# Smooth 4-manifolds with $E_8$ intersection form

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 $16$.

• "Any other" form, viz. $E_8\oplus2U$, is realized by an Enriques surface. (Starting from that, you can do connected sums.) Of course, you also have $S^2\times S^2$. Jan 12, 2015 at 22:20
• Yes, so you can get $E_8\oplus kU$ for $k\ge2$. Jan 12, 2015 at 22:27
• I don't think the fundamental group matters for Donaldson's diagonalisation theorem. Jan 13, 2015 at 1:17
• @AlexDegtyarev: The Enriques surface has intersection form $E_8\oplus U$, not $2U$. Indeed, its second Betti number is 10, since $b_1=0$, and the Euler characteristic is 1/2 the Euler characteristic of the K3 surface. Jan 13, 2015 at 3:56
• There is a proof of Donaldson's diagonalisation theorem (without the assumption on $\pi_1$) in Ozsváth-Szabó's 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). Jan 13, 2015 at 13:09

In particular, $E_8$ can't be the intersection form of a closed, orientable, smooth 4-manifold.