Coglueing Homotopy Equivalences

in which the front square is a pull-back. Then P is often called the fibre-product o f f and p, and it is also said that ~: P ~ X is induced by f from p. The map ~: Q~ P is determined by ~01 and q)2. Our object is to give conditions on the front and back squares which ensure that if q~l, q~2 and ~o are homotopy equivalences, then so also is ~. First of all we shall assume throughout that the back square of (1.1) as well as the front square, is a pull-back. Second recall that a map q: E --* B has the W C H P (weak covering homotopy property) if it has the covering property for all homotopies Z x I ---, B which are stationary on Z x [0, 89 This property has been shown by Dold [3] and Weinzweig [7] to be convenient for studying fiber homotopy equivalences, and our results will extend some of theirs. For the rest of this section we will assume that in (1.1) p and q have the WCHP. Then our main object is the following theorem which will be proved in Sections 2 to 5.