Rokhlin's theorem explained

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

Q _{M\colon} H ^{2(M,\Z) x} H ^{2(M,\Z) →} Z

is unimodular on

\Z 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

CP ^{3} of degree d is spin if and only if d is even. It has signature ( 4- d ^{2)d/3} , which can be seen from Friedrich Hirzebruch's signature theorem. The case d =4 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

\pi ^{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  genus and Rochlin's theorem.

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 Rokhlin invariant is deduced as follows:

For 3-manifold

N and a spin structure s on N , the Rokhlin invariant \mu(N,s) in \Z/ 16 Z is defined to be the signature of any smooth compact spin 4-manifold with spin boundary (N,s) .

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

\operatorname{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 S ^{4} , 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 (N,s) 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 states that if

\Sigma is a characteristic sphere in a smooth compact 4-manifold M, then \operatorname{signature}(M) = \Sigma ⋅ \Sigma \bmod 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 \Sigma to be any small sphere, which has self intersection number 0, so Rokhlin's theorem follows.

The Freedman–Kirby theorem states that if

\Sigma is a characteristic surface in a smooth compact 4-manifold M, then \operatorname{signature}(M) = \Sigma ⋅ \Sigma + 8 \operatorname{Arf}(M,\Sigma) \bmod 16 .where \operatorname{Arf}(M,\Sigma) is the Arf invariant of a certain quadratic form on H _{1(\Sigma,} \Z/ 2 \Z) . This Arf invariant is obviously 0 if \Sigma 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

\operatorname{signature}(M) = \Sigma ⋅ \Sigma + 8 \operatorname{Arf}(M,\Sigma) + 8 \operatorname{ks}(M) \bmod 16 ,where \operatorname{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  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.

proved that if X is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16.

References

• • • • • • (especially page 280)
  • Rokhlin, Vladimir A., New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224.