Editing Atiyah–Singer index theorem (section)

===Â genus and Rochlin's theorem===
The [[Â genus]] is a rational number defined for any manifold, but is in general not an integer. Borel and Hirzebruch showed that it is integral for spin manifolds,  and an even integer if in addition the dimension is 4 mod 8. This can be deduced from the index theorem, which implies that the Â genus for spin manifolds is the index of a Dirac operator. The extra factor of 2 in dimensions 4 mod 8 comes from the fact that in this case the kernel and cokernel of the Dirac operator have a quaternionic structure, so as complex vector spaces they have even dimensions, so the index is even.

In dimension 4 this result implies [[Rochlin's theorem]] that the signature of a 4-dimensional spin manifold is divisible by 16: this follows because in dimension 4 the Â genus is minus one eighth of the signature.