Vol. 35, No. 3, 1970

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Cell-like mappings. II

R. C. Lacher

Vol. 35 (1970), No. 3, 649–660

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.

Mathematical Subject Classification
Primary: 57.05
Secondary: 55.00
Received: 13 June 1969
Revised: 2 February 1970
Published: 1 December 1970
R. C. Lacher