I am referring to the theorem that $\chi(X)=\sum_{n}(-1)^n\text{rank}\,H_n(X)$.
The proof I know (Hatcher pg 146-147) uses the short exact sequences $0\to Z_n\to C_n\to B_{n-1}\to 0$ and $0\to B_n\to Z_n\to H_n\to 0$.
While that proof is very nice, I am just curious if there exists alternate proofs?
Thanks.