# 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
       $Q_{M}\colon H^{2}(M,\mathbb {Z} )\times H^{2}(M,\mathbb {Z} )\rightarrow \mathbb {Z}$
    is unimodular on $\mathbb {Z}$ 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 $\mathbb {CP} ^{3}$ 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 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 $\pi _{3}^{S}$ 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 Rokhlin invariant is deduced as follows:
    For 3-manifold $N$ and a spin structure $s$ on $N$, the Rokhlin invariant $\mu (N,s)$ in $\mathbb {Z} /16\mathbb {Z}$ 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 $\mathbb {Z} /2\mathbb {Z}$, 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 $\mathbb {Z} /2\mathbb {Z}$ 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 (Kervaire & Milnor 1960) states that if $\Sigma$ is a characteristic sphere in a smooth compact 4-manifold M, then
    $\operatorname {signature} (M)=\Sigma \cdot \Sigma {\bmod {1}}6$.
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 (Freedman & Kirby 1978) states that if $\Sigma$ is a characteristic surface in a smooth compact 4-manifold M, then
    $\operatorname {signature} (M)=\Sigma \cdot \Sigma +8\operatorname {Arf} (M,\Sigma ){\bmod {1}}6$.
where $\operatorname {Arf} (M,\Sigma )$ is the Arf invariant of a certain quadratic form on $H_{1}(\Sigma ,\mathbb {Z} /2\mathbb {Z} )$. 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
    $\operatorname {signature} (M)=\Sigma \cdot \Sigma +8\operatorname {Arf} (M,\Sigma )+8\operatorname {ks} (M){\bmod {1}}6$,
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 −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.