Rokhlin's theorem




In 4-dimensional topology, a branch of mathematics, Rokhlin's theorem states that if a smooth, closed 4-manifold M has a spin structure (or, equivalently, the second Stiefel–Whitney class w2(M){displaystyle w_{2}(M)}{displaystyle w_{2}(M)} vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group H2(M){displaystyle H^{2}(M)}H^2(M), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952.




Contents






  • 1 Examples


  • 2 Proofs


  • 3 The Rokhlin invariant


  • 4 Generalizations


  • 5 References





Examples


  • The intersection form on M


QM:H2(M,Z)×H2(M,Z)→Z{displaystyle Q_{M}colon H^{2}(M,mathbb {Z} )times H^{2}(M,mathbb {Z} )rightarrow mathbb {Z} }{displaystyle Q_{M}colon H^{2}(M,mathbb {Z} )times H^{2}(M,mathbb {Z} )rightarrow mathbb {Z} }

is unimodular on Z{displaystyle mathbb {Z} }mathbb {Z} by Poincaré duality, and the vanishing of w2(M){displaystyle w_{2}(M)}{displaystyle 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 w2(M){displaystyle w_{2}(M)}{displaystyle w_{2}(M)} vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem.

  • A complex surface in CP3{displaystyle mathbb {CP} ^{3}}mathbb{CP}^3 of degree d{displaystyle d}d is spin if and only if d{displaystyle d}d is even. It has signature (4−d2)d/3{displaystyle (4-d^{2})d/3}{displaystyle (4-d^{2})d/3}, which can be seen from Friedrich Hirzebruch's signature theorem. The case d=4{displaystyle d=4}d=4 gives back the last example of a K3 surface.


  • Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing w2(M){displaystyle w_{2}(M)}{displaystyle w_{2}(M)} and intersection form E8{displaystyle E_{8}}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 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 w2(M){displaystyle w_{2}(M)}{displaystyle 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 II1,9 of signature −8 (not divisible by 16), but the class w2(M){displaystyle w_{2}(M)}{displaystyle 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 π3S{displaystyle pi _{3}^{S}}{displaystyle pi _{3}^{S}} 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 Rohkhlin invariant is deduced as follows:


For 3-manifold M{displaystyle M}M and a spin structure s{displaystyle s}s on M{displaystyle M}M, the Rokhlin invariant μ(M,s){displaystyle mu (M,s)}mu (M,s) in Z/16Z{displaystyle mathbb {Z} /16mathbb {Z} }{displaystyle mathbb {Z} /16mathbb {Z} } is defined to be the signature of any smooth compact spin 4-manifold with spin boundary (M,s){displaystyle (M,s)}(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{displaystyle operatorname {sign} (M)/8}{displaystyle operatorname {sign} (M)/8} of Z/2Z{displaystyle mathbb {Z} /2Z}{displaystyle mathbb {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{displaystyle E_{8}}E_{8}, so its Rokhlin invariant is 1. This result has some elementary consequences: the Poincaré homology sphere does not admit a smooth embedding in S4{displaystyle S^{4}}S^4, nor does it bound a Mazur manifold.


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



signature⁡(M)=Σ⋅Σmod16{displaystyle operatorname {signature} (M)=Sigma cdot Sigma {bmod {1}}6}{displaystyle operatorname {signature} (M)=Sigma cdot Sigma {bmod {1}}6}.

A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class w2(M){displaystyle w_{2}(M)}{displaystyle w_{2}(M)}. If w2(M){displaystyle w_{2}(M)}{displaystyle w_{2}(M)} vanishes, we can take Σ{displaystyle Sigma }Sigma 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{displaystyle Sigma }Sigma is a characteristic surface in a smooth compact 4-manifold M, then



signature⁡(M)=Σ⋅Σ+8Arf⁡(M,Σ)mod16{displaystyle operatorname {signature} (M)=Sigma cdot Sigma +8operatorname {Arf} (M,Sigma ){bmod {1}}6}{displaystyle operatorname {signature} (M)=Sigma cdot Sigma +8operatorname {Arf} (M,Sigma ){bmod {1}}6}.

where Arf⁡(M,Σ){displaystyle operatorname {Arf} (M,Sigma )}{displaystyle operatorname {Arf} (M,Sigma )} is the Arf invariant of a certain quadratic form on H1(Σ,Z/2Z){displaystyle H_{1}(Sigma ,mathbb {Z} /2mathbb {Z} )}{displaystyle H_{1}(Sigma ,mathbb {Z} /2mathbb {Z} )}. This Arf invariant is obviously 0 if Σ{displaystyle Sigma }Sigma 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)mod16{displaystyle operatorname {signature} (M)=Sigma cdot Sigma +8operatorname {Arf} (M,Sigma )+8operatorname {ks} (M){bmod {1}}6}{displaystyle operatorname {signature} (M)=Sigma cdot Sigma +8operatorname {Arf} (M,Sigma )+8operatorname {ks} (M){bmod {1}}6},

where ks⁡(M){displaystyle operatorname {ks} (M)}{displaystyle operatorname {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.


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.mw-parser-output cite.citation{font-style:inherit}.mw-parser-output .citation q{quotes:"""""""'""'"}.mw-parser-output .citation .cs1-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/6/65/Lock-green.svg/9px-Lock-green.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-limited a,.mw-parser-output .citation .cs1-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/d/d6/Lock-gray-alt-2.svg/9px-Lock-gray-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .citation .cs1-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/a/aa/Lock-red-alt-2.svg/9px-Lock-red-alt-2.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration{color:#555}.mw-parser-output .cs1-subscription span,.mw-parser-output .cs1-registration span{border-bottom:1px dotted;cursor:help}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/thumb/4/4c/Wikisource-logo.svg/12px-Wikisource-logo.svg.png")no-repeat;background-position:right .1em center}.mw-parser-output code.cs1-code{color:inherit;background:inherit;border:inherit;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;font-size:100%}.mw-parser-output .cs1-visible-error{font-size:100%}.mw-parser-output .cs1-maint{display:none;color:#33aa33;margin-left:0.3em}.mw-parser-output .cs1-subscription,.mw-parser-output .cs1-registration,.mw-parser-output .cs1-format{font-size:95%}.mw-parser-output .cs1-kern-left,.mw-parser-output .cs1-kern-wl-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right,.mw-parser-output .cs1-kern-wl-right{padding-right:0.2em}
    0520525
    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, MR 1001966


  • 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 1031992 (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, doi:10.1023/A:1021208007146, MR 1809832




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