Extendability Theorem for Intersection Numbers

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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.

Let there exist a smooth map $g: W \to Y$ such that $g \restriction_X = f$.


Then:

$I \left({f, Z}\right) = 0$

where $I \left({f, Z}\right)$ is the intersection number.


Corollary

Let $f: X \to Y$ be a smooth map of compact oriented manifolds having the same dimension.

Let $X = \partial W$, 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 $\deg \left({f}\right)$ denotes the degree of $f$.


Proof