2007/05/02 by Lukáš Vokřínek, Vokřínek, Lukáš
Mathematics · #53C23 #58K60 #Algebraic Topology (math.AT) #Differential Geometry (math.DG) #FOS: Mathematics #math.AT #math.DG #msc:53C23 #msc:58K60
paper · pdf · doi:10.48550/arxiv.0705.0333
91 pages; PhD thesis, University of Aberdeen, 2006
arxiv created 2007/05/02 · arxiv updated 2009/12/01
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 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 jr:C^∞(M,E)→Γ(Jr(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 Jr(M,E), is a cohomology isomorphism when restricted to certain "nonsingular" subsets (these are defined in terms of a certain subset R⊆ Jr(M,E)). 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.