On odd-primary components of Lie groups

Abstract:

The transfer map $t:{\pi ^s}({P_\infty }{\mathbf {C}}) \to {\pi ^s}({S^0})$ is represented by an element $\tau \in \pi _s^{ - 1}({P_\infty }{{\mathbf {C}}^ + })$. We compute the Adams-e-invariant of $\tau$ and use this and the splitting of the p-localization of ${S^1} \wedge {P_\infty }{\mathbf {C}}$ into a wedge of $(p - 1)$ spaces to prove that for a prime $p \geqslant 5$ the p-component of the element $[G,\mathcal {L}]$ defined by a compact Lie group G in $\pi _ \ast ^s$ is zero in the known part of stable homotopy.