Extendability Theorem for Intersection Numbers
Contents |
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
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Proof of Corollary
This follows immediately from the theorem.
$\blacksquare$
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