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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?

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I thought that definition was already pretty explicit.

You say that one can define a linear functional $H_{n-k}(M,\mathbb{Z})\rightarrow\mathbb{Z}$ by declaring "for $\alpha\in H_{n-k}(M,\mathbb{Z})$, send it to $\alpha\cdot [V]$ where $[V]$ is the homology class of $V$" (where presumably $\alpha\cdot[V]$ has already been defined)

Then the universal coefficient theorem says that $\hom(H_{n-k}(M;\mathbb{Z}),\mathbb{Z})$ is isomorphic to $H^{n-k}(M;\mathbb{Z})$ if Ext$(H_{n-k-1}(M;\mathbb{Z}),\mathbb{Z})=0$. We don't know that it is, but regardless there is a surjection from cohomology to the hom group, so every linear functional comes from a cohomology class. So you can explicitly define $\eta_V\in H^{n-k}(M,\mathbb{Z})$ by

$\eta_V(\alpha)=\alpha\cdot[V]$

(One issue I'm having with this definition is that I'm used to the intersection number being defined using Poincaré duality. What is the definition you/they are using for $\alpha\cdot[V]$? Is this actually what you are asking about?)

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    Do you know a way to describe this cohomology class explicitly? We need to check that this lift is well defined, because we could choose two different classes that map to the same functional.2012-03-03
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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.