Hatcher 0.14

Hatcher, Algebraic Topology, Chapter 0

14. Given positive integers v, e, and f satisfying v-e+f=2, construct a cell structure on S^2 having v 0-cells, e 1-cells, and f 2-cells.

Proof. To begin, consider the triple (v,e,f) of integers satisfying v-e+f=2 and note1 that the following system of equations holds:

\begin{array}{rcl}v&=&1+(v-1)+0 \\ e&=&0+(1-v)+(1-f) \\ f&=&1+0+(f-1)\end{array}.

In particular, introducing the parameters m=v-1 and n=f-1, it follows that the triple (v,e,f) can be written as \{1,m,n\}-linear combinations of the triples (1,0,1),(1,-1,0),(0,-1,1) as follows:

(v,e,f) = (1,0,1) + m(1,-1,0) + n(0,-1,1).\,\,\,\,\,\,\,\,\,\,(1)

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 S^2 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 m and n independently given a base CW-structure for S^2 and by reverting to induction for the general case.

Consider two cases.

Case 1: Increasing m=v-1 by 1 while leaving n fixed.
Notice that in equation (1) above, m is being applied to the triple (1,-1,0), whereby the increase m\mapsto m+1 equates to adding a vertex and an edge. For this construction, consider the sphere S^2 with the cell structure corresponding to one 0-cell e^0_1=x_0 and one 2-cell e^2_1, whereby S^2 is formed via S^2=e^0_1\sqcup_\alpha D^2 and the attachment map \varphi_\alpha:\partial D^2\to x_0. 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 S^2 can have this CW-structure.

Let the added components be denoted as e^0_2=x_1 and e^1_1 for the 0- and 1-cells, respectively, and without loss of generality, suppose that e^1_1 is attached to e^0_2 in such a way that e^0_1\in e^1_1 as shown in Figure 1 below. This describes completely the 1-skeleton X^1 of the CW-structure being constructed.

Hatcher 0.14.1
Figure 1

Next, attach the 2-cell e^2_2 in the obvious way, namely by identifying the boundary \partial D^2 of a disc D^2 with the entirety of X^1=e^1_1\cup e^0_1\cup e^0_2\cong S^1. More precisely, if X denotes the structure at hand, X=X^1\sqcup_\alpha D^2 where \varphi_\alpha:\partial D^2\to X^1. The result, then, is that X is “almost a D^2 in a precise way” as shown in Figure 2 below.

Hatcher 0.14.2
Figure 2

Obviously, then, X can be deformed continuously into a hemisphere by reaching under the space X (think about X as a 2-dimensional space in \mathbb{R}^3 so that “under” makes sense) and pushing up until e^2_2 is rounded. Of course, the map f:\mathbb{C}\to\mathbb{C} for which z\mapsto z^2 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 X can be continuously deformed into S^2. This means S^2 can be given such a cell structure.

Case 2: Increasing n=f-1 by 1 while leaving m fixed.
As in Case 1, notice that equation (1) above indicates the effect of increasing n. In particular, n is applied to the triple (0,1,-1), and so the map n\mapsto n+1 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 S^2 that’s most convenient is the one having two cells in each of dimensions 0, 1, and 2. Letting e^0_i, e^1_i, and e^2_i, i=1,2, denote these cells, the CW-structure of S^2 is shown in Figure 3 below. Note that the cells e^2_i, i=1,2, can be thought of as hemispheres being glued to the equator circle formed by X^1=e^0_i\cup e^1_i, i=1,2.

Hatcher 0.14.3
Figure 3

As per the above remark, increasing n means adding a 1-cell e^1_3 and a 2-cell e^2_3 without adding any 0-cells. In particular, the most logical way to do so would be to add e^1_3 as a loop from one of the vertices e^0_i, i=1,2, and to glue the 2-cell e^2_3 to this 1-cell by identifying the boundary \partial D^2 of a disc D^2 to the (homeomorphic to) S^1 that is e^1_3. A picture of the components of this construction is shown in Figure 4 below.

Hatcher 0.14.4
Figure 4

Using the above identification of the 2-cells e^2_i, i=1,2,3, with hemispheres, it’s easy to see that the space X constructed in this way is homotopic to the space consisting of the wedge sum S^2\vee H where H is a hemisphere of S^2. In particular, then, applying the map z\mapsto z^2 takes the space X = S^2\vee H to the space X'=S^2\vee S^2, and because the wedge of two 2-spheres is again a 2-sphere, it follows that this space X can describe a viable CW-structure on S^2.

To conclude, here’s how to form the general case: Say we’re given v,e,f and told that v-e+f=2. Some sub-collection v',e',f' (not necessarily proper) must generate a copy of S^2 because the collection v,e,f must satisfy v\geq 1, f\geq 1. Using the sphere generated by the cell-complex with v',e',f' 0- 1-, and 2-cells (respectively), consider the values of the integers m and n to accommodate the differences v-v',e-e',f-f': This will yield values m\geq 0,n\geq 0. Following the procedure in cases 1 and 2 above, “count up” to m and n by considering m'=0=n' using the pseudo-algorithm:

Do m'\mapsto m'+1, n' fixed, until m'=m.
Do n'\mapsto n'+1 until n'=n.

The result will be a cell structure on S^2 satisfying the desired parametrization.    \square

    1. The motivation for this technique was borrowed from the very terse solution given here.

2 thoughts on “Hatcher 0.14

  1. 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.

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