Document Type

Dissertation

Date of Award

1975

Keywords

Topological spaces, Manifolds (Mathematics), Kline sphere characterization

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

First Advisor

Dick Wick Hall

Second Advisor

Louis F. McAuley

Third Advisor

Patricia McAuley

Abstract

The classification of topological spaces is an interesting but difficult process. It began with the characterization of the arc as a non-degenerate Peano space having at most two non-cut points; proceeded naturally to S1, and then to S2. Characterization of higher dimensional spheres is still incomplete.

The characterization of the 2-sphere as a Peano space containing a simple closed curve, which is separated by every simple closed curve but by no arc of a simple closed curve was accomplished in 1930 by Leo Zippin [11]. In 1935, E. R. Van Kampen, [8], published a beautiful paper which collected, organized and made generally accessible the then known results in the area.

J. R. Kline conjectured that a non-degenerate Peano space which is separated by every simple closed curve but by no pair of points is a 2-sphere. The problem was studied at length by Kline and Dick Wick Hall, and, in 1942, Hall published a partial solution, [5]. In 1945, R. H. Bing proved the conjecture, [2], which became known as the Bing Sphere Characterization.

It has long been the feeling of a number of mathematicians that the Bing Sphere Characterization should be readily accessible to topologists and other mathematicians. The present paper is an attempt to organize, clarify and explicate the difficult original work of Bing, using some modifications of his methods. Some progress in this direction was made in 1958 by L. F. McAuley, while he was directing a seminar at the University of Wisconsin. McAuley’s notes are included here as Appendix A. They relate most particularly to Theorem 2.5 of this dissertation. They also provide a somewhat more complete history of the problem.

The Bing characterization is clearly the best result of its type for which one could possibly hope. Furthermore, although it is a characterization of S2, which is a 2-manifold, the proofs given by Bing in [1] and [2], and here modified, do not make use of the fact that it is a manifold.

That this may have important implications is clear. There has recently been increasing interest in the study of Peano continua, as is evidenced by recent work of Slocum and Edwards. Edwards shows that, while the suspension of a particular manifold is a Peano continuum, it is not a manifold. The use of brick partitionings provides a very promising method of analyzing Peano spaces.

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