The index sum of an outward field on an even-dimensional manifold

Given: A compact smooth even-dimensional manifold $M$ with nonempty boundary, and a smooth field $X$ on $M$ with only isolated zeros, nonzero and strictly outward along $\mathrm{\partial }M$.

Reduction: the zeros of $X$ lie in the interior at positive distance from $\mathrm{\partial }M$, because $X\mathrm{\ne }0$ on the compact boundary and there are finitely many zeros; hence, by part (iii) of the index-perturbation lemma, they can be perturbed inside disjoint small balls contained in the interior of $M$ and away from a neighbourhood of $\mathrm{\partial }M$, producing a field $X_0$ with only nondegenerate zeros, the same index sum, and still strictly outward on $\mathrm{\partial }M$ (The local index is additive under a transverse perturbation, Isolated zero and local index of a vector field, Inward, outward, and boundary-tangent vectors).

Doubling: choose the global flow collar of $-X$ and use it to define $DM$, a closed smooth $n$-manifold with seam involution $\tau$, and the reflected field $X^{+}$ of The reflection of an outward field extends over the double is smooth, has zeros exactly the two copies of the zeros of $X$, and for every zero $p$, ${\mathrm{ind}}_{\tau p}{X}^{+}={\mathrm{ind}}_p(-X)=(-1{)}^n{\mathrm{ind}}_{p}X$ (The double of a smooth manifold with boundary, The double has a well-defined smooth structure, Negation scales the local index by $(-1{)}^n$).

Poincare-Hopf on $DM$: the index sum of $X^{+}$ on the closed manifold $DM$ equals $\chi (DM)$ (Poincare-Hopf for closed manifolds).

Additivity: $\chi (M{\cup }_{\mathrm{\partial }M}M)=\chi (M)+\chi (M)-\chi (\mathrm{\partial }M)$ by part (iii) of Finiteness and additivity of the Euler characteristic; the inclusion $\mathrm{\partial }M\hookrightarrow M$ is a cofibration, because the collar of Collar neighborhood theorem is a neighbourhood deformation retract structure, whose mapping-cylinder retraction characterizes cofibrations (Cofibrations are characterized by a retraction of the mapping cylinder strip); and $\chi (\mathrm{\partial }M)=0$ because $\mathrm{\partial }M$ is a closed manifold of odd dimension $n-1$ (Closed odd-dimensional manifolds have zero Euler characteristic, Euler characteristic of a compact manifold).