Lectures on Morse Homology by Augustin Banyaga

B is a fibration. 6 of [30] for more details.

Then i : A--+ X is a cofibration if and only if there exists a Str¢m structure on (X, A). Proof: Assume that i : A --+ X is a cofibration. 23 there exists a retraction r : X x I --+ A x I U X x {0}. Let 1r1 : X x I --+ X and 35 Some homotopy theory 1r2 : X x I ----t I be the projections. Since I is compact we have a well defined map a(x) =sup l1r2(r(x, t))- t l, tEl and (a, 1r1 or) is clearly a Str0m structure on (X, A) as long as a is continuous. To see that a is continuous, let f3(x, t) = l1r2(r(x, t)) -tl and f3t(x) = f3(x, t).

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