A generalization of Vassiliev's h-principle
| dc.creator | Vokřínek, Lukáš | |
| dc.date | 2007-05-02 | |
| dc.date.accessioned | 2026-07-07T07:59:09Z | |
| dc.date.available | 2026-07-07T07:59:09Z | |
| dc.description | 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 $j^r:C^\infty(M,E)\toΓ(J^r(M,E))$, 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. | |
| dc.description | 91 pages; PhD thesis, University of Aberdeen, 2006 | |
| dc.identifier | https://arxiv.org/abs/0705.0333 | |
| dc.identifier | http://arxiv.org/abs/0705.0333 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128263 | |
| dc.subject | Algebraic Topology | |
| dc.subject | Differential Geometry | |
| dc.subject | 58K60; 53C23 | |
| dc.title | A generalization of Vassiliev's h-principle | |
| dc.type | text |