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.

This article, or a section of it, needs explaining. In particular: what $\map I {f, Z}$ is the intersection number of: presumably the words will go something like "... the intersection number of $f$ with respect to $Z$", or something. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

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

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