Riemann–Roch theorem for smooth manifolds
# 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

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

where ch is the Chern character, $d(v_f)$ an element of the integral cohomology group $H^2(Y,$ Z) satisfying $d(v_f)$ ≡ $f^*$ $w_2(TY)-w_2(TX)$ mod 2, $f_{K*}$ the Gysin homomorphism for K-theory, and $f_{H*}$ the Gysin homomorphism for cohomology . This theorem was first proven by Atiyah and Hirzebruch.

The theorem is proven by considering several special cases. 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

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

and

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

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

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

and the projection

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

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.

