Define $\mathbb{R}P^k$ as the quotient of $S^k$ by the antipodal map, with smooth structure defined so that the projection $p: S^k \to \mathbb{R}P^k$ is a local diffeomorphism. Suppose that $k$ is odd and $Z_1, Z_2 \subset \mathbb{R}P^k$ are compact submanifolds of positive dimension for which the oriented intersection number $I(Z_1, Z_2)$ is defined. Prove $I(Z_1, Z_2) = 0$. (Hint: What conditions are guaranteed by well-definedness of $I(Z_1, Z_2)$?) Is the corresponding statement true for the mod-2 intersection number $I_2$?
Because $k$ is odd and $Z_1, Z_2$ are of complementary dimension, $I(Z_1, Z_2) = I(Z_2, Z_1)$. But then I found trouble to proceed. Could someone point out what is the missing piece? Or if I am on the right track at all?
Hints: First of all, note that for any $\ell<k$, $\mathbb RP^\ell$ and $\mathbb RP^{k-\ell}$ (generically) intersect in a single point. But the hypotheses of the problem are very specific: If $k$ is odd, $\mathbb RP^k$ is orientable, but to make sense of $I(Z_1,Z_2)$, we need both $Z_1$ and $Z_2$ to be oriented. One more hint: What is the intersection number of complementary dimension (and positive-dimensional) oriented submanifolds of $S^k$?