Riemann–Roch theorem for smooth manifolds explained

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

ch (f _{K*} (x)) = f _{H*} ( 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) satisfyingd(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 linebundles.

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 Xin Y, and excision gives a map

u:H ^{*(B(N),} S(N)) \to H ^{*(Y,} Y - B(N)) \to H ^{*(Y)} andv: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 → Yinto an embedding

i: X \to Y x S ^{2n} and the projectionp: Y x 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 and References

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)