代数拓扑1：解剖几何图形

学完拓扑就学代数拓扑这很合理（确信

动机 motivations

代数拓扑是对拓扑空间的定性分析
$$
\begin{array}{c}
\text{Solution sets } \left { x^2+y^2+z^2 = 1 \right } \
\downarrow \
\text{manifolds}\
\downarrow \
\text{topological spaces}
\end{array}
$$

我们将要做什么

• Have machinery that gives us general information about spaces that is insentive to collapse and various other things.
• We also need to decide on a family of topological space that we are interested in Cell complex and CW complexes.

胞腔复形 cell complex

Definitions/notations

• Open $n$-cell: topolical space homemeomorphic to the open unit ball $\mathbb B ^n$
• Close $n$-cell: topolical space $\overline{\mathbb B ^n} = \mathbb B ^n \cup \mathbb S ^{n-1}$.

Prop:
If $D \subset \mathbb R ^n$ is a compact convex subset weith nonempty interior, then $D$ is a closesd $n$-cell and its interior is an open $n$-cell.
In fact, given any point $p \in \mathrm{Int} D$, there exist a homeomorphism $F:\overline{\mathbb B ^n} \to D$ that sends $0$ to $p$, $\overline{\mathbb B ^n}$ to $\mathrm{Int} D$ and $\mathbb S ^{n-1}$ to $\partial D$.

Example

• Every closed interval in $\mathbb R$ is a closed 1-cell
• Every polygons in $\mathbb R ^2$ are closed 2-cells

Cell decompositions

attaching cells to a given topological space $X$:
Given ${D_{\alpha}}_{\alpha \in A}$ and indexed collection of close $n$-cell foe some fixed $n \ge 1$. we can glue them to X via continous maps $\varphi _\alpha: \partial D \to X$ as follows: Let $\varphi :\bigsqcup_\alpha \partial D_\alpha \to X$ be the map restriction to each $\partial D_\alpha$ is $\varphi_\alpha$ We can then form the cadjunction space $X \cup_\alpha (\sqcup _\alpha D_\alpha)$.

Cell decomposition

Def:
If $X$ is a nonempty topological space, a cell decomposition of $X$ is a partition $\xi$ of $X$ into subspaces that are open cells of various dimensions, such that the following condition is satisfied: for each cell $e \in \xi$ of dimension $n \ge 1$, there exists a continuous map $\varPhi$ from some closed $n$-cell $D$ into $X$ (called a characteristic map for $e$) that restricts to a homeomomorphism from $\mathrm{Int}D$ onto $e$ and maps $\partial D$ into the union of all cells of $\xi$ of dimensions strictly less than $n$.

这个书的定义方式是，先给出一个拓扑空间$X$，然后我们用一些手段去展示这些胞腔之间是怎么拼到一起的。

顺便一提，这里事实上只是限制了，分解出来的空间是个开胞腔，我们对于这个区域在$X$中的边界依旧是一无所知。

这个要求保证，如果存在一个胞腔分解，那么我们就可以用之前的办法来构造这个空间. 事实上，$\varPhi$本身给出了一个将这个子部分与其他地方连接的“连接方式”。在有限的情况下，我们可以认为，这其实给出了用一堆胞腔去粘连出这个cell complex(定义如下)的方法，因此，我们可以用胞腔去分析它 了。

Cell complex

Def:
A cell complex is a Hausdorff space $X$ together with a specific cell decomposition of $X$.

Finite cell complex

Def:
A finite cell complex is one whose cell decomposition has only finitely many cells.

local finite

Def:
A cell complex is called locally finite if the connection of open cell is “locally finite”.

CW complex

CW complex

Def:
A CW complex is a complex $(X,\xi)$ satisfying:
(C, closure finiteness): The closure of each cell is contained in a union of finitely many cells.
(W, weak topology): The topology of X is coherent with the family of closed subspaces ${\overline e | e \in \xi}$