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