Nonexistence of almost complex structures on the product S2m×MS2m×M

In this note we give a necessary condition for having an almost complex structure on the product S2m×MS2m×M, where M   is a connected orientable closed manifold. We show that if the Euler characteristic χ(M)≠0χ(M)≠0, then except for finitely many values of m  , we do not have almost complex structure on S2m×MS2m×M. In the particular case when M=CPn,n≠1M=CPn,n≠1, we show that if n≢3(mod4) then S2m×CPnS2m×CPn has an almost complex structure if and only if m=1,3m=1,3. As an application we obtain conditions on the nonexistence of almost complex structures on Dold manifolds.