Armand Borel and Friedrich Hirzebruch proved the following theorem: If "X" is a smooth compact spin manifold of dimension divisible by 4 then the Â 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 Â 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 Â genus, so in dimension 4 this implies Rokhlin's theorem.