Intersection Pairing and Poincaré Duality

Question

Let $M$ be an $n$-dimensional compact and oriented manifold. Then one can define the intersection pairing $H_k(M,\mathbb Z) \times H_{n-k}(M,\mathbb Z) \to \mathbb Z$. One possible formulation of the Poncaré duality is the folowing:

Every linear functional $H_{n-k}(M,\mathbb Z) \to \mathbb Z$ is given by intersection with some class $\alpha \in H_k(M,\mathbb Z)$ and if $\beta \in H_k(M,\mathbb Z)$ has interesection number $0$ with every class in $H_{n-k}(M,\mathbb Z) $ then $\beta$ is a torsion element.

This formulation is given on Griffiths & Harris "Principles of Algebraic Geometry" and they use this to define the fundamental class of a closed sumbanifold as follows: if $V \subset M$ is a closed and oriented submanifold of dimendion $k$, intersection with $V$ defines a linear funcional $H_{n-k}(M,\mathbb Z) \to \mathbb Z$ and they say

"the corresponding cohomology class $\eta_V \in H^{n-k}(M)$ is the fundamental class of $V$."

What does they mean by "the corresponding cohomology class"? I can see that $\text{Hom} (H_{n-k}(M,\mathbb Z), \mathbb Z)$ is related to $H^{n-k}(M,\mathbb Z)$ by the universal coefficient theorem, but how is $\eta_V$ obtained explicitly?

Answer 1

If $V$ is the closed submanifold of dimension $k$, I would think of $\eta_V$ as the unique closed $n-k$-form with the property that, for all $\gamma \in H_{n-k}(X)$, one has $$ \int_{\gamma} \eta_v = \gamma\cdot V, $$ where $\cdot$ is the intersection product and $\gamma$ is an arbitrary homology class of dimension $n - k$. The integral is well-defined here by Stokes.