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Are the odd-degree Betti numbers of a compact Almost-Kahler Einstein four manifold necessarily even ?

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If some of Betti numbers were odd this would obviously contradict to Goldberg's conjecture, stating that Almost-Kahler Einstein manifolds are Kahler. I assume Goldberg's conjecture is open in all dimensions? – Dmitri Mar 5 '11 at 10:19
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Sorry for the double-retag. I think these tags should work. – Elizabeth S. Q. Goodman Mar 5 '11 at 21:19
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this is W. Watson's conjecture, which it seems is still open: facpub.stjohns.edu/~watsonw/diffgeom.htm (edited "odd" --> "odd-degree" to avoid a contradictio in terminis) – Carlo Beenakker Sep 5 '15 at 10:38

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