Intersection number and compact submanifolds

Question

Let $M^{m}$ and $P^{k}$ be compact submanifolds of $N^{n}$, where $m+k = n$. We define the intersection number $I(M,P)$ as the intersection number of the canonical inclusion $i:M\rightarrow N$ with $P$. Prove that $I(M,P) = (-1)^{mk}I(P,M)$. Thank you in advance for any help.

Answer 1

You must chose orientations. Thus if q is a point of intersection of M and P, then there are positive ordered basis $v_1,\ldots ,v_m$ and $w_1,\ldots ,w_p$ of the tangent spaces to M and P, respectively, at q. Now the orientation of q as a point of intersection is $\pm 1$ depending on whether the basis
$v_1,\ldots ,v_m, w_1,\ldots ,w_p$ is positively or negatively ordered as a basis of N. Changing the order of M and P gives the basis $w_1,\ldots ,w_p, v_1,\ldots ,v_m$ the result follows.