By Dieudonne J.

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Let X be a fibrewise space over B, and let the cylinder X x I be regarded as a fibrewise space over B by precomposing with the second projection. Consider a fibrewise space D over X x I. We regard Dt=DI(Xx{t}) (0 ::; t ::; 1) as a fibrewise space over X in the obvious way. The main step in the proof of the fibrewise homotopy theorem is the demonstration that if D is a fibrewise fibre space over X x I then Do is a fibrewise deformation retract of Dover X. Similarly, Dl is a fibrewise deformation retract of D.

J, and a fibrewise G-homotopy of (J' into ¢/ is given similarly. The next stage is to construct, in infinitely many steps, a fibrewise G-homotopy of ¢ into (J. The first step, indicated by the expression [aI, gl, ta2, g2, (1 - t)a2, g2, ta3, g3, (1 - t)a3, ... J, ends with the fibrewise G-map indicated by the expression The second step is defined similarly, and so on, until after the nth step we reach the fibrewise G-map indicated by the expression 7 Fibrewise fibre bundles 39 By juxtaposing the steps in this series of fibrewise G-homotopies we obtain a fibrewise G-homotopy since at each coordinate place in Ea all but a finite number of the steps are stationary.

Then the closed fibrewise pair (X,X') XB (Y,Y') = (X XB Y,X' XB YUX XB Y') is also fibrewise cofibred. To see this choose fibrewise Str0m structures (a, h) on (X, X') and ({3, k) on (Y, Y'). Define'Y: X XB Y -t I by 'Y(x,y) = min(a(x) , (3(y)), and define f: (X XB Y) x I -t X XB Y by f(x,y,t) = (h(x,min(t, {3(y))),k(y, min(t, a(x)))). Then ('Y, f) constitutes a fibrewise Str0m structure for the closed fibrewise pair (X, XI) XB (Y, yl), as required. For example, consider the endofunctor ~B of our category determined by a closed cofibred pair (D, E).