Equivalences of the form ${\mathrm{\Sigma }}^{V}X\simeq {\mathrm{\Sigma }}^{W}X$ in equivariant stable homotopy theory

We study equivalences of the form ${\mathrm{\Sigma }}^{V}X\simeq {\mathrm{\Sigma }}^{W}X$, where $G$ is a compact Lie group, $X$ is a $G$-spectrum, and $V$ and $W$ are $G$-representations. These equivalences encode a periodicity phenomenon in $G$-equivariant homotopy theory which generalizes the classical James periodicity for $G=C_2$. When $X=C(a_{\lambda })$ is the cofiber of an Euler class, we construct an $RO(G)$-graded $J$-homomorphism $J\text{\hspace{0.17em}⁣}:{\pi }_{\lambda }K{O}_G\to {\pi }_{\star }^{G}C(a_{\lambda }{)}^{\times }$ which gives control over these periodicities. It also produces infinite periodic families in the $G$-equivariant stable stems. We illustrate this with several explicit examples. More generally, our work gives information about $RO(G)$-graded units in equivariant stable cohomotopy rings. We apply this to construct universal periodicities and differentials in the $G$-homotopy fixed point spectral sequence, and other equivariant Atiyah–Hirzebruch spectral sequences.