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. 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:

:$cdots\; o\; H\_\{n+1\}(\; X\_\{n+1\},\; X\_n\; )\; o\; H\_n(\; X\_n,\; X\_\{n-1\}\; )\; o\; H\_\{n-1\}(\; X\_\{n-1\},\; X\_\{n-2\}\; )\; o\; cdots\; .$

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

:$H\_n(X\_n,X\_\{n-1\})\; o\; H\_\{n-1\}(X\_\{n-1\},X\_\{n-2\})\; ,$

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

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 CP^"n" has a cell structure with one cell in each even dimension; it follows that for 0 &le; "k" &le; "n",

:$H\_\{2k\}(mathbb\{CP\}^n;\; mathbb\{Z\})\; cong\; mathbb\{Z\}$

and

:$H\_\{2k+1\}(mathbb\{CP\}^n)\; =\; 0\; .$

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} , &empty;):

:$cdots\; o\; H\_i(\; X\_\{n-1\},\; empty)\; o\; H\_i(\; X\_n,\; empty)\; o\; H\_i(\; X\_\{n\},\; X\_\{n-1\}\; )\; o\; cdots\; .$

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}, &empty;), 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.