De Rham Theorem in the Globally Subanalytic Setting
ABSTRACT
For o-minimal structures there exists a well-established theory on definable singular homology and cohomology. What has been missing so far is a definable version of the de Rham cohomology and of the de Rham theorem. We consider the real field with restricted analytic functions. The definable sets and functions are the globally subanalytic ones. For a globally subanalytic manifold one can define the de Rham complex of globally subanalytic differential forms. But there the de Rham theorem does not hold. Using the larger class of constructible functions where also the global logarithm is involved, we define the de Rham complex of constructible differential forms. Here we can establish the de Rham theorem. This implies that constructible de Rham cohomology groups are canonically isomorphic to the classical ones.
– joint work with Annette Huber and Abhishek Oswal –
