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In this note we give sufficient conditions that certain sphere bundles over spheres, denoted Bn(p), are of the homotopy type of loop spaces mod/? for p an odd prime. The method is to construct a classifying space for the /?-profinite com­ pletion of Bn(p) by collapsing an Eilenberg-Mac Lane space by the action of a certain finite group. We say that a space X has some property mod/? if the localization of X at p has the property. The problem of determining which spheres are of the homotopy type of loop spaces mod/? has been completely solved by Sullivan (9). It is therefore natural to ask which sphere bundles over spheres are of the homotopy type of loop spaces mod/?. In this regard, results of Curtis (2) and Stasheff (7) concerning the question of which sphere bundles over spheres are //-spaces mod/? give some negative infor­ mation. Moreover, in a recent paper (3) we investigated a certain class of sphere bundles over spheres and gave necessary conditions for them to be of the homotopy type of a loop space mod/? for /? an odd prime. In this note we prove that certain of these bundles satisfying the conditions of (3) are of the homotopy type of a loop space mod/? and answer a question posed in (8). For p an odd prime and n a positive integer, the space Bn(p) is an 52n+1-bundle over s2n+w{p~1) classified by the generator of the /?-primary part of iT2n+z(v-i)(S2n+1)- From (5) we have that H*(Bn(p)\Z\p) is an exterior algebra on generators x and y, where degx=2«+l, degj=2«+ 2/?—l and ^xx=y. Although few of the Bn(p) are of the homotopy type of a loop space mod/? (see (3)), we have the following exceptions. THEOREM 1. The space Bn(p) is of the homotopy type of a loop space mod/? ifn and p satisfy any of the following conditions: (i) «=1 ; p=any odd prime, (ii) n =/?—2 ; /?=any odd prime, (iii) «=7;/? = 17, (iv) rc=5;/?=19, (v) H=19;/?=41.

Some examples of sphere bundles over spheres which are loop spaces \bmod𝑝