By M.M. Cohen
Cohen M.M. A direction in simple-homotopy concept (Springer, [1973)(ISBN 3540900551)
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Additional info for A course in simple-homotopy theory
Local Triviality) For each Xo in X there exists an open set V containing Xo and a diffeomorphism \II: -p-i (V) -+ V x G of the form \II(p) = (p(p), 'I/J(p)) , where 'I/J: P -1(V) --+ G satisfies 'I/J(p. g) = 'I/J(p)g for all p E p-1(V) and all 9 E G . 7) P is called the bundle space, X the base space and P the projection of the bundle. For the example we have under consideration, condition (1) follows at once from the way in which we defined P: 8 3 -+ 8 2 • Thus, we need only verify the local triviality condition (2).
1. Topological Spaces 40 One can extend this idea to m x n complex matrices Xl. 17) in the obvious way by stringing out the rows as above ((Zll, . , z1n , . . , zm1, . . , zmn)) and then splitting each zii into real and imaginary parts: (X II , y 11 , . . ,x1n,y1n, . . ,xm1 ,ym1 , . . ,xmn , y mn) . 18) Thus, any set of complex m x n matrices acquires a topology as a subset of 1R 2mn . One can even push this a bit further (and we will need to do so). An m x n matrix whose entries are quaternions = (xii+yiii+Uiij+viik) .
Since I is continuous, l -l(U) is open in X' and therefore X n l-l(U) is open in X. But X n l-l(U) = (J I X) -l(U) so (J I X)-l(U) is open in X and I I X is continuous. The inclusion map is the restriction to X of the identity map id : X' -> X', which is clearly continuous. 1, one may be given a continuous map g : X -> Y and ask whether or not there is a continuous map I : X' -> Y with I IX = g. Should such an I exist it is called a continuous extension of g to X' and g is said to extend continuously to X'.
A course in simple-homotopy theory by M.M. Cohen