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I have a possibly trivial (possibly non-trivial? :rofl:) question. Here it is:

Assumption-Assume I have a closed p-form, whose integral over any p-cycle is always zero.

Statement-The closed p-form is also exact, by what is sometimes called de Rham's first theorem

My question is: what are the topological implications of my statement? (e.g. am I implying that all p-cycles are p-boundaries??)

Further question: de Rham's theorem is often proved for compact manifolds. Is my statement true even for manifolds which are not compact? (assume my manifold is paracompact, but not compact).

Thanks a lot!

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# De Rham's first theorem

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