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 note^{1} 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.

Your equation for e (line 5) is incorrect; I think the original source also made this mistake. The equation should read (v,e,f) = (1,0,1) + m(1,1,0) + n(0,1,1), no minus signs.

Thanks for letting me know! I’ll look into this as soon as I’m back in Algebraic Topology mode!