Hatcher, Algebraic Topology, Chapter 0
14. Given positive integers , , and satisfying , construct a cell structure on having 0-cells, 1-cells, and 2-cells.
Proof. To begin, consider the triple of integers satisfying and note1 that the following system of equations holds:
In particular, introducing the parameters and , it follows that the triple can be written as -linear combinations of the triples as follows:
From the beginning, the problem could have been reduced to the related problem of describing in detail how to modify the base CW-structure of to accommodate for the addition of 0-, 1-, or 2-cells; given this new parameterization, the problem will be solved by showing how to accommodate unit increases of and independently given a base CW-structure for and by reverting to induction for the general case.
Consider two cases.
Case 1: Increasing by 1 while leaving fixed.
Notice that in equation above, is being applied to the triple , whereby the increase equates to adding a vertex and an edge. For this construction, consider the sphere with the cell structure corresponding to one 0-cell and one 2-cell , whereby is formed via and the attachment map . The goal is to “start the CW construction over” now with two 0-cells, one 1-cell, and the above 2-cell and to show that can have this CW-structure.
Let the added components be denoted as and for the 0- and 1-cells, respectively, and without loss of generality, suppose that is attached to in such a way that as shown in Figure 1 below. This describes completely the 1-skeleton of the CW-structure being constructed.
Next, attach the 2-cell in the obvious way, namely by identifying the boundary of a disc with the entirety of . More precisely, if denotes the structure at hand, where . The result, then, is that is “almost a in a precise way” as shown in Figure 2 below.
Obviously, then, can be deformed continuously into a hemisphere by reaching under the space (think about as a 2-dimensional space in so that “under” makes sense) and pushing up until is rounded. Of course, the map for which is a continuous mapping of a hemisphere into a sphere (presuming our hemispheres/spheres are unit hemispheres/spheres, which can be assumed), whereby it follows that the space can be continuously deformed into . This means can be given such a cell structure.
Case 2: Increasing by 1 while leaving fixed.
As in Case 1, notice that equation above indicates the effect of increasing . In particular, is applied to the triple , and so the map is equivalent to adding an edge and a face (i.e., a 1-cell and a 2-cell). For this construction, the CW-model of that’s most convenient is the one having two cells in each of dimensions 0, 1, and 2. Letting , , and , , denote these cells, the CW-structure of is shown in Figure 3 below. Note that the cells , , can be thought of as hemispheres being glued to the equator circle formed by , .
As per the above remark, increasing means adding a 1-cell and a 2-cell without adding any 0-cells. In particular, the most logical way to do so would be to add as a loop from one of the vertices , , and to glue the 2-cell to this 1-cell by identifying the boundary of a disc to the (homeomorphic to) that is . A picture of the components of this construction is shown in Figure 4 below.
Using the above identification of the 2-cells , , with hemispheres, it’s easy to see that the space constructed in this way is homotopic to the space consisting of the wedge sum where is a hemisphere of . In particular, then, applying the map takes the space to the space , and because the wedge of two 2-spheres is again a 2-sphere, it follows that this space can describe a viable CW-structure on .
To conclude, here’s how to form the general case: Say we’re given and told that . Some sub-collection (not necessarily proper) must generate a copy of because the collection must satisfy , . Using the sphere generated by the cell-complex with 0- 1-, and 2-cells (respectively), consider the values of the integers and to accommodate the differences : This will yield values . Following the procedure in cases 1 and 2 above, “count up” to and by considering using the pseudo-algorithm:
Do , fixed, until .
Do until .
The result will be a cell structure on satisfying the desired parametrization.
1. The motivation for this technique was borrowed from the very terse solution given here.