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

Differentiable monotone maps on manifolds. II


Author: P. T. Church
Journal: Trans. Amer. Math. Soc. 158 (1971), 493-501
MSC: Primary 57.20
MathSciNet review: 0278320
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Abstract: Let $ {M^n}$ and $ {N^n}$ be closed manifolds, and let $ G$ be any (nonzero) module. (1) If $ f:{M^3} \to {N^3}$ is $ {C^3}$ $ G$-acyclic, then there is a closed $ {C^3}$ $ 3$-manifold $ {K^3}$ such that $ {N^3}\char93 {K^3}$ is diffeomorphic to $ {M^3}$, and $ {f^{ - 1}}(y)$ is cellular for all but at most $ r$ points $ y \in {N^3}$, where $ r$ is the number of nontrivial $ G$-cohomology $ 3$-spheres in the prime decomposition of $ {K^3}$. (2) If $ f:{M^3} \to {M^3}$ or $ f:{S^3} \to {M^3}$ is $ G$-acyclic, then $ f$ is cellular. In case $ G$ is $ Z$ or $ {Z_p}$ ($ p$ prime), results analogous to (1) and (2) in the topological category have been proved by Alden Wright. (3) If $ f:{M^n} \to {M^n}$ or $ f:{S^n} \to {M^n}$ is real analytic monotone onto, then $ f$ is a homeomorphism.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1971-0278320-X
PII: S 0002-9947(1971)0278320-X
Keywords: Monotone, acyclic, cellular, open, differentiable, real analytic maps
Article copyright: © Copyright 1971 American Mathematical Society