Celebratio Mathematica — Bing

The mathematical work of R H Bing

Works connected to James Cecil Cantrell

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R. H. Bing: “The monotone mapping problem,” pp. 99–115 in Topology of manifolds (University of Georgia, Athens, GA, 11–22 August 1969). Edited by J. C. Cantrell and C. H. Edwards. Markham Mathematics Series. Markham (Chicago), 1970. MR 0275379 Zbl 0283.57004 incollection

It is shown that for \( m = 3,4,\dots \) there is a monotone map of Euclidean \( n \)-space \( E^n \) onto itself that is not compact. This completes the monotone mapping theorem posed by G. T. Whyburn. A key lemma in the treatment shows that there is a monotone map of a cube \( I^2 \) onto itself such that each point inverse intersects a base \( I^2 \) of \( I^3 \). If \( f \) is a map of \( I^3 \) onto \( I^3 \) which is a homeomorphism on \( \operatorname{Int}I^3 \) and takes \( I^2 \) homeomorphically into \( I^2 \), one calls \[ f(\operatorname{Int}I^2\cup\operatorname{Int}I^3) \] a drainage system for \( I^3 \). It is shown that there is a drainage system \( f(\operatorname{Int}I^2 \) \( \cup\operatorname{Int}I^3) \) for \( I^3 \) and a monotone map \( g \) of \[ I^3 - f(\operatorname{Int}I^2\cup\operatorname{Int}I^3) \] onto \( I^3 \) such that \( g \) is the identity on \( \operatorname{Bd}I^3 - \operatorname{Int}I^2 \).

James Cecil Cantrell	Related
C. H. Edwards	Related