|
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 ''w''2(''M'') vanishes), then the signature of its intersection form, a quadratic form on the second cohomology group ''H''2(''M''), is divisible by 16. The theorem is named for Vladimir Rokhlin, who proved it in 1952. ==Examples== *The intersection form on ''M'' :: :is unimodular on by Poincaré duality, and the vanishing of ''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 ''w''2(''M'') vanishes, and the signature is −16, so 16 is the best possible number in Rokhlin's theorem. *Freedman's E8 manifold is a simply connected compact topological manifold with vanishing ''w''2(''M'') and intersection form ''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 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 ''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 ''w''2(''M'') does not vanish and is represented by a torsion element in the second cohomology group. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rokhlin's theorem」の詳細全文を読む スポンサード リンク
|