Jiggling and wrinkling

Abstract

In this thesis we study the topology of the space of geometric structures on a manifold from an h-principle perspective. We loosely define a geometric structure as a section of a fiber bundle over a manifold satisfying some condition R on its derivatives. Accordingly, we refer to such sections as solutions. Rather than studying the space of solutions directly, we introduce the larger space of formal solutions, which is generally easier to study. The core objective in studying h-principles is understanding the connectivity of the inclusion of solutions into formal solutions. If the inclusion is a weak homotopy equivalence, we say that the h-principle holds. The h-principle philosophy has proven successful on open manifolds, but for closed manifolds it often fails. For these we need different sources of flexibility, one of which are singularities. In this thesis we study two types of singularities in the context of h-principles, arising from respectively jiggling and wrinkling sections. In the first part of this thesis, we prove that holonomic approximation of formal solutions by solutions is possible over all manifolds if we allow for mild singularities, known as wrinkles. We apply this result to the study of h-principles using R-microbundles, which are related to Haefliger structures. In the second part, we first provide a modern account of Thurston’s jiggling lemma. We then prove a generalization of jiggling for open and fiberwise dense relations, establishing an h-principle between the spaces of sections and piecewise smooth solutions.