# 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 *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 Friedrich Hirzebruch's signature theorem. The case gives back the last example of a K3 surface.
- Michael 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 the set of merely 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.

## 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.

Robion 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 3-manifold and a spin structure on , the Rokhlin invariant in is defined to be the signature of any smooth compact spin 4-manifold with spin boundary .

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**/2**Z**, where *M* any spin 4-manifold bounding the homology sphere.

For example, the Poincaré homology sphere bounds a spin 4-manifold 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 3-manifold (for example, any **Z**/2**Z** 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 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 **Z**-valued lift of the Rokhlin invariant of integral homology 3-sphere.

## Generalizations

The **Kervaire–Milnor theorem** (Kervaire & Milnor 1960) states that if Σ is a characteristic sphere in a smooth compact 4-manifold *M*, then

- signature(
*M*) = Σ.Σ mod 16.

A characteristic sphere is an embedded 2-sphere 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 4-manifold *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**/2**Z**). This Arf invariant is obviously 0 if Σ 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

- 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 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.

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 0-8218-1432-X
- Kirby, Robion (1989),
*The topology of 4-manifolds*, Lecture Notes in Mathematics,**1374**, Springer-Verlag, doi:10.1007/BFb0089031, ISBN 0-387-51148-2, MR 1001966 - 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 2-spheres in 4-manifolds.*Proc. Natl. Acad. Sci. U.S.A. 47 (1961), 1651-1657. MR0133134 - Michelsohn, Marie-Louise; Lawson, H. Blaine (1989),
*Spin geometry*, Princeton, N.J: Princeton University Press, ISBN 0-691-08542-0, MR 1031992 (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. MR1809832
- Rokhlin, Vladimir A.,
*New results in the theory of four-dimensional manifolds*, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. MR0052101 - Scorpan, Alexandru (2005),
*The wild world of 4-manifolds*, American Mathematical Society, ISBN 978-0-8218-3749-8, MR 2136212. - Szűcs, András (2003), "Two Theorems of Rokhlin",
*Journal of Mathematical Sciences*,**113**(6): 888–892, doi:10.1023/A:1021208007146, MR 1809832