Celebratio Mathematica — Bing

Complete Bibliography

Works connected to Steve Armentrout

Filter the Bibliography List

year	title	people

			clear

S. Armentrout and R. H. Bing: “A toroidal decomposition of \( E^3 \),” Fund. Math. 60 : 1 (1967), pp. 81–87. MR 0206925 Zbl 0145.19702 article

An upper semi-continuous decomposition \( G \) of \( E^3 \) is defined to be a toroidal decomposition if and only if the following condition holds: There is a sequence \( M_1 \), \( M_2 \), \( M_3 \), \( \dots \) of compact 3-manifolds-with-boundary in \( E^3 \) such that (1) for each \( i \), \( M_{i+1}\subset \operatorname{Int} M_i \) and each component of \( M_i \) is a solid torus (cube with one handle) and (2) \( A \) is a non-degenerate element of \( G \) if and only if \( A \) is a non-degenerate component of \( M_1\cdot M_2\cdot M_3\dots \).

Bing proved in [1952] that the union of two solid Alexander horned spheres, sewed together along their boundaries, was homeomorphic to \( S^3 \). A major step in this proof consists of showing that for a certain toroidal decomposition \( H \) of \( E^3 \) into tame arcs and one-point sets, the decomposition space associated with \( H \) is homeomorphic to \( E^3 \). Keldyš raised the following question in [1961]:

Does every toroidal decomposition of \( E^3 \) into tame arcs and one-point sets yield \( E^3 \) as its decomposition space?

In this paper we give a negative answer to this question.

R. H. Bing: Collected papers. Edited by S. Singh, S. Armentrout, and R. J. Daverman. American Mathematical Society (Providence, RI), 1988. In two volumes. MR 950859 Zbl 0665.01012 book

Steve Armentrout	Related
Virendra Singh	Related
Robert J. Daverman	Related