= Homotopy groups as modules over the fundamental group
{title2=$\pi_n(X)\text{ over }\mathbb Z[\pi_1(X)],\quad n\geq2$}
Moving the basepoint around a loop acts on a higher <homotopy group>. Since $\pi_n(X)$ is an <abelian group> for $n\geq2$, this action extends linearly to its integral <group ring>, making $\pi_n(X)$ a module. For a connected <CW complex>, its <universal cover> gives an equivalent description by <deck transformations>, together with change of basepoint upstairs. Left and right conventions are converted by inversion of the loop. The <Hurewicz theorem> respects this action whenever it identifies a higher <homotopy group> with the corresponding homology of the cover.
Back to article page
