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Transactions of the American Mathematical Society

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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
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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.

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