By John Henry Constantine Whitehead

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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 ? < 00, f where d is the boundary operator. 3). Subject to these conditions any homotopy equivalence X -> X' is homotopic to the map determined by a homotopy equivalence of the pair (X,A) to the pair (X\Af). THEOREM Let / ° : X - > X ' be a homotopy equivalence. Since dim A < # and TTk(X\A') = 0 if 1 < k < q-\-l there is obviously a homotopy / ° ~ / x such that fxA c A'.

X, x -> x. x0 is homotopic to the identity. Therefore Sm is an H-space if and only if there is a map SmxSm-> Sm of *yP e (*m> lm)> which is equivalent to the condition [tm, tm] = 0. 44 HOMOTOPY THEORY OF SPHERE BUNDLES OVER SPHERES Let n ^ 1 and let B be a g-sphere bundle over Sn, which admits a crosssection. 23). B is an H-spaee if and only if A(J3) = 0 and Sn, Sq are H-spaces. 2. An expression for J In this section we give, explicitly, the expression for J which is indicated in § 9 of (10). Let rj: EnxEq-+ En+q be the natural homeomorphism, which is defined as on p.

2 . The m a i n t h e o r e m Let X, Y, A be non-vacuous, pathwise connected spaces and let X, Y X A be dominated by OW-complexes. Let N = max(AX, A ( 7 x 4 ) ) < oo, t Numbers in bold type refer to the list of references at the end of this paper. % By this we mean that / satisfies the covering homotopy condition, as stated in § 5 below. This implies t h a t / X = Y. § A space Q is said to dominate a space P if and only if there are maps u: P -> Q, vi Q -> P such that v o u ~ 1, where 1: P -> P is the identity and v o u is the com posite of u followed by v.