An isovariant Elmendorf's theorem (Q2135730)

In equivariant homotopy theory, one studies spaces with group actions together with maps which are equivariant. Such maps induce inclusions on isotropy subgroups, but in general, not equalities. Isovariant maps are defined to be those equivariant maps which strictly preserve isotropy subgroups, and such maps appear in surgery theory and the study of classifications of manifolds. In this paper, the author considers the category of \(G\)-spaces with isovariant maps (for finite groups \(G\)) and equips this with a model structure by replacing the usual orbits used in equivariant homotopy theory, with linking simplices. With this model structure established, the author then proves an analogue of Elmendorf's theorem [\textit{A. D. Elmendorf}, Trans. Am. Math. Soc. 277, 275--284 (1983; Zbl 0521.57027) and \textit{R. J. Piacenza}, Can. J. Math. 43, No. 4, 814--824 (1991; Zbl 0758.55015)] in this setting, showing that the homotopy theory of isovariant spaces is equivalent to the homotopy theory of functors from the link orbit category to non-equivariant spaces.