For \(X\) a finite CW complex and \(F\) a field, show that the Euler characteristic \(\chi(X)\) can also be computed by the formula \(\chi(X) = \sum_n (-1)^n \dim H_n(X; F)\), the alternating sum of the dimensions of the vector spaces \(H_n(X; F)\).
Solution
Let \(C_*^{CW}(X;F)\) be the cellular chain complex over the field \(F\). Since \(X\) is finite, every chain group is finite-dimensional and \[ \dim_F C_n^{CW}(X;F)=c_n, \] the number of \(n\)-cells of \(X\).
Write \[ Z_n=\ker d_n, \qquad B_n=\operatorname{im}d_{n+1}. \] There are short exact sequences \[ 0\to Z_n\to C_n\to B_{n-1}\to0 \] and \[ 0\to B_n\to Z_n\to H_n(X;F)\to0. \] Taking dimensions gives \[ \dim C_n =\dim B_n+\dim H_n(X;F)+\dim B_{n-1}. \] Therefore \[ \begin{aligned} \chi(X) &=\sum_n(-1)^n\dim C_n\\ &=\sum_n(-1)^n\dim H_n(X;F) +\sum_n(-1)^n\dim B_n +\sum_n(-1)^n\dim B_{n-1}. \end{aligned} \] The two boundary sums cancel after shifting the index, leaving \[ \boxed{ \chi(X)=\sum_n(-1)^n\dim_F H_n(X;F).} \]