The \(K\)-theory localization of loops on an odd sphere and applications (Q1312077)

The author constructs the K-theory localization of \(\Omega S^{2n + 1}\) and \(\Omega^ 2 S^{2n + 1}\). That is, maps are constructed from these spaces into K-local spaces which induce isomorphisms in (complex periodic) K-theory. In [\textit{M. Mahowald} and \textit{R. Thompson}, ibid. 31, No. 1, 133-141 (1992; Zbl 0759.55010)] the K-theory localization of \(S^{2n + 1}\) was constructed using maps coming from the reviewer's stable decomposition of the iterated loop spaces, \(\Omega^ m \Sigma^ m S^{2n + 1}\). K-theory localization does not automatically respect taking loops so, while the author's result uses the same maps in its construction, it must be derived independently.