The 3-manifold recognition problem

Authors:
Robert J. Daverman and Thomas L. Thickstun

Journal:
Trans. Amer. Math. Soc. **358** (2006), 5257-5270

MSC (2000):
Primary 57N10, 57P99; Secondary 57M30, 57N60, 57N75

DOI:
https://doi.org/10.1090/S0002-9947-05-03786-4

Published electronically:
December 20, 2005

MathSciNet review:
2238915

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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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Additional Information

**Robert J. Daverman**

Affiliation:
Department of Mathematics, The University of Tennessee at Knoxville, Knoxville, Tennessee 37996-1300

Email:
daverman@math.utk.edu

**Thomas L. Thickstun**

Affiliation:
Department of Mathematics, Texas State University, San Marcos, Texas 78666

Email:
tt04@txstate.edu

DOI:
https://doi.org/10.1090/S0002-9947-05-03786-4

Keywords:
Generalized 3-manifold,
resolvable,
simplicial approximation property,
relative simplicial approximation,
tame embedding,
locally $1$-coconnected

Received by editor(s):
April 21, 2003

Received by editor(s) in revised form:
July 21, 2004

Published electronically:
December 20, 2005

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.