Vol. 97, No. 1, 1981

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Generalized three-manifolds with zero-dimensional nonmanifold set

Matthew G. Brin and Daniel Russell McMillan, Jr.

Vol. 97 (1981), No. 1, 29–58

This paper investigates the statement (GM): “If X is a compact generalized 3-manifold without boundary, and whose nonmanifold set is 0-dimensional, then X is the cell-like image of some closed 3-manifold.” Some necessary and some sufficient conditions on X are given for (GM) to be true. The question of whether (GM) is true in general is shown to be inextricably tangled with the Poincaré Conjecture: (1) If the Poincaré Conjecture fails, then there is an acyclic, monotone union M of handlebodies whose one-point compactification M is a generalized 3-manifold, yet M is not the cell-like image of any compact 3-manifold. (2) If X is a compact generalized 3-manifold with zero-dimensional singular set S and no π1-torsion in any sufficiently tight neighborhood of S, then (modulo the Poincaré Conjecture) X is the cell-like image of a compact 3-manifold.

Mathematical Subject Classification 2000
Primary: 57P05
Secondary: 57N10
Received: 28 March 1980
Published: 1 November 1981
Matthew G. Brin
Daniel Russell McMillan, Jr.