Equivalences of the form $Σ^V X \simeq Σ^W X$ in equivariant stable homotopy theory

Abstract: We study equivalences of the form $\Sigma^{V}X\simeq \Sigma<sup>{W}X$, 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 $$J\colon \pi_\lambda KO_G\rightarrow \pi_\star^G C(a_\lambda)<sup>\times$$ 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.