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 $Y$
$f: X \to Y$ be a smooth map.

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


Then:

$\map I {f, Z} = 0$

where $\map I {f, Z}$ 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:

$\map \deg f = 0$

where $\map \deg f$ denotes the degree of $f$.


Proof