Rokhlin's theorem

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 $$w_2(M)$$ 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
 * $$Q_M\colon H^2(M,\Z)\times H^2(M,\Z)\rightarrow \mathbb{Z}$$
 * 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 &minus;16, so 16 is the best possible number in Rokhlin's theorem.
 * A complex surface in $$\mathbb{CP}^3$$ 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 II1,9 of signature &minus;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 $$\pi^S_3$$ is cyclic of order 24; this is Rokhlin's original approach.

It can also be deduced from the Atiyah–Singer index theorem. See Â 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 $$\Z/16\mathbb{Z}$$ 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
 * $$\operatorname{signature}(M) = \Sigma \cdot \Sigma \bmod 16$$.

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
 * $$\operatorname{signature}(M) = \Sigma \cdot \Sigma + 8\operatorname{Arf}(M,\Sigma) \bmod 16$$.

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
 * $$\operatorname{signature}(M) = \Sigma \cdot \Sigma + 8\operatorname{Arf}(M,\Sigma) + 8\operatorname{ks}(M) \bmod 16$$,

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 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 &minus;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.