# 4.01 CW complexes

## Video

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

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**4.02 Homotopy extension property (HEP)**. - Index of all lectures.

## 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 \(n\geq 0\).

It is possible that you add *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).

If you only add cells up to dimension \(n\) (so that \(X^n=X^{n+1}=\cdots\)) then you don't need to talk about the weak topology. The dimension of a CW complex \(X\) is defined to be the supremum of \(n\) such that \(X\) has an \(n\)-cell (this could be infinite if there are \(n\)-cells for arbitrarily large \(n\)).

*(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).

## Navigation

- Next video:
**4.02 Homotopy extension property (HEP)**. - Index of all lectures.