CW structure of spheres

This article describes how specific information about::spheres can be realized as CW-spaces, i.e., how to construct a CW-complex whose underlying topological space is a sphere.

The case $$n = 0$$
The space here is $$S^0$$, a discrete two-point space. There is an obvious and unique CW-structure for this space: two 0-cells, and no other cells.

The case $$n > 0$$
The sphere $$S^n$$ has one 0-cell and one $$n$$-cell. The The 0-cell gives a point. The attaching map for the boundary of the $$n$$-cell $$D^n$$ sends its boundary (homeomorphic to $$S^{n-1}$$) to the 0-cell point. Specifically, we are using the fact that $$S^n \cong D^n/\partial D^n$$.

Cellular chain complex and cellular homology
Any CW structure on a topological space provides a cellular filtration relative to the empty space. The corresponding cellular chain complex is described below. By excision, the $$k^{th}$$ cellular chain group is $$\mathbb{Z}^d$$ where $$d$$ is the number of $$k$$-cells.

The case $$n = 0$$
In this case, the zeroth chain group is $$\mathbb{Z} \oplus \mathbb{Z}$$ and all higher chain groups are zero. All boundary maps for this chain complex are zero, so the homology groups are the same as the chain groups. In particular, $$H_0(S^0) \cong \mathbb{Z} \oplus \mathbb{Z}$$ and all higher homology groups are zero.

The case $$n > 0$$
In this case, the zeroth and $$n^{th}$$ chain groups are $$\mathbb{Z}$$ and all other chain groups are zero. All boundary maps for this chain complex are zero, so the homology groups are the same as the chain groups. In particular, $$H_0(S^n) \cong H_n(S^n) \cong \mathbb{Z}$$ and all other homology groups are zero.

Note that for $$n > 1$$, the fact that all the chain maps are zero is forced just by looking at the chain groups. This is not the case for $$n = 1$$, but a careful look at the boundary map from the first to the zeroth chain group reveals that it is zero.