Equivalences of the form $\Sigma^V X \simeq \Sigma^W X$ in equivariant stable homotopy theory
Abstract
For a finite group $G$ and virtual $G$representation $\alpha = VW$ of virtual dimension $0$, there is an invertible Thom class $t_\alpha \in \pi_{\alpha}MO_G$ in the $RO(G)$graded coefficients of $G$equivariant cobordism. We introduce and study $t_\alpha$self maps: equivalences $\Sigma^{nV}X\simeq\Sigma^{nW}X$ inducing multiplication by $t_\alpha^n$ in $MO_G$theory. We also treat the variants based on $MU_G$ and $MSp_G$, as well as equivalences not necessarily compatible with cobordism. When $X = C(a_\lambda^m)$ arises as the cofiber of an Euler class, these periodicities may be produced by an $RO(G)$graded $J$homomorphism $\pi_{m\lambda}KO_G\rightarrow (\pi_\star C(a_\lambda^m))^\times$, and we use this to give several examples.
 Publication:

arXiv eprints
 Pub Date:
 June 2023
 DOI:
 10.48550/arXiv.2306.11000
 arXiv:
 arXiv:2306.11000
 Bibcode:
 2023arXiv230611000B
 Keywords:

 Mathematics  Algebraic Topology
 EPrint:
 30 pages