
Before describing the other modification, let us give a definition. Let
$$
\cdots \cup H_{r}^{q_{r}} \cup H_{r+1}^{q_{r+1}} \cup \cdots
$$
be a fragment of a handle decomposition of a triad $\left(M, V_{0}, V_{1}\right)$. Assume that $q_{r}=q, q_{r+1}=q+1$. Both the embedded $b$-sphere $S_{b}$ of $H_{r}^{q}$ (which is diffeomorphic to $S^{m-q-1}$ ) and the embedded $a$-sphere $S_{a}$ of $H_{r+1}^{q+1}$ (which is diffeomorphic to $\left.S^{q}\right)$ are submanifolds of the $(m-1)$-manifold $V_{1, r}$, and $\operatorname{dim} S_{b}+\operatorname{dim} S_{a}=m-1$. By transversality and up to handle sliding, we can assume that $S_{b}$ and $S_{a}$ intersect transversely at a finite number of points.
Definition 9.6. The adjacent handles $H_{r} \cup H_{r+1}$ form a pair of complementary handles provided that $S_{b}$ and $S_{a}$ intersect transversely in $V_{1, r}$ at exactly one point.
Cancelling/inserting pairs of complementary handles. We can state the basic handle cancellation result.
拓扑学代写
数学代写|拓扑学作业代写TOPOLOGY代考|DISSECTIONS CARRIED BY GENERIC MORSE FUNCTIONS
我们在三合会上修复了一个带有项圈的漂亮地图集(米,在0,在1)适应给定的莫尔斯函数F:米→[0,1]. 这意味着以下事实成立：
- The collars are of the form $V_{\left[0, \epsilon_{0}\right]}, V_{\left[1-\epsilon_{0}, 1\right]}$, for some $\epsilon_{0}>0, \epsilon_{0}<$ $c_{0}=f\left(p_{0}\right), c_{s}=f\left(p_{s}\right)<1-\epsilon_{0} .$
- Every critical point $p_{r}$ of $f$ is contained in a unique internal normal chart $\left(W_{r}, \phi_{r}\right)$, in such a way that $B_{r} \cap B_{r^{\prime}}=\emptyset$ if $r \neq r^{\prime}$ (recall that $B_{r}=\phi_{r}^{-1}\left(B^{m}(0,1 / 3)\right)$.
- Every $\left(W_{r}, \phi_{r}\right)$ is such that $\left(f \circ \psi_{r}-c_{r}\right): B^{m}(0,1 / 3) \rightarrow \mathbb{R}$ is in normal form according to Morse’s lemma of Section $1.11$ at $0=$ $\phi_{r}\left(p_{r}\right)$.
- Then, we take $\epsilon>0$ such that the following hold:
- $\epsilon_{0}<c_{0}-\epsilon, c_{0}+\epsilon<c_{1}-\epsilon, \ldots, c_{s-1}+\epsilon<c_{s}-\epsilon, c_{s}+\epsilon<1-\epsilon_{0}$.
- For every $r=0, \ldots, s, V_{c_{r}-\epsilon} \cap B_{r} \neq \emptyset$ and $V_{c_{r}+\epsilon} \cap B_{r} \neq \emptyset$, so that $V_{\left[c_{r}-\epsilon, c_{r}+\epsilon\right]}$ is the union of $V_{\left[c_{r}-\epsilon, c_{r}+\epsilon\right]} \cap B_{r}$ and its complement.
- So we have the dissection of the triad $\left(M, V_{0}, V_{1}\right)$ associated to the Morse function $f$ :
- $V_{\left[0, c_{0}-\epsilon\right]} \cup V_{\left[c_{0}-\epsilon, c_{0}+\epsilon\right]} \cup V_{\left[c_{0}+\epsilon, c_{1}-\epsilon\right]} \cup V_{\left[c_{1}-\epsilon, c_{1}+\epsilon\right]} \cup \cdots \cup V_{\left[c_{s}-\epsilon, c_{s}+\epsilon\right]} \cup V_{\left[c_{s}+\epsilon, 1\right]}$.
- By applying the cylinder and Thom’s lemmas, we have:
- $V_{\left[0, c_{0}-\epsilon\right]} \sim V_{0} \times\left[0, c_{0}-\epsilon\right], V_{\left[c_{s}+\epsilon, 1\right]} \sim\left[c_{s}+\epsilon, 1\right] \times V_{1}$.
- For every $r=0, \ldots, s-1, V_{\left[c_{r}+\epsilon, c_{r+1}-\epsilon\right]} \sim V_{c_{r}+\epsilon} \times\left[c_{r}+\epsilon, c_{r+1}-\epsilon\right]$.
- $V_{\left[0, c_{r}+\epsilon\right]} \sim V_{\left[0, c_{r+1}-\epsilon\right]}$.
- For every $r=0, \ldots, s-1,\left(V_{\left[c_{r}-\epsilon, c_{r}+\epsilon\right]}, V_{c_{r}-\epsilon}, V_{c_{r}+\epsilon}\right)$ is an elementary triad in the sense that it carries a Morse function (the restriction of $f$ ) with only one critical point $\left(p_{r}\right.$ of a certain index $\left.q_{r}\right)$.
数学代写|拓扑学作业代写TOPOLOGY代考|HANDLE DECOMPOSITIONS
让(米,在0,在1)像以前一样成为黑社会。根据定义，三元组的句柄分解是一系列嵌套的三元组，形式为
(米0,在0,在1,0)⊂(米1,在0,在1,1)⊂(米2,在0,在1,2)⊂⋯⊂(米ķ,在0,在1,ķ)
使得以下成立：
- 在1,ķ=在1， 和(米ķ,在0,在1)微分同胚于(米,在0,在1)通过微分同胚，它是邻域中的同一性在0一世在1.
- 对于每一个r=0,…,ķ−1,(米r+1,在0,在1,r+1)是通过附加一个q-处理这Fd一世米和ns一世这n$米$到(米r,在0,在1,r)在在1,r F这rs这米和$q$.
如果句柄分解与微分同胚相关，则它们是微分同胚的，微分同胚是边界附近的同一性并尊重嵌套三元组的序列。我们还可以通过规定它以“正确”项圈开始来规范化给定句柄分解的形式C0的在0并以“左”领结尾C1的在1.
作为命题的直接推论9.3，我们存在每个三元组的句柄分解。
数学代写|拓扑学作业代写TOPOLOGY代考|MOVES ON HANDLE DECOMPOSITIONS
有两种基本方法可以修改三元组的给定句柄分解(米,在0,在1) 在p吨这和q在一世在一种l和nC和这F吨r一世一种ds.
手柄滑动。这与在分解中修改句柄的附加图是同义的，比如说Hr，保持在同一个同位素类别。我们在第 7 章已经注意到，直到微分同胚，这并没有修改米r; 然后我们可以通过用这种微分同胚组合后续的附加映射来继续分解。最后，我们得到一个与给定分解微分同胚的分解p这ss一世bl是b是一种吨吨一种CH一世nG一种F一世n一种lC这ll一种r这F$在1$吨这n这r米一种l一世和和吨H和F这r米.
在描述其他修改之前，让我们给出一个定义。让
$$
\cdots \cup H_{r}^{q_{r}} \cup H_{r+1}^{q_{r+1}} \cup \cdots
$$
be a fragment of a handle decomposition of a triad $\left(M, V_{0}, V_{1}\right)$. Assume that $q_{r}=q, q_{r+1}=q+1$. Both the embedded $b$-sphere $S_{b}$ of $H_{r}^{q}$ (which is diffeomorphic to $S^{m-q-1}$ ) and the embedded $a$-sphere $S_{a}$ of $H_{r+1}^{q+1}$ (which is diffeomorphic to $\left.S^{q}\right)$ are submanifolds of the $(m-1)$-manifold $V_{1, r}$, and $\operatorname{dim} S_{b}+\operatorname{dim} S_{a}=m-1$. By transversality and up to handle sliding, we can assume that $S_{b}$ and $S_{a}$ intersect transversely at a finite number of points.
Definition 9.6. The adjacent handles $H_{r} \cup H_{r+1}$ form a pair of complementary handles provided that $S_{b}$ and $S_{a}$ intersect transversely in $V_{1, r}$ at exactly one point.
取消/插入互补句柄对。我们可以说明基本的句柄取消结果。
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