Riemann–Roch theorem for smooth manifolds

Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.

If dim(X) ≡ dim(Y) mod 2, then where ch is the Chern character, d(vf) an element of the integral cohomology group H2(Y, Z) satisfying d(vf) ≡ f* w2(TY)-w2(TX) mod 2, fK* the Gysin homomorphism for K-theory, and fH* the Gysin homomorphism for cohomology .

Then, using the splitting principle, it suffices to check the theorem via explicit computation for line bundles.

The Gysin map for the projection is the Bott-periodicity isomorphism, which commutes with the Chern character, so the theorem holds in this general case also.

Atiyah and Hirzebruch then specialised and refined in the case X = a point, where the condition becomes the existence of a spin structure on Y. Corollaries are on Pontryagin classes and the J-homomorphism.