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.

Formulation

Let X and Y be oriented smooth closed manifolds, and f: X → Y a continuous map. Let v_f=f^*(TY) − TX in the K-group K(X). If dim(X) ≡ dim(Y) mod 2, then

${\displaystyle \mathrm {ch} (f_{K*}(x))=f_{H*}(\mathrm {ch} (x)e^{d(v_{f})/2}{\hat {A}}(v_{f})),}$

where ch is the Chern character, d(v_f) an element of the integral cohomology group H²(Y, Z) satisfying d(v_f) ≡ f^* w₂(TY)-w₂(TX) mod 2, f_K* the Gysin homomorphism for K-theory, and f_H* the Gysin homomorphism for cohomology .^[1] This theorem was first proven by Atiyah and Hirzebruch.^[2]

The theorem is proven by considering several special cases.^[3] If Y is the Thom space of a vector bundle V over X, then the Gysin maps are just the Thom isomorphism. Then, using the splitting principle, it suffices to check the theorem via explicit computation for line bundles.

If f: X → Y is an embedding, then the Thom space of the normal bundle of X in Y can be viewed as a tubular neighborhood of X in Y, and excision gives a map

${\displaystyle u:H^{*}(B(N),S(N))\to H^{*}(Y,Y-B(N))\to H^{*}(Y)}$

and

${\displaystyle v:K(B(N),S(N))\to K(Y,Y-B(N))\to K(Y)}$.

The Gysin map for K-theory/cohomology is defined to be the composition of the Thom isomorphism with these maps. Since the theorem holds for the map from X to the Thom space of N, and since the Chern character commutes with u and v, the theorem is also true for embeddings. f: X → Y.

Finally, we can factor a general map f: X → Y into an embedding

${\displaystyle i:X\to Y\times S^{2n}}$

and the projection

${\displaystyle p:Y\times S^{2n}\to Y.}$

The theorem is true for the embedding. 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.

Corollaries

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.

Notes

1. M. Karoubi, K-theory, an introduction, Springer-Verlag, Berlin (1978)
2. M. Atiyah and F. Hirzebruch, Riemann–Roch theorems for differentiable manifolds (Bull. Amer. Math. Soc. 65 (1959) 276–281)
3. M. Karoubi, K-theory, an introduction, Springer-Verlag, Berlin (1978)