Rokhlin's theorem

Proofs

Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres ${\displaystyle {\pi }_3^S}$ is cyclic of order 24; this is Rokhlin's original approach.

It can also be deduced from the Atiyah–Singer index theorem. See  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 ${\displaystyle N}$ and a spin structure ${\displaystyle s}$ on ${\displaystyle N}$, the Rokhlin invariant ${\displaystyle \mu (N,s)}$ in ${\displaystyle \mathbb{Z}/16\mathbb{Z}}$ is defined to be the signature of any smooth compact spin 4-manifold with spin boundary ${\displaystyle (N,s)}$.

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 ${\displaystyle \mathrm{sign}(M)/8}$ of ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$, where M any spin 4-manifold bounding the homology sphere.

For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form ${\displaystyle E_8}$, so its Rokhlin invariant is 1. This result has some elementary consequences: the Poincaré homology sphere does not admit a smooth embedding in ${\displaystyle S^4}$, nor does it bound a Mazur manifold.

More generally, if N is a spin 3-manifold (for example, any ${\displaystyle \mathbb{Z}/2\mathbb{Z}}$ 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 ${\displaystyle (N,s)}$ 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 ${\displaystyle \mathrm{\Sigma }}$ is a characteristic sphere in a smooth compact 4-manifold M, then

${\displaystyle \mathrm{signature}(M)=\mathrm{\Sigma }\cdot \mathrm{\Sigma }mod16}$.

A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class ${\displaystyle w_2(M)}$. If ${\displaystyle w_2(M)}$ vanishes, we can take ${\displaystyle \mathrm{\Sigma }}$ 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 ${\displaystyle \mathrm{\Sigma }}$ is a characteristic surface in a smooth compact 4-manifold M, then

${\displaystyle \mathrm{signature}(M)=\mathrm{\Sigma }\cdot \mathrm{\Sigma }+8\mathrm{Arf}(M,\mathrm{\Sigma })mod16}$.

where ${\displaystyle \mathrm{Arf}(M,\mathrm{\Sigma })}$ is the Arf invariant of a certain quadratic form on ${\displaystyle H_1(\mathrm{\Sigma },\mathbb{Z}/2\mathbb{Z})}$. This Arf invariant is obviously 0 if ${\displaystyle \mathrm{\Sigma }}$ 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

${\displaystyle \mathrm{signature}(M)=\mathrm{\Sigma }\cdot \mathrm{\Sigma }+8\mathrm{Arf}(M,\mathrm{\Sigma })+8\mathrm{ks}(M)mod16}$,

where ${\displaystyle \mathrm{ks}(M)}$ 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.

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.