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.
Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres πS3 is cyclic of order 24; this is Rokhlin's original approach.
It can also be deduced from the Atiyah–Singer index theorem. See  genus and Rochlin's theorem.
Kirby (1989) gives a geometric proof.
Since Rokhlin's theorem states that the signature of a spin smooth manifold is divisible by 16, the definition of the Rohkhlin invariant is deduced as follows:
If N is a spin 3-manifold then it bounds a spin 4-manifold M. The signature of M is divisible by 8, and an easy application of Rokhlin's theorem shows that its value mod 16 depends only on N and not on the choice of M. Homology 3-spheres have a unique spin structure so we can define the Rokhlin invariant of a homology 3-sphere to be the element sign(M)/8 of Z/2Z, where M any spin 4-manifold bounding the homology sphere.
For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form E8, so its Rokhlin invariant is 1. This result has some elementary consequences: the Poincaré homology sphere does not admit a smooth embedding in , nor does it bound a Mazur manifold.
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