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# 数学代写|几何组合代写Geometric Combinatorics代考|Cell Complexes and CW Complexes

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## 数学代写|几何组合代写Geometric Combinatorics代考|Cell Complexes and CW Complexes

The main theorems of discrete (and smooth) Morse theory are best stated in the language of CW complexes, so we begin with an overview of the basics of such complexes. J. H. C. Whitehead introduced CW complexes in his foundational work on homotopy theory, and all of the results in this section are due to him. The reader should consult $[\mathbf{6 8}]$ for a very complete introduction to this topic. In these notes we will consider only finite $\mathrm{CW}$ complexes, so many of the subtleties of the subject will not appear.

The building blocks of cell complexes are cells. Let $B^d$ denote the closed unit ball in $d$-dimensional Euclidean space. That is, $B^d=\left{x \in \mathbb{E}^d\right.$ s.t. $\left.|x| \leq 1\right}$. The boundary of $B^d$ is the unit $(d-1)$-sphere $S^{(d-1)}=\left{x \in \mathbb{E}^d\right.$ s.t. $\left.|x|=1\right}$. A d-cell is a topological space which is homeomorphic to $B^d$. If $\sigma$ is $d$-cell, then we denote by $\dot{\sigma}$ the subset of $\sigma$ corresponding to $S^{(d-1)} \subset B^d$ under any homeomorphism between $B^d$ and $\sigma$. A cell is a topological space which is a $d$-cell for some $d$.

The basic operation of cell complexes is the notion of attaching a cell. Let $X$ be a topological space, $\sigma$ a $d$-cell and $f: \dot{\sigma} \rightarrow X$ a continuous map. We let $X \cup_f \sigma$ denote the disjoint union of $X$ and $\sigma$ quotiented out by the equivalence relation that each point $s \in \dot{\sigma}$ is identified with $f(s) \in X$. We refer to this operation by saying that $X \cup_f \sigma$ is the result of attaching the cell $\sigma$ to $X$. The map $f$ is called the attaching map.

We emphasize that the attaching map must be defined on all of $\dot{\sigma}$. That is, the entire boundary of $\sigma$ must be “glued” to $X$. For example, if $X$ is a circle, then Figure 1(i) shows one possible result of attaching a 1-cell to $X$. Attaching a 1-cell to $X$ cannot lead to the space illustrated in Figure 1 (ii) since the entire boundary of the 1-cell has not been “glued” to $X$.

We are now ready for our main definition. A finite cell complex is any topological space $X$ such that there exists a finite nested sequence
(1)
$$\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_n=X$$
such that for each $i=0,1,2, \ldots, n, X_i$ is the result of attaching a cell to $X_{(i-1)}$.

Note that this definition requires that X0 be a 0-cell. If X is a cell complex, we refer to any sequence of spaces as in (1) as a cell decomposition of X. Suppose that in the cell decomposition (1), of the $n+1$ cells that are attached, exactly $c_d$ are $d$-cells. Then we say that the cell complex $X$ has a cell decomposition consisting of $c_d d$-cells for every $d$.

## 数学代写|几何组合代写Geometric Combinatorics代考|The Morse Theory

In this section we introduce the main topic of the first three lectures, namely discrete Morse theory. Morse theory, in the standard setting of smooth manifolds, is usually described in the language of smooth functions on smooth manifolds (e.g. [71]). In practice, though, it is often useful to work with gradient vector fields rather than functions (e.g. $[\mathbf{7 2}],[\mathbf{8 2}]$ ). In the discrete setting, too, one can follow either path. In these notes, we will focus on the notion of a (discrete) gradient vector field. To see how discrete Morse theory can be presented from the function point of view, see $[\mathbf{3 1}]$ or $[\mathbf{3 2}]$

Let $K$ be a CW complex. (Most of our examples will be simplicial complexes, but in a few places, even when our object of study is a simplicial complex, it will be convenient to allow more general cell complexes.)

Definition 8. Let $\beta$ be a $(p+1)$-cell of $K$, with attaching map $h: S^p \rightarrow K_p$, where $K_p$ denotes the union of the cells of dimension $\leq p$.
(i) A cell $\alpha$ is a face of $\beta$, denoted by $\alpha<\beta$ (or $\beta>\alpha$ ) if $\beta \neq \alpha \subset \beta$ (where here we are identifying a cell with its image in $K$ ).
(ii) A face $\alpha$ of $\beta$ is said to be regular if
(a) $h^{-1}(\alpha)$ is homeomorphic to a ball, and
(b) $h$ restricted to $h^{-1}(\alpha)$ is a homeomorphism onto $\alpha$.
(iii) A regular $C W$ complex is a CW complex in which every face is regular. We note that every simplicial complex or polyhedron is a regular $\mathrm{CW}$ complex.

## 数学代写|几何组合代写Geometric Combinatorics代考|Cell Complexes and CW Complexes

$(1)$
$$\emptyset \subset X_0 \subset X_1 \subset \cdots \subset X_n=X$$

## 数学代写|几何组合代写Geometric Combinatorics代考|The Morse Theory

(i) 细胞 $\alpha$ 是一张脸 $\beta$ ，表示为 $\alpha<\beta$ (或者 $\beta>\alpha$ ) 如果 $\beta \neq \alpha \subset \beta$ (在这里我们用它的图像来识别一个单 元格 $K)$.
(二) 做 $\alpha$ 的 $\beta$ 如果
(a) $h^{-1}(\alpha)$ 与球同胚，并且
(b) $h$ 受限于 $h^{-1}(\alpha)$ 是同胚到 $\alpha$.
(iii) 常规 $C W$ complex 是一个 $\mathrm{CW}$ 复形，其中每个面都是规则的。我们注意到每个单纯复形或多面体都是规 则的CW复杂的。

## MATLAB代写

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