Intuition behind CW complexes

Question

These things are the bane of my existence in mathematics. I feel that I can't find any clear examples of these things anywhere. This is a vague question, but how exactly do we intuitively visualize attaching maps and represent them? The examples in Hatcher all seem trivial (like attaching the entire boundary of the 2 cell to a point to make a sphere) or entirely hand waved. Amazingly, there are literally no clear examples of this done online that I can find.

For example, there is an exercise where he says to put a cell complex structure on the sphere with its poles identified. How exactly do you visualize the cell structure and attaching maps and compute the fundamental group?

Answer 1

It's not always possible to visualize things perfectly! But in this case, the spaces involved are quite reasonable. A useful fact here is that a CW pair $(X,A)$ yields a quotient space $X/A$ which inherits a cell structure from $X$ (p.8 in Hatcher). The cells of $X/A$ are the cells of $X -A$ plus one new $0$-cell, the image of $A$ in $X/A$. You get the attaching maps by composing the original attaching maps with the quotient map.

So to get a cell structure on $S^2/A$, where $A$ consists of the north and south poles, we may choose to view $S^2$ as a CW-complex with $0$-skeleton $A$, two distinct arcs connecting those two points, and two discs attached to the resulting circle. In other words, the $0$-skeleton sits inside the $1$-skeleton as the equator, and the $1$-skeleton sits inside $S^2$ as the equator (and this equator passes through the north and south poles).

With this setup, $S^2/A$ has one $0$-cell corresponding to $A$, two $1$-cells and two $2$-cells. The $1$-skeleton is a wedge of two circles, and each disk is attached to this space via a loop $\gamma$ that encircles both circles of the wedge exactly once. Moreover, for the computation of the $\pi_1$ we may choose the basepoint to be the point where the circles meet, i.e., the $0$-cell. By Proposition 1.26 in Hatcher, we have $\pi_1(S^2/A) = \pi_1(S^1 \vee S^1)/N$, where $N$ is the normal subgroup generated by the loop $\gamma$. But we know that $\pi_1(S^1 \vee S^1) \cong \mathbb{Z} * \mathbb{Z} \cong \langle a,b \rangle$. Therefore $\pi_1(S^2/A) \cong \langle a,b \mid ab\rangle \cong \mathbb{Z}$.

Answer 2

CW-complexes are a generalization of the much more readily visualized simplicial complexes: https://en.wikipedia.org/wiki/Simplicial_complex. Besides purely topological research topics involving these complexes (e.g., homotopy theory), simplicial complexes are used in many other ways. One way not mentioned in the above Wikipedia article is the Finite Element Method, aimed at solving a PDE in a domain and requires first approximating that domain by a simplicial complex. Other numerical methods (e.g., the Ordered Upwind Methods by Vladimirsky et. al.) have a similar requirement. The 2-D case of a simplicial complex is a triangulated mesh. It is not a partition: the constituend triangles (simplices) share faces and vertices.

Answer 3

I am not someone familiar with even simplicial complexes, but CW complex drew my attention from my own field of interest, computational fluid dynamics. We work with a concept of a mesh or grid which has cell, cell faces, edges and nodes. They seem to be exactly the 3 skeleton, 2 skeleton, 1 skeleton and 0 skeleton which is spoken of in context of CW complexes. We cannot allow an edge to contain a node anywhere else than at its endpoints. Two edges cannot intersect anywhere other than at a node. I am not able to think through the definition of a CW complex properly, but looks like our "mesh" could possibly be an example of a CW complex.