1Definitions
II Algebraic Topology
1.2 Cell complexes
In topology, we can construct some really horrible spaces. Even if we require them
to be compact, Hausdorff etc, we can often still produce really ugly topological
spaces with weird, unexpected behaviour. In algebraic topology, we will often
restrict our attention to some nice topological spaces, known as cell complexes.
To build cell complexes, we are not just gluing maps, but spaces.
Definition (Cell attachment). For a space
X
, and a map
f
:
S
n−1
→ X
, the
space obtained by attaching an n-cell to X along f is
X ∪
f
D
n
= (X q D
n
)/∼,
where the equivalence relation
∼
is the equivalence relation generated by
x ∼ f
(
x
)
for all x ∈ S
n−1
⊆ D
n
(and q is the disjoint union).
Intuitively, a map
f
:
S
n−1
→ X
just picks out a subset of
X
that looks like
the sphere. So we are just sticking a disk onto
X
by attaching the boundary of
the disk onto a sphere within X.
Definition (Cell complex). A (finite) cell complex is a space X obtained by
(i) Start with a discrete finite set X
(0)
.
(ii)
Given
X
(n−1)
, form
X
(n)
by taking a finite set of maps
{f
α
:
S
n−1
→
X
(n−1)
} and attaching n-cells along the f
α
:
X
(n)
=
X
(n−1)
q
a
α
D
n
α
!
/{x ∼ f
α
(x)}.
For example, given the
X
(0)
above, we can attach some loops and lines to
obtain the following X
(1)
We can add surfaces to obtain the following X
(2)
(iii) Stop at some X = X
(k)
. The minimum such k is the dimension of X.
To define non-finite cell complexes, we just have to remove the words “finite” in
the definition and remove the final stopping condition.
We have just had an example of a cell complex above, and it is easy to
produce many more examples, such as the
n
-sphere. So instead let’s look at a
non-cell complex.
Example. The following is not a cell complex: we take
R
2
, and add a circle with
radius
1
2
and center (0
,
1
2
). Then we add another circle with radius
1
4
and center
(0
,
1
4
), then a circle with radius
1
8
and center (0
,
1
8
) etc. We obtain something like
This is known as the Hawaiian Earring.
Why is this not an (infinite) cell complex? We did obtain it by attaching
lots of 1-cells to the single point (0
,
0). However, in the definition of a cell
complex, the cells are supposed to be completely unrelated and disjoint, apart
from intersecting at the origin. However, here the circles clump together at the
origin.
In particular, if we take the following sequence (0
,
1)
,
(0
,
1
2
)
,
(0
,
1
4
)
, · · ·
, it
converges to (0
,
0). If this were a cell complex, then this shouldn’t happen
because the cells are unrelated, and picking a point from each cell should not
produce a convergent sequence (if you are not convinced, if we actually did
produce by attaching cells, then note that during the attaching process, we
needn’t have attached them this way. We could have made it such that the
n
th
cell has radius
n
. Then clearly picking the topmost point of each cell will not
produce a convergent sequence).
We will see that the Hawaiian Earring will be a counterexample to a lot of
our theorems here.