$p$-compact groups as framed manifolds


We describe a natural way to associate to any $p$-compact group an element of the $p$-local stable stems, which, applied to the $p$-completion of a compact Lie group $G$, coincides with the element represented by the manifold $G$ with its left-invariant framing. To this end, we construct a $d$-dimensional sphere $S_G$ with a stable $G$-action for every $d$-dimensional $p$-compact group $G$, which generalizes the one-point compactification of the Lie algebra of a Lie group. The homotopy class represented by $G$ is then constructed by means of a transfer map between the Thom spaces of spherical fibrations over $BG$ associated with $S_G$.

Topology 43/3 (2004), pp. 569–597

The original thesis version with more exposition is available here.