Let \(X\) be a finite CW complex with \(c_n\) cells of dimension \(n\). The Euler characteristic of \(X\) is \[ \chi(X)\coloneqq\sum_{n\geq0}(-1)^nc_n. \]
For a finite CW complex \(X\), \[ \chi(X)=\sum_{n\geq0}(-1)^n\rank H_n(X;\ZZ), \] where \(H_n(X;\ZZ)\) is singular homology [@Hat02]. Consequently \(\chi(X)\) does not depend on the CW structure, and finite CW complexes that are homotopy equivalent have the same Euler characteristic.