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A closed $1$-form in a simply connected set in $R^n$ is exact. I would like a similar condition (with a reference) on sets in $R^n$ that closed $2$-forms are exact. De Rham cohomology gives an algebraic answer, but I am interested in a condition like simply connected with geometric appeal.

The condition should not exclude "too many" sets. By analogy, closed $1$-forms on contractible sets are exact. But this condition excludes too many sets, e.g., the simply connected set $R^3 - \{{0\}}$.

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    A simply connected compact manifold $M\subset \mathbb R^n$ of dimension $3$ has the property you require, by Poincaré duality. Unfortunately some Russian guy has proved that these manifolds are fairly rare , so I'm not making this an answer.2012-05-31
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    @GeorgesElencwajg Why 'unfortunately'? ;-)2012-05-31
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    Dear @Thomas: oh, I meant it's unfortunate for me, because my class of examples is consequently not a very large one. So you have heard of that Russian too?2012-05-31

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