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 \(n\geq 0\).
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).
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