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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Mappings from $ 3$-manifolds onto $ 3$-manifolds


Author: Alden Wright
Journal: Trans. Amer. Math. Soc. 167 (1972), 479-495
MSC: Primary 57A10
MathSciNet review: 0339186
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Abstract: Let f be a compact, boundary preserving mapping from the 3-manifold $ {M^3}$ onto the 3-manifold $ {N^3}$. Let $ {Z_p}$ denote the integers mod a prime p, or, if $ p = 0$, the integers. (1) If each point inverse of f is connected and strongly 1-acyclic over $ {Z_p}$, and if $ {M^3}$ is orientable for $ p > 2$, then all but a locally finite collection of point inverses of f are cellular. (2) If the image of the singular set of f is contained in a compact set each component of which is strongly acyclic over $ {Z_p}$, and if $ {M^3}$ is orientable for $ p \ne 2$, then $ {N^3}$ can be obtained from $ {M^3}$ by cutting out of $ \operatorname{Int} \;{M^3}$ a compact 3-manifold with 2-sphere boundary, and replacing it by a $ {Z_p}$-homology 3-cell. (3) If the singular set of f is contained in a 0-dimensional set, then all but a locally finite collection of point inverses of f are cellular.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0339186-3
PII: S 0002-9947(1972)0339186-3
Keywords: Monotone mapping, 3-manifold, acyclic mapping, decomposition space, strongly acyclic
Article copyright: © Copyright 1972 American Mathematical Society