# 4.01 CW complexes

Below the video you will find accompanying notes and some pre-class questions.

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# Notes

## Intuition for CW complexes

*(0.00)* In this section, we will introduce a construction which
yields a huge variety of spaces called *CW complexes* or *cell
complexes*. Most of the spaces we study in topology are (homotopy
equivalent to) CW complexes. The construction relies heavily on the
quotient topology.

*(0.35)* A CW complex is a space built out of smaller spaces,
iteratively by a process called *attaching cells*. A \(k\)-cell is a
\(k\)-dimensional disc \[D^k=\{x\in\mathbf{R}^k\ :\ |x|\leq 1\}.\]
*Attaching* a \(k\)-cell to another space \(X\) means, intuitively,
forming the union of \(X\) and \(D^k\) where we glue the boundary of
\(D^k\) to \(X\).

*(1.46)*Let \(X\) be a single point \(p\) and attach a 1-cell \(D^1=[-1,1]\) to \(X\) so that the two endpoints attach at the point \(p\). The result is a circle. Alternatively, one could attach a 2-cell to \(X\) by collapsing its boundary circle to \(p\); the result is a 2-sphere.

*(3.00)*You could attach several cells. For example, attaching two 1-cells to a single point yields the figure 8.

*(3.15)*The 2-torus is built by attaching a square to a figure 8. Since the square is topologically a disc, this is a 2-cell attachment. The boundary of the square (disc) is attached in a more interesting way than the previous examples: its boundary runs along the two loops, \(a,b\), of the figure 8 in the order \(b^{-1}a^{-1}ba\).

## Attachment of cells

*(5.26)*Let \(X\) be a space and let \(D^k\) be the \(k\)-dimensional disc. Let \(\varphi\colon\partial D^k\to X\) be a continuous map from the boundary \(\partial D^k\) of the \(k\)-cell to \(X\). Consider the space \((X\cup_\varphi D^k=X\coprod D^k)/\sim\) where \(\coprod\) denotes disjoint union and \(\sim\) is the equivalence relation identifies each point \(z\in\partial D^k\) with its image \(\varphi(x)\in X\). We call \(X\cup_{\varphi} D^k\)

*the result of attaching a \(k\)-cell to \(X\) along the map \(\varphi\)*.

*(7.35)*The map \(\varphi\) is an important part of this definition. Different \(\varphi\) will yield different spaces:

*(7.52)*In the example of the 2-torus, we attached the 2-cell along a map \(\varphi\colon S^1\to 8\) which represented the homotopy class \(b^{-1}a^{-1}ba\) of loops in the figure 8 space. Suppose instead that we had attached using the constant map \(\varphi'\colon S^1\to 8\) which sends the circle to the cross-point of the figure 8. In this case, \(X\cup_{\varphi'} D^2=S^1\vee S^1\vee S^2\). That is not homotopy equivalent to \(T^2\): the torus has abelian fundamental group, whereas \(X\cup_{\varphi'} D^2\) has fundamental group \(\mathbf{Z}\star\mathbf{Z}\), a nonabelian group.

*(9.44)*As another example, let \(X\) be a pair of points and attach a 1-cell in two different ways:

- in the first case, attach the two endpoints of the 1-cell to different points, for example taking \(\varphi_0\colon\{2\mbox{ points}\}\to\{2\mbox{ points}\}\) to be the identity. The result is an interval.
- in the second case, attach both endpoints of the 1-cell to the
same point in \(X\), for example taking \(\varphi_1\colon\{2\mbox{
points}\}\to\{2\mbox{ points}\}\) to be a constant map. The result
is a disjoint union of a circle with a point.

## CW complexes

*(11.23)*A CW complex is any topological space \(X\) built in the following way.

- You start with the empty set, and attach a collection of 0-cells
(points: the ``boundary of a point'' is the empty set, so the
attaching map is the unique map from the empty set to the empty
set!). The result is a discrete space (just a bunch of points)
called \(X^0\) (the
*0-skeleton*of \(X\)). - You add 1-cells \(e\) (possibly infinitely many) by specifying attaching maps \(\partial e\to X^0\). The result is called the 1-skeleton \(X^1\).
- You add 2-cells \(e\) (possibly infinitely many) by specifying attaching maps \(\partial e\to X^1\). The result is called the 2-skeleton \(X^2\).
- You continue in this manner, constructing a nested sequence of skeleta \[X^0\subset X^1\subset X^2\subset\cdots\subset X^n\subset\cdots\].
*(14.30)*You take the union \(X=\bigcup_{n\geq 0}X^n\) of all skeleta and equip it with the*weak topology*, in which a subset \(U\subset X\) is open if and only if \(U\cap X^n\) is open for all \(n\geq 0\).

*no*\(k\)-cells for some \(k\). You can still add higher-dimensional cells: for example, we saw the 2-sphere is a CW complex by attaching a 2-cell to a point (no 1-cells).

*(16.38)*The weak topology is responsible for the ``W'' in the name ``CW complex''. It is not related to the weak star topology which you may have encountered in courses on functional analysis.

*(17.36)*The circle \(S^1\) has a cell structure with two 0-cells and two 1-cells. The 2-sphere can be obtained from this by adding the North and South hemispheres (2-cells). The 3-sphere can be obtained from the 2-sphere by adding the ``North and South hemispheres'' (3-cells). And so on, ad infinitum. By taking the weak topology on the nested union of these spheres, you get the

*infinite-dimensional sphere*.

*(19.17)*CW complexes have very nice homotopical properties, as we shall see in the section on the homotopy extension property.

# Pre-class questions

- Consider the figure 8 with the two loops labelled \(a,b\). Attach a
2-cell \(e\) to this using an attaching map \(\varphi\colon\partial
e\to 8\) which is a loop representing the homotopy class
\(ba^{-1}ba\). What topological space do you get? (Hint: Try
modifying the example of the torus).

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