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$X$ is a compact, oriented $n$-manifold and $Y$ is a $k$-submanifold of $X$. Let $\eta_{Y}$ be the Poincare dual class to the fundamental cycle $[Y] \in H_{k}(X,\mathbb{Z})$ and $\phi \in H^{k}_{dR}(X)$.

Why is $\int_{Y} \phi = \int_{X} \eta_{Y} \wedge \phi$ ?

lnth
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    Do you mean $\phi \in H^{\color{red}k}_{\text{dR}}(X)$ and $\int_Y \phi = \int_X {\color{red}\eta_Y} \wedge \phi$? – Henry T. Horton Jun 08 '13 at 19:36
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    This is basically the definition of Poincaré duality. What's more interesting is that if $\phi=\eta_Z$ for a codimension $k$ submanifold $Z$, then this integral computes the intersection number of $Y$ and $Z$. BTW, you need $X$ compact and oriented. – Ted Shifrin Jun 08 '13 at 23:22
  • @Ted Shifrin I see, thank you very much. – lnth Jun 09 '13 at 01:56

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