Extendability Theorem for Intersection Numbers

Theorem
Let $X = \partial W$ be a smooth manifold which is the boundary of a smooth compact manifold $W$.

Let $Y$ be a smooth manifold, $Z$ be a closed smooth submanifold of $Y$, and $f: X \to Y$ a smooth map.

If there is a smooth map $g: W \to Y$ such that $g \restriction_X = f$, then the intersection number $I(f,Z)=0$.

Corollary
Suppose $f: X \to Y$ is a smooth map of compact oriented manifolds having the same dimension.

Suppose that $X = \partial W$, where $W$ is compact.

If there is a smooth map $g: W \to Y$ such that $g\restriction_X = f$, then:
 * $\deg \left({f}\right) = 0$

where the $\deg \left({f}\right)$ denotes the degree of $f$.

Proof of Corollary
This follows immediately from the theorem.