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 w2(M) vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.

Examples

Q_M : 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 w2(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.

Proofs

Rokhlin's theorem can be deduced from the fact that the third stable homotopy group of spheres πS3 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.

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 M and a spin structure s on M, the Rokhlin invariant \mu(M,s) in \mathbb{Z}/16\mathbb{Z} is defined to be the signature of any smooth compact spin 4-manifold with spin boundary (M,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 sign(M)/8 of Z/2Z, where M any spin 4-manifold bounding the homology sphere.

For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form E8, 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/2Z 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 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 w2(M). If w2(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 H1(Σ, Z/2Z). 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  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.

References

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