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 *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* .

(3) *The inclusion of into the mapping cylinder of is collared*.

(4) *The mapping cylinder triad is homeomorphic to the product triad* rel .

Condition (2) is slightly weaker than (point) is homogeneously collapsible in . Condition (4) when stated more precisely implies 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 to have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence relation generated by the relation , where means for some transverse cellular map.

**[A]**Ethan Akin,*Manifold phenomena in the theory of polyhedra*, Trans. Amer. Math. Soc.**143**(1969), 413–473. MR**0253329**, 10.1090/S0002-9947-1969-0253329-1**[C]**Marshall M. Cohen,*Simplicial structures and transverse cellularity*, Ann. of Math. (2)**85**(1967), 218–245. MR**0210143****[C]**Marshall M. Cohen,*A general theory of relative regular neighborhoods*, Trans. Amer. Math. Soc.**136**(1969), 189–229. MR**0248802**, 10.1090/S0002-9947-1969-0248802-6**[C]**Marshall M. Cohen,*Homeomorphisms between homotopy manifolds and their resolutions*, Invent. Math.**10**(1970), 239–250. MR**0275439****[D]**J. Dancis, Unpublished correspondence.**[H]**J. F. P. Hudson,*Piecewise linear topology*, University of Chicago Lecture Notes prepared with the assistance of J. L. Shaneson and J. Lees, W. A. Benjamin, Inc., New York-Amsterdam, 1969. MR**0248844****[L]**W. B. R. Lickorish,*The piecewise linear unknotting of cones*, Topology**4**(1965), 67–91. MR**0203736**

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DOI:
https://doi.org/10.1090/S0002-9947-1972-0326745-7

Keywords:
Collapsing,
transverse cellularity,
cone,
cone complex,
cell-like,
mapping cylinder,
thickening

Article copyright:
© Copyright 1972
American Mathematical Society