An interval has nonzero Euler characteristic despite being odd-dimensional

Given: The closed interval $[0,1]$ as a compact smooth $1$-manifold with boundary (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and the field $X(t)=t-\frac{1}{2}$.

A contractible nonempty space has the rational homology of a point, so $H_0([0,1];\mathbb{Q})\cong \mathbb{Q}$, all higher groups vanish, and $\chi ([0,1])=1$ (Contractible nonempty spaces have the homology of a point, Euler characteristic of a compact manifold).

The field $X(t)=t-\frac{1}{2}$ is strictly outward on $\mathrm{\partial }[0,1]$ at the endpoint $1$ (where $X(1)=\frac{1}{2}>0$, pointing out of $[0,1]$) and strictly outward at the endpoint $0$ (where $X(0)=-\frac{1}{2}<0$, pointing out of $[0,1]$); its only zero is $t=\frac{1}{2}$, nondegenerate with index $\mathrm{sign}\det (1)=+1$ (The index of a nondegenerate vector-field zero, Inward, outward, and boundary-tangent vectors).

The outward-boundary form of Poincare-Hopf applies to this smooth field and gives ${\sum }_p{\mathrm{ind}}_{p}X=\chi ([0,1])$, i.e. $1=1$ (Poincare-Hopf with outward-pointing boundary).