Rokhlin's theorem
In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, compact 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class w2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.
Examples
The intersection form on M
A K3 surface is compact, 4 dimensional, and w2(M) vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.
Freedman's E8 manifold is a simply connected compact topological manifold with vanishing w2(M) and intersection form E8 of signature 8. Rokhlin's theorem implies that this manifold has no smooth structure. This manifold shows that Rokhlin's theorem fails for topological (rather than smooth) manifolds.
If the manifold M is simply connected (or more generally if the first homology group has no 2-torsion), then the vanishing of w2(M) is equivalent to the intersection form being even. This is not true in general: an Enriques surface is a compact smooth 4 manifold and has even intersection form II1,9 of signature −8 (not divisible by 16), but the class w2(M) does not vanish and is represented by a torsion element in the second cohomology group.
