intersection number of twocompact oriented manifolds 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? 

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 $-1$.
