Algebraic and Classical Topology. The Mathematical Works of by John Henry Constantine Whitehead

By John Henry Constantine Whitehead

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Then # n /S*- 1 c (3^ and a map T: fifn-1 -> G, which characterizes the structure of the bundle B, is defined by Tx = unx for every x e S71'1. Notice that Tan~l = e, the identity in G, since &nan-x = 60. 2) n be defined by h(x,y) = un(x)yy for every x e V , y e Y. Since ph(x,y) =punx = unx, n n l h maps (V —S ~ )xY homeomorphically onto B—Yv Notice that h{x, y) = T(x)y if x e S"-1. , where y0 is a fixed point in Y. group and fibre are the subgroup of stability, H c G, of y0. e. £? = / £ » , where/: £ n -► B is a map such that p 0 / = 1.

Then£ 0 = l , { 1 Z = I , ^ ( ? c C and ^determines a homotopy equivalence (Z, C) -> (X9A), which is a homotopy inverse of the identical map i: (X, A) -> (Z, C). #' = #' (a: e X, x* e X'), is obviously a homotopy equiva­ lence. So therefore is g o i. But g o i = / . 1). Let A, A', as well as X, X ' , be connected and let X , X ' be of the same homotopy type. L e t | q = max (dim A, dirndl') < 00 and let (a) 7Tk(X,A) = 0, 7Tk(X',A') = 0 ( l < f c < g + l ) | (3 2) (6) dnq+1(X,A) = 09 d7rq+1(X'iA') = 0 if ?

N. 6) t h a t g' is a homotopy equivalence. 1) below, it follows t h a t h' is a homotopy equivalence of the pairf (X,A) to the pair (YxA,y0xA). Let Z be any CW-complex and let C be a sub-complex of Z. I t follows from the homotopy extension theorem^ t h a t C is a retract of Z if there is a map r: Z -> C such t h a t r\C c^. 1. Hence it obviously follows t h a t C is a retract of Z if the pair (Z, C) is of the same homotopy type as any pair of spaces (Z', 0"), such t h a t C is a retract of Z'.

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