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This paper is an addendum to
the previous paper, Celllike Mappings, I. Therein, the category of cell-like maps
between ENR’s was established, homotopy-theoretic characterizations of cell-like
maps were given, and the image of a cell-like map on an ENR was studied.
In the present paper, three related topics are considered: the relationship
between (sometimes global) properties of a map and local properties of its
mapping cylinder; limits of cell-like maps; and preservation of tameness
properties under cell-like maps. Loose descriptions of some of the results
follow.
If an onto map between metric spaces has its image locally collared in
its mapping cylinder, then the two spaces are stably homeomorphic. If a
proper, onto map between ENR’s has its mapping cylinder locally k-connected
mod its image for all k, then the map is cell-like (hence a proper homotopy
equivalence).
The limit of a sequence of cell-like maps between ENR’s is cell-like. Likewise, if a
proper map between ENR’s is “concordantly” approximated by cell-like maps, it is
cell-like.
The property of having ULCl complements (for compact sets in ENR’s) is
preserved under monotone maps.
In an appendix, the nonexistence of two types of isolated singularities is
proved.
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