Hi, I am trying to show that ## \mathbb CP^4 ## and ## \mathbb CP^2 \times \mathbb CP^2 ## are not homeomorphic. None of the standard methods --comparing the fundamental group of the product with the product of fundamental groups, nor (co)homology seem to work. So I am trying to work with the cup products and show these are different. Please let me know if this is correct: we need to compute the cohomology groups on each side, and then we just compute all possible cup products using Kunneth's formula. Is this correct? Thanks.