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I am trying to solve the following question as stated by my teacher: Let $V \subset X$ be an inclusion of oriented real compact manifolds. Suppose that codim $_X V = r$. Using the bilinear pairing through the wedge product between $H^k_{DR}(X, \mathbb{R})$ and $H^{n-k}_{DR}(X, \mathbb{R})$ show via integration of forms over $V$ , that $V$ determines naturally a cohomology class $[V] \in H^r_{DR}(X, \mathbb{R})$.

I have no clue on how to get started on this problem. The question essentially asks to canonically associate a closed $r-$ form upto exactness using the bilinear pairing, but I am not able to come up with such a procedure.

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  • $\begingroup$ No, the question doesn't ask what you say it essentially asks. Rather, it asks you to come up with an element of the dual space to $H^{n-k}$, and tells you how to do it: by integrating representative $n-k$ forms over $V$. $\endgroup$
    – tracing
    Dec 13, 2016 at 23:53
  • $\begingroup$ @tracing: No, the OP is absolutely right. You are not wrong but one should use Poincaré duality after having done what you describe. See my answer. $\endgroup$ Dec 14, 2016 at 9:26
  • $\begingroup$ @GeorgesElencwajg: I agree, but my point (maybe made too emphatically --- sorry!) to the OP was (as you know) that they don't have to think about how to construct a closed $r$-form, since Poincare duality (which they seem to be assuming) will do that for them, but rather how to construct a functional on $k-r$ forms. $\endgroup$
    – tracing
    Dec 14, 2016 at 12:11

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Let $k=\operatorname {dim}(V)=n-r$. There is a canonical map $$res: H^k(X)\to H^k(V): [\omega ]\mapsto [\omega \vert V] $$ obtained by restricting differential $k$-forms on $X$ to $V\subset X$.
Integrating on $V$ yields a linear form $$\int_V:H^k(V)\to \mathbb R$$ and the composition $\int_V\circ res:H^k(X)\to \mathbb R$ is a linear form $l\in (H^k(X)^*$.
The crucial point now is that we have a Poincaré isomorphism $$\operatorname {Poinc}: H^{r }(X)\stackrel {\sim}{\to} (H^k(X))^*:[\eta]\mapsto \langle [\phi] \mapsto \int_X \eta\wedge \phi\rangle$$ Your required class is then $[V]=(\operatorname {Poinc})^{-1}(l)$

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