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 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:Xto Ytimes S^{2n}} and the projection {displaystyle p:Ytimes 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) hide vte Bernhard Riemann Cauchy–Riemann equationsGeneralized Riemann hypothesisGrand Riemann hypothesisGrothendieck–Hirzebruch–Riemann–Roch theoremHirzebruch–Riemann–Roch theoremLocal zeta functionMeasurable Riemann mapping theoremRiemannRiemann Xi functionRiemann curvature tensorRiemann hypothesisRiemann integralRiemann invariantRiemann mapping theoremRiemann formRiemann problemRiemann series theoremRiemann solverRiemann sphereRiemann sumRiemann surfaceRiemann zeta functionRiemann's differential equationRiemann's minimal surfaceRiemannian circleRiemannian connection on a surfaceRiemannian geometryRiemann–Hilbert correspondenceRiemann–Hilbert problemsRiemann–Lebesgue lemmaRiemann–Liouville integralRiemann–Roch theoremRiemann–Roch theorem or smooth manifoldsRiemann–Siegel formulaRiemann–Siegel theta functionRiemann–Silberstein vectorRiemann–Stieltjes integralRiemann–von Mangoldt formula Category Categories: Theorems in differential geometryAlgebraic surfacesBernhard Riemann

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