Rokhlin's theorem
Encyclopedia
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.
It can also be deduced from the Atiyah–Singer index theorem
.
gives a geometric proof.
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
, 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 structure
s on N, and which evaluates to the Rokhlin invariant of the pair
where s is a spin structure on N.
The Rokhlin invariant of M is equal to half the Casson invariant
mod 2.
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 states that if Σ is a characteristic surface in a smooth compact 4-manifold M, then
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
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.
proved that if X is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16.
Differentiable manifold
A differentiable manifold is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since...
, compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
4-manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
M has a spin structure
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
(or, equivalently, the second Stiefel–Whitney class
Stiefel–Whitney class
In mathematics, in particular in algebraic topology and differential geometry, the Stiefel–Whitney class is an example of a \mathbb Z_2characteristic class associated to real vector bundles.-General presentation:...
w2(M) vanishes), then the signature
Signature (topology)
In the mathematical field of topology, the signature is an integer invariant which is defined for an oriented manifold M of dimension d=4k divisible by four ....
of its intersection form
Intersection form
Intersection form may refer to:*Intersection theory *intersection form...
, a quadratic form
Quadratic form
In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example,4x^2 + 2xy - 3y^2\,\!is a quadratic form in the variables x and y....
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 is unimodularUnimodular latticeIn mathematics, a unimodular lattice is a lattice of determinant 1 or −1.The E8 lattice and the Leech lattice are two famous examples.- Definitions :...
by Poincaré dualityPoincaré dualityIn mathematics, the Poincaré duality theorem named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds...
, and the vanishing of w2(M) implies that the intersection form is even. By a theorem of Cahit ArfCahit ArfCahit Arf was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 in topology, the Hasse–Arf theorem in ramification theory, Arf semigroups, and Arf rings.-Biography:Cahit Arf was born on 11 October 1910 in Selanik , which was then...
, 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 surfaceK3 surfaceIn mathematics, a K3 surface is a complex or algebraic smooth minimal complete surface that is regular and has trivial canonical bundle.In the Enriques-Kodaira classification of surfaces they form one of the 5 classes of surfaces of Kodaira dimension 0....
is compact, 4 dimensional, and w2(M) vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem. - Freedman's E8 manifoldE8 manifoldIn mathematics, the E8 manifold is the unique compact, simply connected topological 4-manifold with intersection form the E8 lattice.The E8 manifold was discovered by Michael Freedman in 1982...
is a simply connectedSimply connected spaceIn topology, a topological space is called simply connected if it is path-connected and every path between two points can be continuously transformed, staying within the space, into any other path while preserving the two endpoints in question .If a space is not simply connected, it is convenient...
compact topological manifoldTopological manifoldIn mathematics, a topological manifold is a topological space which looks locally like Euclidean space in a sense defined below...
with vanishing w2(M) and intersection form E8 of signature 8. Rokhlin's theorem implies that this manifold has no smooth structureSmooth structureIn mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows one to perform mathematical analysis on the manifold....
. 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 surfaceEnriques surfaceIn mathematics, Enriques surfaces, discovered by , are complex algebraic surfacessuch that the irregularity q = 0 and the canonical line bundle K is non-trivial but has trivial square...
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) 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 πS3 is cyclic of order 24; this is Rokhlin's original approach.It can also be deduced from the Atiyah–Singer index theorem
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
.
gives a geometric proof.
The Rokhlin invariant
If N is a spinSpin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
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
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
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
, nor does it bound a Mazur manifoldMazur manifold
In differential topology, a branch of mathematics, a Mazur manifold is a contractible, compact, smooth 4-dimensional manifold which is not diffeomorphic to the standard 4-ball...
.
More generally, if N is a spin
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
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 structure
Spin structure
In differential geometry, a spin structure on an orientable Riemannian manifold \,allows one to define associated spinor bundles, giving rise to the notion of a spinor in differential geometry....
s on N, and which evaluates to the Rokhlin invariant of the pair
where s is a spin structure on N.The Rokhlin invariant of M is equal to half the Casson invariant
Casson invariant
In 3-dimensional topology, a part of the mathematical field of geometric topology, the Casson invariant is an integer-valued invariant of oriented integral homology 3-spheres, introduced by Andrew Casson....
mod 2.
Generalizations
The Kervaire–Milnor theorem 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 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
Arf invariant
In mathematics, the Arf invariant of a nonsingular quadratic form over a field of characteristic 2 was defined by when he started the systematic study of quadratic forms over arbitrary fields of characteristic2. The Arf invariant is the substitute, in...
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
Armand Borel
Armand Borel was a Swiss mathematician, born in La Chaux-de-Fonds, and was a permanent professor at the Institute for Advanced Study in Princeton, New Jersey, United States from 1957 to 1993...
and Friedrich Hirzebruch
Friedrich Hirzebruch
Friedrich Ernst Peter Hirzebruch is a German mathematician, working in the fields of topology, complex manifolds and algebraic geometry, and a leading figure in his generation.-Life:He was born in Hamm, Westphalia...
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
Atiyah–Singer index theorem
In differential geometry, the Atiyah–Singer index theorem, proved by , states that for an elliptic differential operator on a compact manifold, the analytical index is equal to the topological index...
: Michael Atiyah
Michael Atiyah
Sir Michael Francis Atiyah, OM, FRS, FRSE is a British mathematician working in geometry.Atiyah grew up in Sudan and Egypt but spent most of his academic life in the United Kingdom at Oxford and Cambridge, and in the United States at the Institute for Advanced Study...
and Isadore Singer
Isadore Singer
Isadore Manuel Singer is an Institute Professor in the Department of Mathematics at the Massachusetts Institute of Technology...
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.
proved that if X is a compact oriented smooth spin manifold of dimension 4 mod 8, then its signature is divisible by 16.