Riemann Roch Theorem for Surfaces
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In their paper they define, for oriented smooth closed manifolds X and Y and a continuous mapping f: Y X that f is a c1-map if there is c1 in the integral cohomology group H2(Y, Z) such that for the Stiefel-Whitney classes w2 we have c1 = w2(Y) f (w2(X)) modulo 2 in H2(Y, Z/2Z). Writing ch(X) for the image in H (X, Q) they showed that for f a c1-map there is f!: ch(Y) ch(X) which is a homomorphism of abelian groups, and satisfying f!(y)A^(X) = f (y.exp(c1)/2)A^(Y)), w...
Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In their paper they define, for oriented smooth closed manifolds X and Y and a continuous mapping f: Y X that f is a c1-map if there is c1 in the integral cohomology group H2(Y, Z) such that for the Stiefel-Whitney classes w2 we have c1 = w2(Y) f (w2(X)) modulo 2 in H2(Y, Z/2Z). Writing ch(X) for the image in H (X, Q) they showed that for f a c1-map there is f!: ch(Y) ch(X) which is a homomorphism of abelian groups, and satisfying f!(y)A^(X) = f (y.exp(c1)/2)A^(Y)), where A^ is the A-hat genus and f the Gysin homomorphism. This mimics the GRR theorem; but f! has only an implicit definition. This they 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.
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