A Riemann- Roch theorem for compact spaces

Further to Grothendieck's works that proof that we have a Riemann-Roch theorem for certain morphisms of algebraic varieties and of Hirzebruch and Atiyah for certain morphisms of differentiable manifolds we will proof that we have a Riemann-Roch theorem for continuous applications between compact spaces verifying certain conditions in the context of topological K-theory of compact spaces.The Riemann-Roch theorem that we have in mind involves the K functor defined by K (X): =-1K°(X) K (X) where K°(X) denotes the Grothendieck group of complex fiber bundles over X -1 where K (X): = K°(S(X)) where S(X) denotes the reduced suspension of X and the H* functor k defined by par H*(X): = H (X;Q) .These two functors will apply to the category where the objects are compact spaces and the morphisms are applications that we will call using Lang and Fulton terminology regular.