Workshop on the h-principle and beyond

Abstract: I will discuss a remarkable generalization of Mather’s theorem by Thurston that relates the identity component of diffeomorphism groups to the classifying space of Haefliger structures. The homotopy type of this classifying space played a fundamental role in foliation theory. However, it is notoriously difficult to determine its homotopy groups. Mather-Thurston theory relates the homology of diffeomorphism groups to these homotopy groups. Hence, this h-principle type theorem has been used as the main tool to get at the homotopy groups of Haefliger spaces. We talk about generalizing Thurston's method to prove analogue of MT for other subgroups of diffeomorphism groups that was conjectured to hold. Most h-principle methods use the local statement for open ball M=R^n to prove a statement about compact manifolds. Thurston’s method unlike the later proof by McDuff-Segal has not been widely used. The interesting point about Thurston's original method is it is intrinsically compactly supported method and it is suitable when the local statement for M=R^n is hard to prove e.g. Mather-Thurston’s theory for contactomorphism, foliation preserving diffeomorphisms etc.