Complex sentence: The Â genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies Rochlin's theorem)--for general manifolds, the Â genus is not always an integer.
Split into simpler sentences: The Â genus of a spin manifold is an integer, and an even integer if the dimension is 4 mod 8 (which in dimension 4 implies Rochlin's theorem). For general manifolds, the Â genus is not always an integer.