The outward boundary hypothesis cannot be replaced by nonzero on the boundary

Assume the Axiom of Choice (The Axiom of Choice) for the applications of the general index theorems below.

The hypothesis in Poincare-Hopf with outward-pointing boundary is strictly stronger than "$X\mathrm{\ne }0$ on $\mathrm{\partial }M$": a field that is nonzero on the boundary but not outward, with only isolated zeros, contributes a boundary correction term. On the closed unit ball $D^n\subseteq {\mathbb{R}}^n$, $n\ge 3$ odd, the inward radial field $X(u)=-u$ is nonzero on $\mathrm{\partial }{D}^n$ and has the single zero $0$, which is nondegenerate with linearization $-I_n$ (Inward, outward, and boundary-tangent vectors, Isolated zero and local index of a vector field); by The index of a nondegenerate vector-field zero its index is $\mathrm{sign}\det (-I_n)=(-1{)}^n=-1$. On the other hand $\chi (D^n)=1$, because $D^n$ is contractible with the rational homology of a point (Contractible nonempty spaces have the homology of a point, The Axiom of Choice, Euler characteristic of a compact manifold). Hence ${\sum }_p{\mathrm{ind}}_{p}X=-1\mathrm{\ne }1=\chi (D^n)$: nonzero on the boundary does not suffice, and the outwardness in the boundary form is a genuine hypothesis rather than a convenience.

In even dimensions the inward radial field on $D^n$ has index $+1$ and happens to agree with $\chi (D^n)=1$, despite not being outward. Thus equality of the index sum with $\chi$ does not imply outwardness. For a compact smooth full-dimensional Euclidean domain $N\subset {\mathbb{R}}^n$, $n\ge 1$, the boundary lemma identifies the index sum of a smooth field with only isolated zeros and nonzero on $\mathrm{\partial }N$ with the degree of its normalized boundary map (reduced degree when $n=1$). Outwardness is sufficient to identify that degree with the Gauss degree, which equals $\chi (N)$ by the outward-boundary theorem. This sphere-map description uses the Euclidean tangent trivialization and is not asserted for an arbitrary manifold with a possibly nontrivial tangent bundle.