Intersection number on $S^k$

Question

Is the intersection number on $S^k$ is always zero? Choose $X$ a compact submanifold of $S^k$, and $Z$ a closed submanifold of complementary dimension, viewing $I_2(X,Z) = I_2(i, Z)$ where $i: X \hookrightarrow Y$ is the inclusion. Then $X,Z$ can always be homotopped to be disjoint?

Answer 1

Yes, intersection number is zero unless either X or Z is a finite set. This follows from vanishing of homology groups of the sphere except in dimension 0 and k. You can also homotope the submanifolds to be disjoint. Hint: first do this for $R^k$ instead of the sphere.