Abstract
In this paper, we study smooth complex projective 4-folds which are topologically equivalent. First we show that Fano fourfolds are never oriented homeomorphic to Ricci-flat projective fourfolds and that Calabi-Yau manifolds and hyperkahler manifolds in dimension ≥ 4 are never oriented homeomorphic. Finally, we give a coarse classification of smooth projective fourfolds which are oriented homeomorphic to a hyperkahler fourfold which is deformation equivalent to the Hilbert scheme S[2] of two points of a projective K3 surface S.
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