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.