The K-theory localization of loops on an odd sphere and applications

Abstract

K-theory localization of an unstable odd sphere is constructed by Mahowald and Thompson in [8]. Earlier work on unstable K-theory localization includes Bousfield’s construction of the K-theory localization of infinite loop spaces in [4] and Mislin’s construction of the K-theory localization of Eilenberg-MacLane spaces in [13]. The con- struction of Mislin demonstrates that the unstable K-theory localization functor does not commute with taking loops on a space, nor does it preserve the connectivity of a space. This being the case, we note that the K-theory localization of loops on an odd sphere cannot be directly given by knowing the K-theory localization of an odd sphere. In recent work Bousfield avoids this difficulty by defining a new vi localization functor which behaves well with respect to fibrations and then shows that it is closely related to the unstable K-theory localization functor. In this paper a more direct approach will be taken to construct the K-theory localiza- tion of loops on an odd sphere and double loops on an odd sphere. The construction and proof is a generalization of the results of Mahowald and Thompson in [IS]. As an application, these results will be used to construct the K-theory localization of the fiber of the Hopf map. Construction and Results: We will begin by summarizing the construction of the K-theory localization of loops on an odd sphere done in [S]. By [14] the localization with respect to real and complex K-theory give the same result. Throughout this paper K-theory will refer to complex K-theory. For simplicity, we will take the localization with respect to KCpJ at each prime separately, where K,,, denotes p local complex K-theory. These localiza- tions may then be pieced together to give the K-theory localization. Let F, denote the homotopy fiber of the Snaith map where QX = C2= C”X, q = 2p - 2, and B, denotes a stunted BXP, localized at p with bottom cell in dimension m. The inclusion map ii :S2”+’ -+ QS2”+’ lifts to a map ji:S”‘+i -+ F,. Mahowald and Thompson show in [S] that the map jr induces an isomor- phism in p local K-theory. This is the first step in showing that their construction is the K-theory localization of S2”+ ‘. Now let G, denote the homotopy fiber of the localized Snaith map