Hatcher, Algebraic Topology, Chapter 0
17. (a) Show that a mapping cylinder of every map is a CW complex.
(b) Construct a 2-dimensional CW complex that contains both an annulus and a Möbius band as deformation retracts.
Proof. (a) Let be arbitrary and consider the mapping cylinder . In particular, the space “looks like” a “hollow cylinder” attached to a copy of by the identification for all in the “top copy” of . The process to construct a CW structure on is as follows:
- Begin with two points and .
- Construct the 1-skeleton of by adding three 1-cells , :
- Add two 1-cells and as disjoint circles for which , .
- Let be the 1-cell formed from the “graph” of , drawn “around” the “cylindrical part” of .
- Attach a 2-cell by the attachment map that identifies with the 1-skeleton formed above.
Figure 1 below shows a (very crude) illustration of what such a thing might look like. Note that the red curve in Figure 1 represents the “graph” of , where the thickened red line represents the part of the graph “on the front” of the cylinder and the thinner line represents the part “on the back” of the cylinder.
(b) This problem follows a pattern similar to that in part (a). First, let denote a Möbius band and let denote an annulus. Here, note that both and retract onto their center circles and , respectively. Let and denote these retractions.
Let and be defined to be the mapping cylinders associated with and , respectively; that is,
Note, in particular, that both spaces , , deformation retract onto their “range” subspaces as noted on page 2 of the text, i.e., deformation retracts onto and deformation retracts onto . Finally, one can form the desired space by “gluing together” the spaces , , via the identity map on . Visually, then, looks like a paper towel roll with a Möbius band glued to one end and an annulus glued to the other, and by previous remarks, deformation retracts onto the spaces and .