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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Compact sets definable by free $ 3$-manifolds

Author: W. H. Row
Journal: Trans. Amer. Math. Soc. 197 (1974), 225-244
MSC: Primary 57A10; Secondary 54C56
MathSciNet review: 0362314
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Abstract: Shape conditions are given that force a compactum (i.e., a compact metric space) embedded in the interior of a nonclosed, piecewise-linear 3-manifold to have arbitrarily close, compact, polyhedral neighborhoods each component of which is a 3-manifold with free fundamental group (i.e., to be definable by free 3-manifolds). For compact, connected ANR's these conditions reduce to the criterion of having a free fundamental group. Additional conditions are given that insure definability by handlebodies or cubes-with-handles. An embedding of Menger's universal 1-dimensional curve in Euclidean 3-space is shown to have the property that all tame surfaces, separating in 3-space a fixed pair of points, cannot be adjusted (by a small space homeomorphism) to intersect the embedded curve in a 0-dimensional set.

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Keywords: 3-manifold, handlebody, cube-with-handles, definable by free 3-manifolds, definable by thin handlebodies, arc pushing properties, UV properties, ANR-sequence, fundamental shape, fundamental domination, free group
Article copyright: © Copyright 1974 American Mathematical Society

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