CW structure of spheres

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

Description of cells and attaching maps

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.