 Solution

Let $m=\dim X$ be odd. Use mod-two Poincare duality, which applies even when the closed manifold is nonorientable. It gives $b_k^{(2)}=b_{m-k}^{(2)}$, where $b_k^{(2)}={\dim }_{{\mathbb{F}}_2}{H}_k(X;{\mathbb{F}}_2)$. The Euler characteristic can be computed with any field of coefficients: for a finite cell model, the alternating sum of chain dimensions equals the alternating sum of homology dimensions, and the former counts cells independently of the field.

Consequently $\chi (X)={\sum }_{k=0}^m(-1{)}^k{b}_k^{(2)}$. Pair each term with index $m-k$. The paired Betti numbers are equal and their signs opposite because $m$ is odd; there is no unpaired middle index. ThereforeThis argument applies componentwise if the manifold is disconnected and does not require an orientation.