Shape fibrations I

Abstract

The homotopy lifting property is not a very useful notion when applied to maps p:E→B between spaces with bad local properties. The approximate homotopy lifting property, introduced by D.S. Coram and P.F. Duvall, is useful only when E and B are ANR's. This paper introduces a new class of maps p:E→B between locally compact metric spaces called shape fibrations. Shape fibrations are defined in the spirit of the ANR-sequence approach to shape theory. It is shown that shape fibrations coincide with approximate fibrations whenever the base space and total space are ANR's. The following are typical results:

• 1.

  (i) fibers have the same shape whenever the base space is path connected,

• 2.

  (ii) any proper cell-like map between finite-dimensional locally compact metric spaces is a

• 3.

  shape fibration, and

• 4.

  (iii) the Taylor map is a cell-like map which fails to be a shape fibration.