A topological space X is invertible at p ∈ X if for every· neighborhood U of p in X, there is a homeomorphism h on X onto X such that h(X - U) ⊆ U. X is continuously invertible at p ∈ X if for every neighborhood U of p in X there is an isotopy {h<sub>t</sub> , 0 ≤ t ≤ 1, on X onto X such that h₁(X - U) ⊆ U.
It is proved that, if X is a locally compact space which is invertible at a point p which has an open cone neighborhood, and if the inverting homeomorphisms may be taken to be the identity at p, then X is continuously invertible at p.
A locally compact Hausdorff space X, invertible at two or more points which have open cone neighborhoods in X, is characterized as a suspension. A locally compact Hausdorff space X which is invertible at exactly one point p, which has an open cone neighborhood U such that U - p has two components, while X - p is connected, is characterized as a suspension with suspension points identified.
Let Cⁿ be an n-conplex with invert point p. Let U be an open cone neighborhood of p in Cⁿ, and let L be the link of U in Cⁿ. Then it is shown that H<sub>p</sub>(Cⁿ) is isomorphic to a subgroup of H<sub>p-1</sub>(L).
Invertibility properties of the i-skeleton of an n-complex are discussed, for i < n. Also, a method is described by which an n-complex which is invertible at certain points may be expressed as the union of subcomplexes, ca.ch of which is invertible at the same points.
One-complexes with invert points are characterized as either a suspension over a finite set of points or a union of simple closed curves [n above ⋃ and i = 1 below that symbol], such that Sᵢ ⋂ Sⱼ = p, i ≠ j.
It is proved that, if C² may be expressed as the monotone union of closed 2-cells. Also if the link of an open cone neighborhood of an invert point in a 2 - complex C² is planar, C² may be embedded in E³. / Doctor of Philosophy
| Identifer | oai:union.ndltd.org:VTETD/oai:vtechworks.lib.vt.edu:10919/88665 |
| Date | January 1965 |
| Creators | Klassen, Vyron Martin |
| Contributors | Mathematics |
| Publisher | Virginia Polytechnic Institute |
| Source Sets | Virginia Tech Theses and Dissertation |
| Language | en_US |
| Detected Language | English |
| Type | Dissertation, Text |
| Format | 45 leaves, application/pdf, application/pdf |
| Rights | In Copyright, http://rightsstatements.org/vocab/InC/1.0/ |
| Relation | OCLC# 20269135 |
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