TITLE: Euler characteristic: dependence on coefficients
QUESTION [10 upvotes]: Let $X$ be a finite CW complex and $\chi(X)$ its Euler characteristic (defined using integer coefficients). When is it true that $\chi(X)=\sum (-1)^i \dim H_i(X;F)$, where $F$ is a field?
I thought it would be true for all fields, but I noticed that for $X$ the Klein bottle and $F=\mathbb{Z}/2$ this is false! In fact Bredon (Geometry and topology) claims it to be true, but it isn't..

REPLY [7 votes]: There is also a way showing the independence of $\chi (X)$ without using the universal coefficient theorem, but using cellular homology:
Let $a_n$ denote the number of cells of $X$ in dimension $n$. Then we get the cellular chain complex 
$$\ldots \longrightarrow F^{a_k} \longrightarrow F^{a_{k-1}} \longrightarrow \ldots $$
Using that the Euler characteristic of a chain complex equals that of its Homology (which is proved using the dimension formula for vector spaces), we get 
$$\chi_F=\sum (-1)^i a_i$$ and 
$$\chi_F=\sum (-1)^i \dim H_i(X;F)$$
which is then independent of the field.