A Generalization of Vassiliev's H-principle

This thesis consists of two parts which share only a slight overlap. The first part is concerned with the study of ideals in the ring C ∞ (M, R) of smooth functions on a compact smooth manifold M or more generally submodules of a finitely generated C ∞ (M, R)-module V. We define a topology on the space M d (V) of all submodules of V of a fixed finite codimension d. Its main property is that it is compact Hausdorff and, when V = C ∞ (M, R), it contains as a subspace the configuration space of d distinct unordered points in M and therefore gives a " compactification " of this configuration space. We present a concrete description of this space for low codimensions. The main focus is then put on the second part which is concerned with a generalization of Vassiliev's h-principle. This principle in its simplest form asserts that the jet prolongation map j r : C ∞ (M, V) → Γ(J r (M, V)), defined on the space of smooth maps from a compact manifold M to a Euclidean space V and with target the space of smooth sections of the jet bundle J r (M, V), is a coho-mology isomorphism when restricted to certain " nonsingular " subsets (these are defined in terms of a certain subset R ⊆ J r (M, V)). Our generalization then puts this theorem in a more general setting of topological C ∞ (M, R)-modules. As a reward we get a strengthening of this result asserting that all the homotopy fibres have zero homology.