= Solution
\b[True.] The <Künneth theorem> gives the <Poincaré polynomial>
$$
P_{S^4\times T^4}(t)=(1+t^4)(1+t)^4,
$$
so $b_2=6$ and $b_4=2$. The proposed equality also gives $b_0=b_8=1$. Even if <compactness> was not stated separately, these two numbers force the putative eight-manifold to be connected and compact: the top ordinary real <cohomology> of a connected noncompact manifold is zero.
The auxiliary <Hard Lefschetz theorem on de Rham cohomology> says that, on a compact <Kähler manifold> of complex dimension $n$, $L^{n-k}:H^k\to H^{2n-k}$ is an <isomorphism>, where $L\alpha=[\omega]\smile\alpha$. Here $n=4$, so $L^2:H^2\to H^6$ is <injective>. If $L\alpha=0$, then $L^2\alpha=0$ and therefore $\alpha=0$. Thus $L:H^2\to H^4$ is <injective>, forcing $b_2\leq b_4$. The inequality $6\leq2$ is impossible. Hence \b[these <Betti numbers> cannot occur for a Kähler eight-manifold].
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