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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles 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 1.48.

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Some mapping theorems
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by R. C. Lacher PDF
Trans. Amer. Math. Soc. 195 (1974), 291-303 Request permission

Abstract:

Various mapping theorems are proved, culminating in the following result for mappings f from a closed $(2k + 1)$-manifold M to another, N: If “almost all” point-inverses of f are strongly acyclic in dimensions less than k and if “almost all” point-inverses of f have Euler characteristic equal to one, then all but finitely many point-inverses are totally acyclic. (Here “almost all” means “except on a zero-dimensional set in N".) More can be said when $k = 1$: If f is a monotone map between closed 3-manifolds and if the Euler characteristic of almost-all point-inverses is one, then all but finitely many point-inverses of f are cellular in M; consequently M is the connected sum of N and some other closed 3-manifold and f is homotopic to a spine map. Other results include an acyclicity criterion using the idea of “nonalternating” mapping and the following result for PL maps $\phi$ between finite polyhedra X and Y: If the Euler characteristic of each point-inverse of $\phi$ is the integer c then $\chi (X) = c\chi (Y)$.
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Additional Information
  • © Copyright 1974 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 195 (1974), 291-303
  • MSC: Primary 57A15
  • DOI: https://doi.org/10.1090/S0002-9947-1974-0350743-2
  • MathSciNet review: 0350743