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? 

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 $M$.
