Enter Michael Freedman

The first thunderbolt struck 4-manifold theory in 1982: The topological classification of simply-connected 4-manifolds indeed reflects the algebraic classification of quadratic forms.

Theorem (Freedman's classification theorem). For any integral quadratic form $Q$ there exists a closed, simply-connected topological 4-manifold that has $Q$ as its intersection form.

  • If $Q$ is even, up to homeomorphism, there is exactly one such manifold $M_Q$.
  • If $Q$ is odd, there are, up to homeomorphism, exactly two manifolds $M_Q$ and $N_Q$ with $Q$ as its intersection form. They are distinguished by the fact that $S^1\times M_Q$ carries a smooth structure, whereas $S^1\times N_Q$ admits no smooth structure.

Because of Rokhlin's theorem, the manifolds $M_{E_8}$ and $\overline{M_{E_8}}:=M_{-E_8}$ cannot carry a smooth structure. However, the connected sum $$M_{-E_8} \,\#\,\overline{M_{E_8}}=M_{8H}$$ carries a smooth structure as a connected sum of 8 copies of $S^2\times S^2$.

This little observation already makes it clear that the construction of these manifolds necessarily involves highly non-smoothable and counter-intuitive methods.