Cellular Homology - Euler Characteristic

Euler Characteristic

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

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

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

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 (Xn - 1, Xn - 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.


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