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| In 4-dimensional topology, a branch of mathematics, '''Rokhlin's theorem''' states that if a [[Differentiable manifold|smooth]], [[Compact space|compact]] 4-[[manifold]] ''M'' has a [[spin structure]] (or, equivalently, the second [[Stiefel–Whitney class]] ''w''<sub>2</sub>(''M'') vanishes), then the [[Signature (topology)|signature]] of its [[intersection form]], a [[quadratic form]] on the second [[cohomology group]] ''H''<sup>2</sup>(''M''), is divisible by 16. The theorem is named for [[Vladimir Rokhlin (Soviet mathematician)|Vladimir Rokhlin]], who proved it in 1952.
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| ==Examples==
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| *The intersection form is [[unimodular lattice|unimodular]] by [[Poincaré duality]], and the vanishing of ''w''<sub>2</sub>(''M'') implies that the intersection form is even. By a theorem of [[Cahit Arf]], any even unimodular lattice has signature divisible by 8. Rokhlin's theorem improves this by a factor of 2.
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| *A [[K3 surface]] is compact, 4 dimensional, and ''w''<sub>2</sub>(''M'') vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.
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| *Freedman's [[E8 manifold]] is a [[Simply connected space|simply connected]] compact [[topological manifold]] with vanishing ''w''<sub>2</sub>(''M'') and intersection form ''E''<sub>8</sub> of signature 8. Donaldson's work implies that this manifold has no [[smooth structure]]. This shows that Rokhlin's theorem fails for topological (rather than smooth) manifolds.
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| *If the manifold ''M'' is simply connected (or more generally if the first homology group has no 2-torsion), then the vanishing of ''w''<sub>2</sub>(''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<sub>1,9</sub> of signature −8 (not divisible by 16), but the class ''w''<sub>2</sub>(''M'') does not vanish and is represented by a [[Torsion (algebra)|torsion element]] in the second cohomology group.
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| ==Proofs==
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| Rokhlin's theorem can be deduced from the fact that the third [[stable homotopy group of spheres]] π<sup>''S''</sup><sub>3</sub> is cyclic of order 24; this is Rokhlin's original approach.
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| It can also be deduced from the [[Atiyah–Singer index theorem]].
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| {{harvtxt|Kirby|1989}} gives a geometric proof.
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| ==The Rokhlin invariant==
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| If ''N'' is a [[spin structure|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''.
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| 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'''/2'''Z''', where ''M'' any spin 4-manifold bounding the homology sphere.
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| For example, the [[Poincaré homology sphere]] bounds a spin 4-manifold with intersection form ''E''<sub>8</sub>, so its Rokhlin invariant is 1. This result has some elementary consequences: the [[Poincaré homology sphere]] does not admit a smooth embedding in <math>S^4</math>, nor does it bound a [[Mazur manifold]].
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| More generally, if ''N'' is a [[spin structure|spin]] 3-manifold (for example, any '''Z'''/2'''Z''' 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''.
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| On a topological 3-manifold ''N'', the '''generalized Rokhlin invariant''' refers to the function whose domain is the [[spin structure]]s on ''N'', and which evaluates to the '''Rokhlin invariant''' of the pair <math>(N,s)</math> where ''s'' is a spin structure on ''N''.
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| The Rokhlin invariant of M is equal to half the [[Casson invariant]] mod 2.
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| ==Generalizations==
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| The '''Kervaire–Milnor theorem''' {{harv|Kervaire|Milnor|1960}} states that if Σ is a characteristic sphere in a smooth compact 4-manifold ''M'', then
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| :signature(''M'') = Σ.Σ mod 16. | |
| A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class ''w''<sub>2</sub>(''M''). If ''w''<sub>2</sub>(''M'') vanishes, we can take Σ to be any small sphere, which has self intersection number 0, so Rokhlin's thorem follows.
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| The '''Freedman–Kirby theorem''' {{harv|Freedman|Kirby|1978}} states that if Σ is a characteristic surface in a smooth compact 4-manifold ''M'', then
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| :signature(''M'') = Σ.Σ + 8Arf(''M'',Σ) mod 16.
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| where Arf(''M'',Σ) is the [[Arf invariant]] of a certain quadratic form on H<sub>1</sub>(Σ, '''Z'''/2'''Z'''). This Arf invariant is obviously 0 if Σ is a sphere, so the Kervaire–Milnor theorem is a special case.
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| A generalization of the Freedman-Kirby theorem to topological (rather than smooth) manifolds states that
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| :signature(''M'') = Σ.Σ + 8Arf(''M'',Σ) + 8ks(''M'') mod 16,
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| where ks(''M'') is the [[Kirby–Siebenmann invariant]] of ''M''. The Kirby–Siebenmann invariant of ''M'' is 0 if ''M'' is smooth.
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| [[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.
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| {{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.
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| ==References==
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| * [[Michael Freedman|Freedman, Michael]]; [[Robion Kirby|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
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| * {{citation
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| | mr=1001966
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| | last = Kirby|first= Robion
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| | authorlink = Robion Kirby
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| | title = The topology of 4-manifolds
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| | year = 1989
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| | series = Lecture Notes in Mathematics|volume= 1374|publisher= Springer-Verlag
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| | isbn =0-387-51148-2
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| | doi=10.1007/BFb0089031
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| }}
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| * [[Michel Kervaire|Kervaire, Michel A.]]; [[John Milnor|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}}
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| * [[Michel Kervaire|Kervaire, Michel A.]]; [[John Milnor|Milnor, John W.]], ''On 2-spheres in 4-manifolds.'' Proc. Nat. Acad. Sci. U.S.A. 47 (1961), 1651-1657. {{MR|0133134}}
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| * {{Citation
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| |author=[[Marie-Louise Michelsohn|Michelsohn, Marie-Louise]]; Lawson, H. Blaine
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| |title=Spin geometry
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| |publisher=[[Princeton University Press]]
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| |location=Princeton, N.J
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| |year=1989 |pages=
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| |isbn=0-691-08542-0 |doi=
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| |mr= 10319928}} (especially page 280)
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| * 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}}
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| * [[Vladimir Rokhlin (Soviet mathematician)|Rokhlin, Vladimir A.]], ''New results in the theory of four-dimensional manifolds'', Doklady Acad. Nauk. SSSR (N.S.) 84 (1952) 221–224. {{MR|0052101}}
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| * {{citation
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| |last= Scorpan
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| |first= Alexandru
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| |year= 2005
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| |title= The wild world of 4-manifolds
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| |publisher= [[American Mathematical Society]]
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| |isbn= 978-0-8218-3749-8
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| |mr= 2136212
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| }}.
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| *{{citation
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| |first=András|last= Szűcs
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| |title=Two Theorems of Rokhlin
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| |doi= 10.1023/A:1021208007146
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| |journal=Journal of Mathematical Sciences
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| |volume =113
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| |issue= 6
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| |year= 2003
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| |pages= 888–892
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| |mr=1809832
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| }}
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| [[Category:Geometric topology]]
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| [[Category:4-manifolds]]
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| [[Category:Differential structures]]
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| [[Category:Surgery theory]]
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| [[Category:Theorems in topology]]
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I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. What me and my family adore is doing ballet but I've been taking on new things recently. North Carolina is exactly where we've been living for years and will never move. Distributing production has been his occupation for some time.
Feel free to visit my page :: psychic phone (official site)