Vol. 30, No. 3, 1969

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Cell-like mappings. I

R. C. Lacher

Vol. 30 (1969), No. 3, 717–731

Cell-like mappings are introduced and studied. A space is cell-like if it is homeomorphic to a cellular subset of some manifold. A mapping is cell-like if its point-inverses are celllike spaces. It is shown that proper, cell-like mappings of ENR’S (Euclidean NR’s) form a category which includes both proper, contractible maps of ENR’s and proper, cellular maps from manifolds to ENR’s. It is difficult to break out of the category: The image of a proper, cell-like map on an ENR, is again an ENR, provided the image is finite-dimensional and Hausdorff.

Some applications to (unbounded) manifolds are given. For example: A cell-like map between topological manifolds of dimension 5 is cellular. The property of being an open n-cell, n 5, is preserved under proper, cell-like maps between topological manifolds. The image of a proper, cellular map on an n-manifold is a homotopy n-manifold.

Mathematical Subject Classification
Primary: 55.25
Secondary: 54.00
Received: 26 June 1968
Published: 1 September 1969
R. C. Lacher