Definitions

Rokhlin's theorem

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 Abramovich Rokhlin, who proved it in 1952.

Examples

  • The intersection form is unimodular 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.
  • A K3 surface is compact, 4 dimensional, and w2(M) vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.
  • Freedman's E8 manifold is a simply connected compact topological manifold with vanishing w2(M) and intersection form E8 of signature 8. Rokhlin's theorem implies that this manifold has no smooth structure. This manifold shows that Rokhlin's theorem fails for 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 w2(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 −8 (not divisible by 16), but the class w2(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 πS3 is cyclic of order 24; this is Rokhlin's original approach.

It can also be deduced from the Atiyah-Singer index theorem. gives a geometric proof.

The Rokhlin invariant

If N is a homology 3-sphere, 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. So we can define the Rokhlin invariant of M to be the element sign(M)/8 of Z/2Z. For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form E8, so its Rokhlin invariant is 1.

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.

The Rokhlin invariant of M is equal to half the Casson invariant mod 2.

Generalizations

The Kervaire-Milnor theorem 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 thorem follows.

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

References

  • Freedman, Michael; Kirby, Robion, "A geometric proof of Rochlin's theorem", in: Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 85--97, Proc. Sympos. Pure Math., XXXII, Amer. Math. Soc., Providence, R.I., 1978. ISBN 082181432X

 | last = Kirby|first= Robion
 | authorlink = Robion Kirby
 | title = The topology of 4-manifolds
 | year = 1989
 | series = Lecture Notes in Mathematics|volume= 1374|publisher= Springer-Verlag
 | isbn =0-387-51148-2
 | doi=10.1007/BFb0089031
}}

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