A Modern Overview of Local Sections of Flows
| dc.contributor.advisor | Markley, Nelson | |
| dc.contributor.author | Colston, Helen Marie | |
| dc.contributor.department | Mathematics | |
| dc.contributor.publisher | University of Maryland (College Park, Md) | |
| dc.contributor.publisher | Digital Repository at the University of Maryland | |
| dc.date.accessioned | 2022-12-01T19:37:50Z | |
| dc.date.available | 2022-12-01T19:37:50Z | |
| dc.date.issued | 1990 | |
| dc.description.abstract | This paper examines local cross sections of a continuous flow on a locally compact metric space. Sane of the history of the study of local cross sections is reviewed, with particular attention given to H. Whitney's work. The paper presents a modern proof that local cross sections always exist at noncritical points of a flow. Whitney is the primary source for the key idea in the existence proof; he also gave characterizations of local cross sections on 2- and 3-dimensional manifolds. We show various topological properties of local cross sections, the most important one being that local cross sections on the same orbit are locally homeomorphic. A new elementary proof using the Jordan Curve Theorem shows that when a flow is given on a 2-manifold, a local cross section will be an arc. Whitney is cited for a similar result on 3-maniforlds. Finally, the so-called "dob=bone" space of R. Bing is used to construct a flow on a 4-manifold with a point at which every local cross section is not homeomorphic to a 3-dimensional disk. | en_US |
| dc.identifier | https://doi.org/10.13016/de9l-qszx | |
| dc.identifier.other | ILLiad # 1542312 | |
| dc.identifier.uri | http://hdl.handle.net/1903/29477 | |
| dc.language.iso | en_US | en_US |
| dc.title | A Modern Overview of Local Sections of Flows | en_US |
| dc.type | Thesis | en_US |