A generalization of Vassilievs 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 $C^\infty(M,R)$ of smooth functions on a compact smooth manifold M or more generally submodules of a finitely generated $C^\infty(M,R)$-module V. We define a topology on the space of all submodules of V of a fixed finite codimension d. Its main property is that it is compact Hausdorff and, in the case of ideals in the ring itself, 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 $j^r:C^\infty(M,E)\to\Gamma(J^r(M,E))$, defined on the space of smooth maps from a compact manifold M to a Euclidean space E and with target the space of smooth sections of the jet bundle $J^r(M,E)$, is a cohomology isomorphism when restricted to certain “nonsingular” subsets (these are defined in terms of a certain subset $R\subseteq J^r(M,E)$). Our generalization then puts this theorem in a more general setting of topological $C^\infty(M,R)$-modules. As a reward we get a strengthening of this result asserting that all the homotopy fibres have zero homology.

💡 Research Summary

The thesis is divided into two largely independent parts, each addressing a different aspect of smooth function theory on a compact manifold M.

Part I – Ideals, submodules and a compactification of configuration spaces
The author studies the collection M_d(V) of all submodules B⊂V of a fixed finite codimension d, where V is a finitely generated C^∞(M,ℝ)‑module (the most important case being V=C^∞(M,ℝ) itself). By introducing the notion of a transversal finite‑dimensional subspace D⊂V – i.e. a subspace that meets every B∈M_d(V) in a complementary way – one obtains a natural map
 M_d(V) → Gr_d(D)
into the Grassmannian of codimension‑d subspaces of D. The existence of such a D is proved by a careful interpolation argument: if D is transversal, every v∈V can be written uniquely as v = d (mod B) with d∈D, which yields a continuous “interpolation” map V×M_d(V)→D. This construction supplies M_d(V) with a topology that makes the map into the Grassmannian continuous. Since the Grassmannian is compact Hausdorff, M_d(V) inherits these properties.

When V=C^∞(M,ℝ), the ideals m_Y={f | f vanishes on a finite set Y⊂M} lie in M_d(V) and give an embedding of the unordered configuration space M^{