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

The intersection form on M

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.

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.
A complex surface in $\mathbb {C} P^{3}$ 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 Hirzebruch's signature theorem. The case $d=4$ gives back the last example of a K3 surface.
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 II_1,9 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 π^S₃ 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.

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

M

, the Rokhlin invariant

\mu (M,s)

{\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 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.

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. MR 0520525 ISBN 0-8218-1432-X
Kirby, Robion (1989), The topology of 4-manifolds, Lecture Notes in Mathematics, 1374, Springer-Verlag, ISBN 0-387-51148-2, MR 1001966, 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. MR 0121801
Kervaire, Michel A.; Milnor, John W., On 2-spheres in 4-manifolds. Proc. Natl. Acad. Sci. U.S.A. 47 (1961), 1651-1657. MR 0133134
Michelsohn, Marie-Louise; Lawson, H. Blaine (1989), Spin geometry, Princeton, N.J: Princeton University Press, ISBN 0-691-08542-0, 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. MR 1809832
Rokhlin, Vladimir A., New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. MR 0052101
Scorpan, Alexandru (2005), The wild world of 4-manifolds, American Mathematical Society, ISBN 978-0-8218-3749-8, MR 2136212 .
Szűcs, András (2003), "Two Theorems of Rokhlin", Journal of Mathematical Sciences, 113 (6): 888–892, MR 1809832, doi:10.1023/A:1021208007146

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