Rokhlin's theorem
In 4dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, compact 4manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class w_{2}(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H^{2}(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.
Examples
 The intersection form on M

 is unimodular on by Poincaré duality, and the vanishing of w_{2}(M) implies that the intersection form is even. By a theorem of Cahit Arf, any even unimodular lattice has signature divisible by 8, so Rokhlin's theorem forces one extra factor of 2 to divide the signature.
 A K3 surface is compact, 4 dimensional, and w_{2}(M) vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.
 A complex surface in of degree is spin if and only if is even. It has signature , which can be seen from Hirzebruch's signature theorem. The case gives back the last example of a K3 surface.
 Freedman's E8 manifold is a simply connected compact topological manifold with vanishing w_{2}(M) and intersection form E_{8} 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 2torsion), then the vanishing of w_{2}(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 II_{1,9} of signature −8 (not divisible by 16), but the class w_{2}(M) does not vanish and is represented by a torsion element in the second cohomology group.
Proofs
Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres π^{S}_{3} is cyclic of order 24; this is Rokhlin's original approach.
It can also be deduced from the Atiyah–Singer index theorem. See Â genus and Rochlin's theorem.
Kirby (1989) gives a geometric proof.
The Rokhlin invariant
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:
 For 3manifold and a spin structure on , the Rokhlin invariant in is defined to be the signature of any smooth compact spin 4manifold with spin boundary .
If N is a spin 3manifold then it bounds a spin 4manifold 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 3spheres have a unique spin structure so we can define the Rokhlin invariant of a homology 3sphere to be the element sign(M)/8 of Z/2Z, where M any spin 4manifold bounding the homology sphere.
For example, the Poincaré homology sphere bounds a spin 4manifold with intersection form E_{8}, 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.
More generally, if N is a spin 3manifold (for example, any Z/2Z homology sphere), then the signature of any spin 4manifold M with boundary N is well defined mod 16, and is called the Rokhlin invariant of N. On a topological 3manifold 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 where s is a spin structure on N.
The Rokhlin invariant of M is equal to half the Casson invariant mod 2. The Casson invariant is viewed as the Zvalued lift of the Rokhlin invariant of integral homology 3sphere.
Generalizations
The Kervaire–Milnor theorem (Kervaire & Milnor 1960) states that if Σ is a characteristic sphere in a smooth compact 4manifold M, then
 signature(M) = Σ.Σ mod 16.
A characteristic sphere is an embedded 2sphere whose homology class represents the Stiefel–Whitney class w_{2}(M). If w_{2}(M) vanishes, we can take Σ 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 Σ is a characteristic surface in a smooth compact 4manifold M, then
 signature(M) = Σ.Σ + 8Arf(M,Σ) mod 16.
where Arf(M,Σ) is the Arf invariant of a certain quadratic form on H_{1}(Σ, Z/2Z). This Arf invariant is obviously 0 if Σ is a sphere, so the Kervaire–Milnor theorem is a special case.
A generalization of the FreedmanKirby theorem to topological (rather than smooth) manifolds states that
 signature(M) = Σ.Σ + 8Arf(M,Σ) + 8ks(M) mod 16,
where ks(M) is the Kirby–Siebenmann invariant of M. The Kirby–Siebenmann invariant of M is 0 if M is smooth.
Armand Borel and Friedrich Hirzebruch proved the following theorem: If X is a smooth compact spin manifold of dimension divisible by 4 then the Â 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 Â genus is the index of the Atiyah–Singer operator, which is always integral, and is even in dimensions 4 mod 8. For a 4dimensional manifold, the Hirzebruch signature theorem shows that the signature is −8 times the Â 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.
References
 Freedman, Michael; Kirby, Robion, "A geometric proof of Rochlin's theorem", in: Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 85–97, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. MR0520525 ISBN 082181432X
 Kirby, Robion (1989), The topology of 4manifolds, Lecture Notes in Mathematics, 1374, SpringerVerlag, ISBN 0387511482, MR 1001966, doi:10.1007/BFb0089031
 Kervaire, Michel A.; Milnor, John W., "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", 1960 Proc. Internat. Congress Math. 1958, pp. 454–458, Cambridge University Press, New York. MR0121801
 Kervaire, Michel A.; Milnor, John W., On 2spheres in 4manifolds. Proc. Natl. Acad. Sci. U.S.A. 47 (1961), 16511657. MR0133134
 Michelsohn, MarieLouise; Lawson, H. Blaine (1989), Spin geometry, Princeton, N.J: Princeton University Press, ISBN 0691085420, MR 10319928 (especially page 280)
 Ochanine, Serge, "Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la Kthéorie réelle", Mém. Soc. Math. France 1980/81, no. 5, 142 pp. MR1809832
 Rokhlin, Vladimir A., New results in the theory of fourdimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. MR0052101
 Scorpan, Alexandru (2005), The wild world of 4manifolds, American Mathematical Society, ISBN 9780821837498, MR 2136212.
 Szűcs, András (2003), "Two Theorems of Rokhlin", Journal of Mathematical Sciences, 113 (6): 888–892, MR 1809832, doi:10.1023/A:1021208007146