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I've heard somewhere that a compact submanifold of a manifold gives a homology class of the whole manifold and vice versa. Can somebody explain this to me?

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    I don't know a lot about this, but in one direction it seems that one should take the inclusion $i\colon N \to M$ and push forward the [fundamental class](http://en.wikipedia.org/wiki/Fundamental_class) $[N]$ of $N$. In the other direction, maybe you could use [Poincaré duality](http://en.wikipedia.org/wiki/Poincaré_duality)? This last bit I'm not so sure of. – 2011-11-12

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