CW complex which is not a polyhedron

Question

I understand that a CW complex is a generalization of the concepts of simplicial complex, and it is simple to visualize how any polyhedron can be seen as a CW complex, but since they're different concepts there must be CW complexes that are not polyhedrons. Can anyone give a example of the construction of a such CW complex? I've searched a few books for an example but I couldn't find anything except for a problem in Rotman's Introduction to Algebraic Topology where he suggests attaching a 2-cell to a circle via the mapping $x\sin(\frac{1}{x})$ gives a CW complex with this property, but I couldn't construct it myself even with the hint.

Answer 1

Intuitively, when you attach a disk to a space $X$ along a map $f:S^1\to X$, what you are doing locally is gluing a rectangle $[0,1]\times[0,1]$ to $X$ by having the bottom side of the rectangle trace out the path given by $f$. (Globally, the path $f$ is a loop so the left and right sides of this rectangle glue together, and then the top end of the rectangle comes together at a single point, the center of the disk you are attaching, but let's not worry about that.)

In particular, as in Rotman's example, imagine that the path $f:S^1\to X$ passes through a single point $p$ infinitely many times while oscillating around it. When you attach a disk along $f$ to get a new space $Y$, the local topology will now be very bad near $p$. It will look like $X$ near $p$, but with infinitely many two-dimensional "flaps" attached at $p$, one for each time the path passes through $p$. (To visualize this, imagine taking a rectangle and folding its bottom edge on itself infinitely many times with the folds getting smaller and smaller.)

Proving rigorously that the resulting space is not a polyhedron is a bit tricky, but hopefully it should be geometrically clear that a (finite) polyhedron cannot look like this locally, since you would need separate 2-cells for each of the infinitely many "flaps". (As for the finiteness assumption, since this space is compact, if it were a polyhedron it would have to be a finite polyhedron.)