I am somewhat struggling to see the difference between a regular CW complex and a non-regular CW complex.
The difference is all the attaching maps are homeomorphisms – i.e. there are no identifications made on the boundary. So I guess if I produce a 1-sphere (circle) by a single zero cell and a single one cell, this is not regular (as both endpoints of the 1-cell get mapped to the zero cell)? However, if we use two 1-cells and two 0-cells we can get a regular CW structure?
How about:
.
I guess this is not regular (the 2-cell intersecting the 1-cell at the top is the problem.
The 'thoughtful' question coming from this – we have seen the sphere admits both a regular and non-regular CW complex. To me, the regular CW complex seems easier to work with, as the "degree term" in the cellular boundary formula is either $-1,0,1$.
What type of spaces admit a CW structure, but not a regular one? I am thinking of a pathological example, such as attaching a 2-cell to the 1-sphere with attaching map like $x \sin(1/x)$ (what would that look like?!)
Best Answer
By and large, lack of regularity is for convienience. The "standard" CW-decomposition of a 3-dimensional lens space $L_{p,q}$ has one 0-cell, one 1-cell, one 2-cell and one 3-cell. But it's impossible to make such a simple CW-decomposition into a regular one, since $H_1 L_{p,q} \simeq \mathbb Z_p$. A regular CW-decomposition with one cell in every dimension has $H_1$ free abelian.
Of course, the lens space has a regular CW-decomposition, but it's more work and more fuss to find it. This is much like how every manifold has a triangulation but you maybe don't want to work with a triangulation. The cellular boundary "degree term" is simpler, but there's far more cells, so the benefit of having a simple degree term is killed by having a complicated chain complex.
Presumably there are spaces that have non-regular CW-decompositions and lack regular CW-decompositions. But this is very much a fussy point-set topological curiosity -- the real reason one cares about regular vs. non-regular is the one given above. I think an example of a space where there is a CW-decomposition but no regular decomposition would be the interval $[0,1]$ attach a 2-cell, where the attaching map $f : S^1 \to [0,1]$ is given by:
write $z \in S^1$ as $z=e^{i\theta}$ with $\theta \in [0,2\pi]$.
then $f(z) = (\theta/2\pi) |\sin((2\pi)^2/\theta)|$
A little argument tells you if there was a regular CW-structure then there would have to be infinitely-many cells. But then you can argue this space does not have the weak topology of such a complex. Anyhow, something like that should work.