By Wraith, Gavin

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149], the "telescope" has the homotopy type of write i. SI Since [ ~ - l , ~ ] ) c X x [0,®) cat X~ _< k, we may is closed and contractible in X~, hence 25 in X. , k. Y = ^

O(M) , the normal bundle to f : Mn ~ G / F L and a specific G/PL M . , a homotopy equi- ~ , so the diagram IM(id)>IMP2 commutes. Now make desired manifold transverse regular to ~ : ~n ~ M n 4k+2 , and ~ ([30]). ~-l(. X M) is the M . If we assume ing (* × M) M is simply connected, there are only two obstructions to assum- is a homotopy equivalence: a Kervaire invariant the differences of the indexes ~ (I(M)-I(M)) if K(~) c ~ n = ~k if n = ([13], [5]). Sullivan noted that these obstructions actually do not depend on the homotopy class of f but rather on the unoriented bordism class in the case of the Kervaire invariant and on the oriented bordism class in the case of the index.

Sq [±i I±i S n-! 4) affirms that Plainly if k e ±i mod p, then p, p ~ 2, it follows readily that ~ = ±~ B1 or and k ~ ±I mod p. B2 are diffeomorphlc. 3 we may obtain the following statement. 4 q >_ 3 Let we may find n be a given odd integer and such that n÷ ® as q ÷ ® d a given positive integer. , d such that B I × S q = B 2 × S q = ... = B d x Sq, but B i ~ B j, i # J Proof. , d. 3 establish the conclusions. 3. SPHERE-BUNDLES AND LOCALIZATION The attempt to apply the arguments of the preceding section to the case of Sq-bundles with q even, is obstructed by the fact that such bundles are not, in general, quasi-prlncipal.