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 $$g|_X = f$$, then the intersection number $$I(f,Z)=0$$.

= Proof =

= Corollary =

Suppose $$f:X \to Y$$ is a smooth map of compact oriented manifolds having the same dimension and that $$X = \partial W$$, where $$W$$ is compact. If there is a smooth map $$g:W \to Y$$ such that $$g|_X = f$$, then the degree $$\text{deg}(f)=0$$.

= Proof =

This follows immediately from the theorem.