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Extra info for Algebraic and Classical Topology. The Mathematical Works of J. H. C. Whitehead

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17) (see also p. 18). Let S ~ admit an (n — \)-field, where n ^ 4. Then Pan J* 0, Pfin 7* 0. 19) Pan j*0, P0n5*O ifn - 4 or 8. We return to the case in which n is odd and prove (see also , p. 20) We have Pfo = 0. 16) occurs on p. 80 of . ET7(SZ), 22 NOTE ON THE WHJTEHEAD PRODUCT where 7 € T7(S*) is the Hopf element and y*a7 = 7 © «7 5^ 0. 2) of . 2), it follows that E2w7(Sz) = 0. 1) that Pft = Pi 5 o &h = E(y o «7 o ft). 15), we have #(27) = Bfy. Also TTII(58) = Z24 and is generated by E*y.

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. 46 HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES un may be defined by composing this with the identical map S0 -> S.

Y1 = Conversely, let G be an effective topological transformation group of Y> let T: S71'1 -> G be a given map, and let B be the space which is obtained from Vn X Y by identifyingf (x, y) with T(x)y if x e S*- 1 . 2) be the identification map and let p: B -+ Sn be defined by ph(xiy) = unx. , and let <£p: J7p x F -> B be defined by <£iK*i>3/) = %i> x 2(uy) = T'ix^y), h(*2>y)> where unxp e Up. /) = y if a; e /S n_1 it follows that x is a homeomorphism of t ^ x Y onto p~1]Uv If unxe ^0 U2) then i1(f>2(^n^y) = ^ixHx,y) = (wn(s), T'(^)y).