Odd-dimensional closed manifolds have zero Euler characteristic
= Odd-dimensional closed manifolds have zero Euler characteristic
{title2=$\chi(M^{2r+1})=0$}
Use <Poincare duality> with coefficients in $\mathbb F_2$, valid also without orientability. The ranks in complementary degrees agree, and their signs in the alternating sum are opposite. They therefore cancel in pairs. Euler characteristic is independent of the coefficient field, giving the stated integer equality.