Hatcher, Algebraic Topology, Chapter 0
15. Enumerate all the subcomplexes of , with the cell structure on that has as its -skeleton.
Proof. Recall that by definition, (see page 7 in Chapter 0). Moreover, recall that for finite, can be given a cell structure in which each of the spheres , , is a subcomplex: This is done by way of attaching two -cells to the considered as the equator as for all . More precisely:
Begin with as two points and ; obtain by attaching two “arcs” (i.e., closed segments homeomorphic to the 1-dimensional “disk”) and to create a circle; next, obtain by attaching two 2-cells (i.e., closed regions homeomorphic to the 2-dimensional disk) and . Continue inductively by attaching two -cells and , each homeomorphic to the -disk , to the resulting . This yields a CW-complex structure on each which has two cells in each dimension and which has as a subcomplex for each . The construction for , , is shown in this solution to a previous exercise (see Figure 3 in that solution).
Therefore, we have that and that is the union of all the . Also, for any , a subcomplex of is automatically closed (by definition of subcomplex) and therefore must contain the boundary of any -cell it contains, . What does this mean? It means that any (closed) subcomplex of which contains a single -cell must also include every single -cell for . Thus, it follows that the only -dimensional subcomplexes of are the hyper-hemispheres1 of the form and so enumerating these for all yields the final result.
1. Thanks to Tarun Chitra for coming up with a good prose way to describe these objects geometrically. His work can be found here.