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

Abstract

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 completion 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 information. 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 5-bundle over s~ classified by the generator of the /?-primary part of iT2n+z(v-i)(S)From [5] we have that H*(Bn(p)\Z\p) is an exterior algebra on generators x andy , where d e g x = 2 « + l , d e g j = 2 « + 2/?—l and ^x=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 = 1 9 ; / ? = 4 1 . AMS (MOS) subject classifications (1970). Primary 55F25, 55F35. Copyright © American Mathematical Society 1974