Topological classification of complete intersections

We show that the homeomorphism type of the complete intersection X(n)(d(1), d(2), ..., d(r)) is determine by its Pontryagin classes, Euler characteristic and the product d(1)d(2) ... d(r), if d(1)d(2) .. d(r) has no prime factors less then n+2/2. We also prove that, when n is even the homotopy type X(n) (d(1), d(2), ..., d(r)) is determined by its Euler characteristic, signature and the product d(1)d(2) ... d(r), in the case d(1)d(2) ... d(r) has no prime factors less than n+3/2. The latter confirms a conjecture of Libgober and Wood [8].