The two following theorems appear to be contradictory. Both are proved in Milnor's "Topology from the differentiable viewpoint." I'm sure I'm overlooking something incredibly trivial:
Let $f$ be a smooth map from a smooth compact oriented manifold $K$ to a smooth connected oriented manifold $N$, both of dimension $m-1$. If $K$ is the oriented boundary of some smooth compact oriented $m$-manifold (with boundary) $Y$ and if $f$ extends to a smooth map from $Y$ to $N$ then the degree of $f$ (i.e. the sum of the signs of the determinants of the Jacobians at the elements of the preimage of any regular value) is zero.
(Poincaré-Hopf) If $M$ is a smooth compact manifold and $V$ is a smooth vector field on $M$ with isolated singularities then the degree of the vector field is equal to the Euler characteristic of $M$.
Now consider the setting of the second statement and assume $M$ is oriented. delete a small open ball around each zero of $V$. This gives a manifold with boundary $Y$ and by normalizing $V$ we get a map $g$ from $Y$ to the $(m-1)$-sphere. Now if $K$ is the boundary of $M$ then the first statement says the degree of $g$ restricted to $K$ is zero while the second statement says it is the Euler characteristic of $M$.