The 3-manifold recognition problem

Daverman, Robert; Thickstun, Thomas

doi:10.1090/S0002-9947-05-03786-4

The 3-manifold recognition problem
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by Robert J. Daverman and Thomas L. Thickstun PDF

Trans. Amer. Math. Soc. 358 (2006), 5257-5270 Request permission

Abstract:

We introduce a natural Relative Simplicial Approximation Property for maps from a 2-cell to a generalized 3-manifold and prove that, modulo the Poincaré Conjecture, 3-manifolds are precisely the generalized 3-manifolds satisfying this approximation property. The central technical result establishes that every generalized 3-manifold with this Relative Simplicial Approximation Property is the cell-like image of some generalized 3-manifold having just a 0-dimensional set of nonmanifold singularities.

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