In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class
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
is cyclic of order 24; this is Rokhlin's original approach.
See  genus and Rochlin's theorem.
Since Rokhlin's theorem states that the signature of a spin smooth manifold is divisible by 16, the definition of the Rokhlin 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
For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form
This result has some elementary consequences: the Poincaré homology sphere does not admit a smooth embedding in
homology sphere), then the signature of any spin 4-manifold M with boundary N is well defined mod 16, and is called the Rokhlin invariant of N. On a topological 3-manifold N, the generalized Rokhlin invariant refers to the function whose domain is the spin structures on N, and which evaluates to the Rokhlin invariant of the pair
The Kervaire–Milnor theorem (Kervaire & Milnor 1960) states that if
is a characteristic sphere in a smooth compact 4-manifold M, then A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class
to be any small sphere, which has self intersection number 0, so Rokhlin's theorem follows.
The Freedman–Kirby theorem (Freedman & Kirby 1978) states that if
is a sphere, so the Kervaire–Milnor theorem is a special case.
A generalization of the Freedman-Kirby theorem to topological (rather than smooth) manifolds states that where
Armand Borel and Friedrich Hirzebruch proved the following theorem: If X is a smooth compact spin manifold of dimension divisible by 4 then the  genus is an integer, and is even if the dimension of X is 4 mod 8.
This can be deduced from the Atiyah–Singer index theorem: Michael Atiyah and Isadore Singer showed that the  genus is the index of the Atiyah–Singer operator, which is always integral, and is even in dimensions 4 mod 8.
For a 4-dimensional manifold, the Hirzebruch signature theorem shows that the signature is −8 times the  genus, so in dimension 4 this implies Rokhlin's theorem.
Ochanine (1980) proved that if X is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16.