Cellular homology

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In mathematics, cellular homology in algebraic topology is a homology theory for CW-complexes.

It agrees with singular homology, and can provide an effective means of computing homology modules. If X is a CW-complex with n-skeleton Xn, the cellular homology modules are defined as the homology groups of the cellular chain complex:

\cdots \to H_{n+1}( X_{n+1}, X_n ) \to H_n( X_n, X_{n-1} ) \to H_{n-1}( X_{n-1}, X_{n-2} ) \to \cdots .

The module

H_n( X_n, X_{n-1} ) \,

is free, with generators which can be identified with the n-cells of X. The boundary maps

H_n(X_n,X_{n-1}) \to H_{n-1}(X_{n-1},X_{n-2}) \,

can be determined by computation of the degrees of the attaching maps of the cells.

An important consequence of the cellular perspective is that if a CW-complex has no cells in consecutive dimensions, all its homology modules are free. For example, complex projective space \mathbb{CP}^n has a cell structure with one cell in each even dimension; it follows that

H_{2k}(\mathbb{CP}^n; \mathbb{Z}) \cong \mathbb{Z}

for 0\leq k \leq n, and

H_{2k+1}(\mathbb{CP}^n) = 0 for all k.

Also, one sees from the cellular chain complex that the n-skeleton determines all lower-dimensional homology:

H_k(X) \cong H_k(X_n)

for k < n.

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