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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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Transverse cellular mappings of polyhedra
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by Ethan Akin PDF
Trans. Amer. Math. Soc. 169 (1972), 401-438 Request permission

Abstract:

We generalize Marshall Cohen’s notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let $f:K \to L$ be a proper simplicial map of locally finite simplicial complexes. The following are equivalent: (1) The dual cells of the map are all cones. (2) The dual cells of the map are homogeneously collapsible in $K$. (3) The inclusion of $L$ into the mapping cylinder of $f$ is collared. (4) The mapping cylinder triad $({C_f},K,L)$ is homeomorphic to the product triad $(K \times I;K \times 1,K \times 0)$ rel $K = K \times 1$. Condition (2) is slightly weaker than ${f^{ - 1}}$(point) is homogeneously collapsible in $K$. Condition (4) when stated more precisely implies $f$ is homotopic to a homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications are first, to develop the proper theory of “attaching one polyhedron to another by a map of a subpolyhedron of the former into the latter". Second, we classify when two maps from $X$ to $Y$ have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation $f \sim g$, where $f,g:X \to Y$ means $f = gr$ for $r:X \to X$ some transverse cellular map.
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Additional Information
  • © Copyright 1972 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 169 (1972), 401-438
  • MSC: Primary 57C05
  • DOI: https://doi.org/10.1090/S0002-9947-1972-0326745-7
  • MathSciNet review: 0326745