Rokhlin's theorem

Remarks

Vladimir Rokhlin proved the theorem in 1952. One modern proof uses the Atiyah–Singer index theorem: the index of the Dirac operator of $M$ equals $-\sigma(M)/8$. In dimension $4$ the spinor bundles are quaternionic, so the kernel and cokernel of the Dirac operator are quaternionic vector spaces and the index is even. Hence $\sigma(M)/8$ is even. The K3 surface is a spin $4$-manifold of signature $-16$, so $16$ cannot be replaced by a larger number. Without smoothness the theorem fails: the $E_8$ manifold is a closed topological $4$-manifold with even form and signature $8$. Without the spin condition it fails: the complex projective plane has signature $1$. Algebra alone gives less: an even unimodular form always has signature divisible by $8$. For a simply connected $M$, being spin is the same as having an even intersection form.

Going deeper

For a student

The theorem is the first sign that smooth $4$-manifolds are more rigid than topological ones. It shows that the $E_8$ manifold has no smooth structure. It defines the Rokhlin invariant of an integral homology $3$-sphere: the signature of a smooth spin $4$-manifold it bounds, divided by $8$, modulo $2$. The 11/8 conjecture and Furuta's $10/8$ theorem refine it for spin $4$-manifolds.