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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Differentiable monotone maps on manifolds. II
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by P. T. Church PDF
Trans. Amer. Math. Soc. 158 (1971), 493-501 Request permission

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 ${N^3}\# {K^3}$ 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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Additional Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 158 (1971), 493-501
  • MSC: Primary 57.20
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0278320-X
  • MathSciNet review: 0278320