Does there exist a closed orientable smooth 4-manifold $M$ whose intersection form is the $E_8$-form? Here by the intersection form I mean the $\mathbb{Z}$-valued bilinear form on $H^2(M;\mathbb{Z})/\text{torsion}$ induced by the cup product map $$H^2(M;\mathbb{Z}) \times H^2(M;\mathbb{Z}) \longrightarrow H^4(M;\mathbb{Z}) \cong \mathbb{Z}.$$

By Rochlin's theorem, this is not possible if $M$ is spin. It is almost the case that if the intersection form on a 4-manifold $M$ is even, then $M$ is spin. However, this is not necessarily the case if $H_1(M;\mathbb{Z}/2)$ is nonzero. So we're looking for a manifold with an interesting fundamental group. It's not clear to me whether or not such a thing exists.

I'd also be interested in realizing any other even form whose signature is nonzero modulo $16$.

Absolutely graded Floer homologies and intersection forms for four-manifolds with boundary, which is an adaptation of Frøyshov's proof (for which I do not have a reference). $\endgroup$7more comments