The \(K\)-theory localization of an unstable sphere
Let \(K_ *\) denote (periodic) complex \(K\)-theory. A space, \(X\), is \(K\)- local if, whenever \(f: Y\to Z\) induces on isomorphism in \(K\)-theory, \(f^*: [Z,X]\to [Y,X]\) is bijective. The \(K_ *\)- localization of \(X\) is a \(K\)-local space, \(X_ K\), together with a map, \(X\to X_ K\), inducing an isomorphism in \(K\)-theory. The authors identify \(S^{2n+1}_ K\), the \(K\)-localization of an odd sphere. In addition, if \(v_ 1\) is the Adams map between \(\mathbb{Z}/p^ r\) Moore spaces (\(p\) prime) one may localize the \(\bmod p^ r\) homotopy by inverting the self-map induced by \(v_ 1\). It is shown that the \(K\)- theory localization induces an automorphism on \(v_ 1\)-localized the homotopy \[ v^{-1}_ 1\pi_ *(S^{2n+1};\mathbb{Z}/p^ r)@>\equiv>> v^{-1}_ 1\pi_ *(S^{2n+1}_ K;\mathbb{Z}/p^ r). \] Finally, the results are strengthened to determine the integral homotopy groups of \(S^{2n+1}_ K\). The significance of \(\pi_ *(X_ K)\) is that it is often the target of the Adams spectral sequence for \(X\), based on \(K_ *\)-theory. The proofs use a number of \(K_ *\)-calculations of the reviewer -- and his stable decomposition of iterated loop spaces. Thank you, gentlemen!
scientific article; zbMATH DE number 54634