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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 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 Abramovich Rokhlin, who proved it in 1952.
## Examples

## 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. ## The Rokhlin invariant

If N is a homology 3-sphere, 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. So we can define the
Rokhlin invariant of M to be the element sign(M)/8 of Z/2Z.
For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form E_{8}, so its Rokhlin invariant is 1. ## Generalizations

_{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 thorem follows. _{1}(Σ, Z/2Z). This Arf invariant is obviously 0 if Σ is a sphere, so the Kervaire-Milnor theorem is a special case. ## References

- The intersection form is unimodular 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. - 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 2-torsion), 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.

It can also be deduced from the Atiyah-Singer index theorem. gives a geometric proof.

More generally, if N is a spin 3-manifold (for example, any Z/2Z 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.

The Rokhlin invariant of M is equal to half the Casson invariant mod 2.

The Kervaire-Milnor theorem states that if Σ is a characteristic sphere in a smooth compact 4-manifold M, then

- signature(M) = Σ.Σ mod 16.

The Freedman-Kirby theorem states that if Σ is a characteristic surface in a smooth compact 4-manifold M, then

- signature(M) = Σ.Σ + 8Arf(M,Σ) mod 16.

A generalization of the Freedman-Kirby theorem to topological (rather than smooth) manifolds states that

- signature(M) = Σ.Σ + 8Arf(M,Σ) +8ks(M) mod 16,

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 4-dimensional manifold, the Hirzebruch signature theorem shows that the signature is −8 times the Â genus, so in dimension 4 this implies Rokhlin's theorem. proved that if X is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16.

- 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. ISBN 082181432X

| last = Kirby|first= Robion

| authorlink = Robion Kirby

| title = The topology of 4-manifolds

| year = 1989

| series = Lecture Notes in Mathematics|volume= 1374|publisher= Springer-Verlag

| isbn =0-387-51148-2

| 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.
- Kervaire, Michel A.; Milnor, John W., On 2-spheres in 4-manifolds. Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651-1657.
- Michelsohn, Marie-Louise; Lawson, H. Blaine (1989).
*Spin geometry*. Princeton, N.J: Princeton University Press. (especially page 280) - Ochanine, Serge, "Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle", Mém. Soc. Math. France 1980/81, no. 5, 142 pp.
- Rokhlin, Vladimir A, New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221-224.
- .
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