Let's consider closed simply connected manifold $M^n$ and a $a\in H^k(M)$ and $a*\in H^{n-k}(M)$ is the dual to $a$.
Is it true that dual to $a$ is realisable as a immersed sphere or $a*=bc$ for some $b,c\in H^*(M)$ ?
Edit: it is more natural to ask about possibility to decompose dual to $a$ as a product, see example in the answer below.
Edit2: let's assume that $M$ is not decompasable, so there is no $X,Y$ such that $M = X\times Y$