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.