Transverse Cellular Mappings of Polyhedra

We generalize Marshall Cohen's notion of transverse cellular map to the polyhedral category. They are described by the following: Proposition. Let f : K —* 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,, K, L) is homeomorphic to the product triad (Kxl; Kxl.KxO) rel K = K x 1. Condition (2) is slightly weaker than /"* (point) is homogeneously collapsible in K. Condition (4) when stated more precisely implies / is homotopic toa homeomorphism. Furthermore, the homeomorphism so defined is unique up to concordance. The two major applications ate first, to develop the proper theory of "attaching one polyhedron to another by a map of a subpolyhedron of the former into the lattet". Second, we„classify when two maps ftom X to Y have homeomorphic mapping cylinder triads. This property turns out to be equivalent to the equivalence telation generated by the relation fr^g, where /, g: X —» Y means /= gr for r: X —* X some transverse cellular map. Marshall Cohen has developed (see [C.]) a theory of transverse cellular mappings defined on manifolds. They satisfy a slightly weaker condition than collapsibility of point-inverses. They are close to homeomorphisms in that they share with homeomorphisms the property that their mapping cylinder is a product. Their interest is that they are precisely all the maps which satisfy this property. In this paper, we generalize the notion of transverse cellularity to proper maps of locally compact polyhedra. Proposition. Let f: K—>L be a proper, simplicial map of locally finite simplicial complexes. Then the following are equivalent: (1) For every A £ L, the dual cell of A with respect to f, DÍA; f) idejined to be f~ DÍA; L)) is homeomorphic to the cone on DÍA; f) rel DÍA; f) (where DÍA; f)^f-YDÍA; L)). Received by the editots June 11, 1970 and, in revised form, May 4, 1971. AMS 1969 subject classifications. Primary 5701, 5705.