Rokhlin's theorem

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Short description: On the intersection form of a smooth, closed 4-manifold with a spin structure

In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, orientable, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class [math]\displaystyle{ w_2(M) }[/math] vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group [math]\displaystyle{ H^2(M) }[/math], is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.

Examples

[math]\displaystyle{ Q_M\colon H^2(M,\Z)\times H^2(M,\Z)\rightarrow \mathbb{Z} }[/math]
is unimodular on [math]\displaystyle{ \Z }[/math] by Poincaré duality, and the vanishing of [math]\displaystyle{ w_2(M) }[/math] 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 [math]\displaystyle{ w_2(M) }[/math] vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.
  • A complex surface in [math]\displaystyle{ \mathbb{CP}^3 }[/math] of degree [math]\displaystyle{ d }[/math] is spin if and only if [math]\displaystyle{ d }[/math] is even. It has signature [math]\displaystyle{ (4-d^2)d/3 }[/math], which can be seen from Friedrich Hirzebruch's signature theorem. The case [math]\displaystyle{ d=4 }[/math] gives back the last example of a K3 surface.
  • Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing [math]\displaystyle{ w_2(M) }[/math] and intersection form [math]\displaystyle{ E_8 }[/math] of signature 8. Rokhlin's theorem implies that this manifold has no smooth structure. This manifold shows that Rokhlin's theorem fails for the set of merely 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 [math]\displaystyle{ w_2(M) }[/math] 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 [math]\displaystyle{ w_2(M) }[/math] 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 [math]\displaystyle{ \pi^S_3 }[/math] 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 [math]\displaystyle{ N }[/math] and a spin structure [math]\displaystyle{ s }[/math] on [math]\displaystyle{ N }[/math], the Rokhlin invariant [math]\displaystyle{ \mu(N,s) }[/math] in [math]\displaystyle{ \Z/16\mathbb{Z} }[/math] is defined to be the signature of any smooth compact spin 4-manifold with spin boundary [math]\displaystyle{ (N,s) }[/math].

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 [math]\displaystyle{ \operatorname{sign}(M)/8 }[/math] of [math]\displaystyle{ \Z/2\Z }[/math], where M any spin 4-manifold bounding the homology sphere.

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

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

[math]\displaystyle{ \operatorname{signature}(M) = \Sigma \cdot \Sigma \bmod 16 }[/math].

A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class [math]\displaystyle{ w_2(M) }[/math]. If [math]\displaystyle{ w_2(M) }[/math] vanishes, we can take [math]\displaystyle{ \Sigma }[/math] to be any small sphere, which has self intersection number 0, so Rokhlin's theorem follows.

The Freedman–Kirby theorem (Freedman Kirby) states that if [math]\displaystyle{ \Sigma }[/math] is a characteristic surface in a smooth compact 4-manifold M, then

[math]\displaystyle{ \operatorname{signature}(M) = \Sigma \cdot \Sigma + 8\operatorname{Arf}(M,\Sigma) \bmod 16 }[/math].

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

[math]\displaystyle{ \operatorname{signature}(M) = \Sigma \cdot \Sigma + 8\operatorname{Arf}(M,\Sigma) + 8\operatorname{ks}(M) \bmod 16 }[/math],

where [math]\displaystyle{ \operatorname{ks}(M) }[/math] 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

  • Freedman, Michael; Kirby, Robion (1978), "A geometric proof of Rochlin's theorem", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), Part 2, pp. 85–97, Proceedings of Symposia in Pure Mathematics, XXXII, Providence, Rhode Island: American Mathematics Society, ISBN 0-8218-1432-X 
  • Kirby, Robion (1989), The Topology of 4-Manifolds, Lecture Notes in Mathematics, 1374, Springer-Verlag, doi:10.1007/BFb0089031, ISBN 0-387-51148-2 
  • Kervaire, Michel A.; Milnor, John W. (1960), "Bernoulli numbers, homotopy groups, and a theorem of Rohlin", Proceedings of the International Congress of Mathematicians, 1958, New York: Cambridge University Press, pp. 454–458 
  • Kervaire, Michel A.; Milnor, John W. (1961), "On 2-spheres in 4-manifolds", Proceedings of the National Academy of Sciences, 47, pp. 1651–1657 
  • Matsumoto, Yoichirou (1986) (in en), An elementary proof of Rochlin's signature theorem and its extension by Guillou and Marin, https://www.maths.ed.ac.uk/~v1ranick/papers/matumoto5.pdf 
  • Spin geometry, Princeton, New Jersey: Princeton University Press, 1989, ISBN 0-691-08542-0  (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 
  • Rokhlin, Vladimir A., New results in the theory of four-dimensional manifolds, Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. MR0052101
  • Scorpan, Alexandru (2005), The wild world of 4-manifolds, American Mathematical Society, ISBN 978-0-8218-3749-8 
  • Szűcs, András (2003), "Two Theorems of Rokhlin", Journal of Mathematical Sciences 113 (6): 888–892, doi:10.1023/A:1021208007146 



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