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Transactions of the American Mathematical Society

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

 
 

 

Transverse cellular mappings of polyhedra


Author: Ethan Akin
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
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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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Keywords: Collapsing, transverse cellularity, cone, cone complex, cell-like, mapping cylinder, thickening
Article copyright: © Copyright 1972 American Mathematical Society