Cellular homology

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.

Definition

If X is a CW-complex with n-skeleton X_n, 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. Let $e_n^{\alpha}$ be an n-cell of X, let $\chi_n^{\alpha} : \partial e_n^{\alpha}\cong S^{n-1} \to X_{n-1}$ be the attaching map, and consider the composite maps

$\chi_n^{\alpha\beta}:S^{n-1} \to X_{n-1} \to X_{n-1}/(X_{n-1}-e_{n-1}^{\beta})\cong S^{n-1}$

where $e_{n-1}^{\beta}$ is an (n − 1)-cell of X and the second map is the quotient map identifying $(X_{n-1}-e_{n-1}^{\beta})$ to a point.

The boundary map

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

is then given by the formula

$d_n(e_n^{\alpha})=\sum_{\beta}deg(\chi_n^{\alpha\beta})e_{n-1}^{\beta}\,$

where $deg(\chi_n^{\alpha\beta})$ is the degree of $\chi_n^{\alpha\beta}$ and the sum is taken over all (n − 1)-cells of X, considered as generators of $H_{n-1}(X_{n-1},X_{n-2})\,$.

Other properties

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.

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 CPⁿ has a cell structure with one cell in each even dimension; it follows that for 0 ≤ k ≤ n,

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

and

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

Generalization

The Atiyah-Hirzebruch spectral sequence is the analogous method of computing the (co)homology of a CW-complex, for an arbitrary extraordinary (co)homology theory.

Euler characteristic

For a cellular complex X, let X_j be its j-th skeleton, and c_j be the number of j-cells, i.e. the rank of the free module H_j(X_j, X_j-1). The Euler characteristic of X is defined by

$\chi (X) = \sum _0 ^n (-1)^j c_j.$

The Euler characteristic is a homotopy invariant. In fact, in terms of the Betti numbers of X,

$\chi (X) = \sum _0 ^n (-1)^j \; \mbox{rank} \; H_j (X).$

This can be justified as follows. Consider the long exact sequence of relative homology for the triple (X_n, X_{n - 1} , ∅):

$\cdots \to H_i( X_{n-1}, \empty) \to H_i( X_n, \empty) \to H_i( X_{n}, X_{n-1} ) \to \cdots .$

Chasing exactness through the sequence gives

$\sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, \empty) = \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, X_{n-1}) \; + \; \sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_{n-1}, \empty).$

The same calculation applies to the triple (X_{n - 1}, X_{n - 2}, ∅), etc. By induction,

$\sum_{i = 0} ^n (-1)^i \; \mbox{rank} \; H_i (X_n, \empty) = \sum_{j = 0} ^n \; \sum_{i = 0} ^j (-1)^i \; \mbox{rank} \; H_i (X_j, X_{j-1}) = \sum_{j = 0} ^n (-1)^j c_j.$

References

• A. Dold: Lectures on Algebraic Topology, Springer ISBN 3-540-58660-1

• A. Hatcher: Algebraic Topology, Cambridge University Press ISBN 978-0-521-79540-1