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.