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Transactions of the American Mathematical Society
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
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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DOI: http://dx.doi.org/10.1090/S0002-9947-1972-0326745-7
PII: S 0002-9947(1972)0326745-7
Keywords: Collapsing, transverse cellularity, cone, cone complex, cell-like, mapping cylinder, thickening
Article copyright: © Copyright 1972 American Mathematical Society