Topology, Homeomorphisms of Some Direct Limit Spaces, directed by Richard E. Heisey.
Topology, Homeomorphisms of Some Direct Limit Spaces, directed by Richard E. Heisey.
The dissertation deals with the topology of some direct limit spaces and their related homeomorphism groups. In particular, let R denote the real numbers and Q the Hilbert cube. Let F be either of lim Rn or lim Qn with the direct limit topology. The spaces considered are F-manifolds M and their spaces of homeomorphisms H(M). Selected results follow.
For closed subsets of a connected F-manifold, metrizability is equivalent to each of local compactness and first countability. In addition, such subsets are negligible in F: if A is a closed, locally compact subset of F, then F - A is homeomorphic to F.
In the limitation topology, the path-components of H(M) are points. In the compact-open topology, H(M) is regular, Lindelof, paracompact, F-stable, and non-first countable. Further, M is a k-space, but H(M) is not.
All free Zq-actions on F are conjugate. Any free Zq-action of an F-deficient subset of F extends to a free Zq-action of F.
Concerning factors of homeomorphisms, we prove the following general result. Suppose a group G acts continuously on a space X. If there is a subspace H of H(X) and a continuous map r: H ® G satisfying certain algebraic properties, then G is a topological factor of H. Corollaries are that H(M) is F-stable, that the linear homeomorphism group of lim Rn has a Hilbert space factor, and that H(lim Rn) is homeomorphic to its two subgroups obtained by fixing one point and two points, respectively.
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Topology. Qualifying paper: Two Theorems from Hilbert Cube Manifold Theory. This is an exposition of two theorems of T.A. Chapman, the "Triangulation Theorem" and the "Classification Theorem." |
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Mathematics, graduation with honors. |