Show that if $f: X \rightarrow S^k$ is smooth, $X$ compact and $0 < dim(X) < k$, then for all closed $Z \subseteq S^k$ of dimension complementary to $X$, we have $I_2(X, Z) = 0$.
An idea：let p be a regular value of f, using Sard,which is to say a point missed by X.Then we can stereographically project away from that point,so pretend the $f: X \rightarrow \mathbb{R}^k$ ,and then homotope f to a constant map by shrinking in $\mathbb{R}^k$.
A detailed explain to this idea or another solution to this problem in detail would be appreciated a lot!