Operations which detect \(P^ 1\) in odd primary connective K-theory (Q1081888)

Let G denote the Adams summand of the connective unitary K-theory spectrum at the odd prime p. We study maps \(\phi\) : \(G\to G\) which have two properties (1) \(\phi _ *=0: \pi _ 0(G)\to \pi _ 0(G)\) and (2) \(\phi _ *(v)=p\epsilon v\) with the unit \(\epsilon \in Z^{\times}_{(p)}\), where \(\pi _ *(G)=Z_{(p)}[v]\) and \(| v| =2(p-1).\) An example of such operations is the Adams operation \(\psi ^{p+1}-1\) and it gives us an elementary proof of nonexistence of elements of mod p Hopf invariant one. Furthermore, these operations are useful in the analysis of the action of the mod p Steenrod algebra on certain spectra with few cells.