Michael Freedman's E8 manifold is a simply connected compact topological manifold with vanishing and intersection form 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 manifolds.
If the manifold M is simply connected, then the vanishing of 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, but the class does not vanish and is represented by a torsion element in the second cohomology group.
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: 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 of, where M any spin 4-manifold bounding the homology sphere. For example, the Poincaré homology sphere bounds a spin 4-manifold with intersection form, 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, 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 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 states that if is a characteristic sphere in a smooth compact 4-manifold M, then A characteristic sphere is an embedded 2-sphere whose homology class represents the Stiefel–Whitney class. If vanishes, we can take to be any small sphere, which has self intersection number 0, so Rokhlin's theorem follows. The Freedman–Kirby theorem states that if is a characteristic surface in a smooth compact 4-manifold M, then where is the Arf invariant of a certain quadratic form on. This Arf invariant is obviously 0 if is a sphere, so the Kervaire–Milnor theorem is a special case. A generalization of the Freedman-Kirby theorem to topological manifolds states that where 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.
