There does not exist closed manifold along the line of projective planes
above the dimension of octonions due to the inexistence of hopf invariant
1 map in higher dimensions. I investigated the existence dimension of such
manifold in the rational sense, such that the rational cohomology is rank
one in dimension 0, 2k and 4k and is zero otherwise. Applying rational
surgery, the problem can be reduced to finding possible Pontryagin classes
satisfying the Hirzebruch signature formula and a set of congruence relations
determined by the Riemann-Roch integrality conditions, which is eventually
equivalent to solving a system of Diophantine equations. After a negative
answer in dimension 24, the first existence dimension of such manifold is 32.