Cell Complex

Notation

$e^n$ is an n cell, homeomorphic to the open n disk $D^n-\partial D^n$ or $D^n-S^{n-1}$ or $Int(D^n)$

Construction

• We start from a discrete set $X^0$ (0-cells)
• form the n-skeleton $X^n$ from $X^{n-1}$ by attaching $n$-cells $e^n$ via the attaching map

$${\phi }_{\alpha }:S^{n-1}\to X^{n-1}$$

For example we construct $X^1$ from $X^0$ by adding edges ($1$-cells) between discrete points. Each attaching map is ${\phi }_{\alpha }:S^0\to X^0$ (remember that $S^0$ is two points). Here $S^0$ is the boundary of the $1$-cells that will be attached We now have the new space

$$X^n=X^{n-1}\coprod _{\alpha }{e}_{\alpha }^n\text{, for each }\alpha$$

or more formally

$$X^n=(X^{n-1}\bigsqcup \coprod _{\alpha }{D}_{\alpha }^n)/\sim ,x\sim {\phi }_{\alpha }(x)\text{ with }x\in D_{\alpha }^n$$

In the end we get $X^0\subset X^1\subset X^2\subset \cdots \subset X$

Example

a 1-dimensional cell complex $X=X^1$ is a graph

A 2-dimensional cell complex $X=X^2$

characteristic map

$${\Phi }_{\alpha }:D_{\alpha }^n\to X$$

This can be seen as a composition of maps

$$D_{\alpha }^n\hookrightarrow X^{n-1}\coprod _{\alpha }{D}_{\alpha }^n\to X^n\hookrightarrow X^n$$

the middle map is the quotient map defining $X^n$

Subcomplex

A subcomplex of a CW complex $X$ is a subspace $A$ formed by a collection of cells of $X$ such that it is still a complex. This means once you include a cell, you must also include all lower-dimensional cells on which its boundary is attached.

We call $(X,A)$ a CW pair

Operations

Products

If $X$ and $Y$ are cell complexes, then $X\times Y$ is also a cell complex, whose cells are exactly the products $e_{\alpha }^m\times e_{\beta }^n$, where $e_{\alpha }^m$ is a cell of $X$ and $e_{\beta }^n$ is a cell of $Y$.

Warning

The topology on $X\times Y$ as a cell complex could be finer than the product topology. The two topology coincide if either $X$ or $Y$ has finitely many cells, or both $X$ and $Y$ have countably many cells. We could ignore the difference in most of the cases.

Quotients

Let $(X,A)$ be a CW pair $X/A$ means collapsing the subcomplex $A$ into a single point (a $0$-cell) This single point is also the image of $A$ in $X/A$

Others

The suspension of $X$ is obtained by stretching $X$ between two new points: a north pole and a south pole. Or stretching $X$ into a cylinder and then collapsing both end faces to points. Formally,

$$\Sigma X=(X\times I)/(X\times \{0\}\sim {\ast }_S,X\times \{1\}\sim {\ast }_N).$$

The join $X\ast Y$ connects every point of $X$ to every point of $Y$ by a line segment. Formally,

$$X\ast Y=(X\times Y\times I)/\sim ,$$

where $(x,y_1,0)\sim (x,y_2,0),(x_1,y,1)\sim (x_2,y,1).$

The wedge sum glues two based spaces together at their basepoints:

$$X\vee Y=(X\bigsqcup Y)/({\ast }_X\sim {\ast }_Y).$$

The smash product is like the product $X\times Y$, but we collapse the parts where one coordinate is the basepoint:

$$X\wedge Y=(X\times Y)/(X\vee Y).$$