By Dieudonne J.

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We deduce that for any space B there is a natural equivalence between mapB(XO x B, Yo x B) and map(Xo, Yo) x B, as fibrewise spaces over B, provided Xo is locally compact regular. The space of fibrewise maps Returning to the general case, let us compare the space r(mapB(X, Y)) of sections s : B -+ mapB(X, Y) of the fibrewise mapping-space with the space MAPB(X, Y) of fibrewise maps

This condition implies, in particular, that X admits a section, since we can take B' and V to be empty. Unlike fibrewise contractibility, the section extension property is not natural, in our sense. However, if X has the property then so does any fibrewise space which is fibrewise dominated by X. In particular, X has the property if X is fibrewise contractible. If the fibrewise space X over B has the section extension property then so does the restriction XB' of X to any numerically defined open set B' of B.

This proves the first assertion. To prove the second let (J, (J' : E ~ Ea be fibrewise G-maps, expressed in the form We start by showing that (J and G-maps ¢ and ¢t given by (J' are fibrewise G-homotopic to the fibrewise ¢ = [aI, gl, 0, C, a2, g2, 0, c, ... J, ¢t = [O,c,a~,g~,O,c,a~,g~, .. ). In fact a fibrewise G-homotopy H t : E ~ Ea of (J into ¢ is given by the expression [(1 - t)a1, gl, tal, gl, (1 - t)a2, g2, ta2, g2, ... 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.

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Algebraic geometry by Dieudonne J.

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