Riemann–Roch theorem for smooth manifolds
In mathematics, a Riemann–Roch theorem for smooth manifolds is a version of results such as the Hirzebruch–Riemann–Roch theorem or Grothendieck–Riemann–Roch theorem (GRR) without a hypothesis making the smooth manifolds involved carry a complex structure. Results of this kind were obtained by Michael Atiyah and Friedrich Hirzebruch in 1959, reducing the requirements to something like a spin structure.
[edit] Formulation
In their paper[1] they define, for oriented smooth closed manifolds X and Y and a continuous mapping
- f: Y → X
that f is a c1-map if there is c1 in the integral cohomology group
- H2(Y, Z)
such that for the Stiefel-Whitney classes w2 we have
- c1 = w2(Y) − f*(w2(X)) modulo 2
in
- H2(Y, Z/2Z).
Writing ch(X) for the image in H*(X, Q) they showed that for f a c1-map there is
- f!: ch(Y) → ch(X)
which is a homomorphism of abelian groups, and satisfying
- f!(y)A^(X) = f*(y.exp(c1)/2)A^(Y)),
where A^ is the A-hat genus and f* the Gysin homomorphism. This mimics the GRR theorem; but f! has only an implicit definition.
This they 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.
[edit] Notes
- ^ M. Atiyah and F. Hirzebruch, Riemann–Roch theorems for differentiable manifolds (Bull. Amer. Math. Soc. 65 (1959) 276–281)
