Equivariant periodicity for abelian group action (Q2725175)

One of the most fundamental phenomena in the homotopy classification theory of topological manifolds is Siebenmann's periodicity theorem [\textit{R. C. Kirby} and \textit{L. C. Siebenmann}, Foundational essays on topological manifolds, smoothings and triangulations, Ann. Math. Stud. 88 (1977; Zbl 0361.57004)] which states that there is a 1-1 correspondence between the manifolds homotopy equivalent (relative to the boundary) to a manifold \(M\) and the same thing for \(M\times D^4\), i.e. the structure set of manifolds homotopy equivalent to \(M\) relative to the boundary is 4-periodic. This periodicity has been extended to topological manifolds with homotopically stratified group actions by odd order groups, with \(D^4\) replaced by the unit ball of any 4-fold permutation representation [\textit{M. Yan}, Commun. Pure Appl. Math. 46, No. 7, 1013-1040 (1993; Zbl 0833.57020)]. In this nice paper the authors extend such equivariant periodicity to the case where the group is compact abelian, and \(D^4\) is replaced by the unit ball of twice of any complex representation. More precisely, let \(S_G(M,\text{rel}\partial M)\) be the space of \(G\)-isovariant homotopy structures of \(M\) relative to the boundary \(\partial M\) (here \(G\) is a compact Lie group). Let \(k:G\to S^1\) be a character of \(G\), and \(V=\mathbb{C}^2\) the twice of the \(G\)-representation induced from \(k\). The authors prove that if \(M\) is a homotopically stratified \(G\)-manifold with codimension \(\geq 3\) gap and nontrivial \(G\)-action, then there is a periodicity equivalence \(S_G(M,\text{rel} \partial M)\simeq S_G(M\times DV,\text{rel} \partial (M\times DV))\), where \(DV\) is the unit disk of an orthogonal representation \(V\).