I'm trying to understand the geometric meaning of (symmetric) bilinear forms.
I'm reading parts of "Symmetric Bilinear Forms", in particular, the appendix mentions what I'm interested in: on page 100 they write
"Let $M = M^{2n}$ be a closed manifold of dimension $2n$, and let $F_2$ be the field with two elements. If $x,y$ are homology classes in $H_n(M, F_2)$, the intersection number $ x \cdot y = y \cdot x \in F_2$ is defined. The Poincaré duality theorem, see e.g. [Spanier], implies that $H_n(M,F_2)$ is an inner product space over $F_2$ using the intersection number as inner product."
What exactly is the intersection number? For example: I don't know how to think about $n>1$ but I think if $n=1$ we think of the elements in $H_1$ as equivalence classes of paths, namely, two paths are equivalent if they differ by a boundary which means that $p_1 - p_2 = \partial U$ where $U \subset M$ is a submanifold (is this the correct term?) of $M$. If $M$ is for example the torus, we see that any two cycles around it form the boundary of a cylinder, hence are equivalent. Similarly, any two cycles around the centre hole form the boundary of an annulus, hence are equivalent. Hence the first homology group is generated by two elements and hence we get $H_1(T) = \mathbb Z \oplus \mathbb Z$. Now assume we pick two arbitrary representatives $x,y$ of each equivalence class. Now I don't know the actual definition of intersection number but assuming it means number of points of intersection, is it correct that $x \cdot y = 1$ in the example of the torus? And what about a sphere? Then it should be $x \cdot y = 0$ because we don't have any holes. Can you please give me a rigorous definition of intersection number?
Where does the Poincaré duality come in here? As far as I know it tells us that $H^k (M^n) = H_{n-k}(M^n)$. I dont's see where we need this to compute intersection numbers.
Thank you for your help.