L'Application Canonique $J:H^2(X) \otimes H^2(X)->H^1(X\otimes X)$ n'est   pas Surjective en G\'en\'eral
  We introduce the $H^1$-projective property, and use it to construct a Banach space $X$ such that the natural map   $J:H^2(X)\otimes H^2(X) -> H^1(X\otimes X)$ is not onto. 
Omran Kouba