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 "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 Abramovich Rokhlin, who proved it in 1952.

Examples

*The intersection form is unimodular 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.
*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 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 π^"S"₃ is cyclic of order 24; this is Rokhlin's original approach.

It can also be deduced from the Atiyah-Singer index theorem.

harvtxt|Kirby|1989 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 theRokhlin 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 "E"₈, 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 harv|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 thorem follows.

The Freedman-Kirby theorem harv|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.

harvtxt|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

* 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. MathSciNet|id=0520525 ISBN 082181432X
* citation
id=MR|1001966
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
* Kervaire, Michel A.; Milnor, John W., "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", 1960 Proc. Internat. Congress Math. 1958, pp. 454--458, Cambridge University Press, New York. MathSciNet|id=0121801
* Kervaire, Michel A.; Milnor, John W., "On 2-spheres in 4-manifolds." Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651-1657. MathSciNet|id=0133134
* cite book
author=Michelsohn, Marie-Louise; Lawson, H. Blaine
title=Spin geometry
publisher=Princeton University Press
location=Princeton, N.J
year=1989 |pages=
isbn=0-691-08542-0 |oclc= |doi=
id= MR|10319928 (especially page 280)
* Ochanine, Serge, "Signature modulo 16, invariants de Kervaire généralisés et nombres caractéristiques dans la K-théorie réelle", Mém. Soc. Math. France 1980/81, no. 5, 142 pp. MathSciNet|id=1809832
* Rokhlin, Vladimir A, "New results in the theory of four-dimensional manifolds", Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221-224. MathSciNet|id=0052101
* citation
last= Scorpan
first= Alexandru
year= 2005
title= The wild world of 4-manifolds
publisher= American Mathematical Society
isbn= 978-0-8218-3749-8
id= MR|2136212 .
*citation
first=András|last= Szűcs
title=Two Theorems of Rokhlin
doi= 10.1023/A:1021208007146
journal=Journal of Mathematical Sciences
volume =113
issue= 6
year= 2003
pages= 888–892
id=MR|1809832