intersection number of twocompact oriented manifolds

Question

I have an oriented manifold M of n dimension and 2 oriented submanifolds, one of dimension k and the other of dimension n-k , I have to understand which is the intersection number of those manifolds. I know that every submanifold represent an homology class and I have this definition of intersection: $H_{k}(M) \times H_{n-k}(M) \longrightarrow H_{0}(M)\simeq \mathbb{Z}$, where the last isomorphism is the function which counts the points of intersection. If the submanifolds are transversals the intersection number is a sum of 1 and -1 on the points of intersection, my question is when it is 1 and when it is -1?

Answer 1

I an not 100% sure about the details, but I believe that things are more or less as follows.

An orientation on a manifold amounts to fix consistently an orientation of a base of the tangent space at each of its points. If $A$ and $B$ are subvarieties of $M$ of complementary dimensions intersecting at $P$ transversally, there's a decomposition of tangent spaces $$ T_p(M)=T_p(A)\oplus T_p(B) $$ Choose bases $\cal B$ of $T_p(A)$ and $\cal B^\prime$ of $T_P(B)$ corresponding to the given orientations and consider the basis ${\cal B}\cup{\cal B}^\prime$ of $T_p(M)$. If its orientation corresponds to that of $M$ then the intersection pairing value is $+1$, else $-1$.

Answer 2

Let $N$ be the submanifold of dimension $k$ and $N^\prime$ the submanifold of dimension $n-k$. For each point $x \in N\cap N^\prime$, the fact that the intersection is transversal implies that $T_x N \times T_x N^\prime = T_x M$. This is true on the level of vector spaces, but not necessarily on the level of oriented vector spaces. If the orientation of $T_x N \times T_x N^\prime$ is the same as $T_x M$, then it is $+1$, if not, $-1$. You can check this easily (depending on what information you have) by concatenating an oriented basis for $T_x N$ with an oriented basis for $T_x N^\prime$ to yield a basis for $T_x M$ and checking to see if this basis has the same orientation or the opposite orientation as that of $M$.