Semifree Isovariant Poincaré Spaces and the Gap Condition

Abstract: We introduce the notion of a semifree isovariant $G$-Poincar\'e space, a homotopical notion interpolating between semifree closed smooth $G$-manifolds and the equivariant Poincar\'e spaces of [HKK24b]. It carries the additional structure of an equivariant Poincar\'e embedding of the fixed points of a semifree $G$-Poincar\'e space. Under suitable gap conditions on the codimension, we show that the space of isovariant structures on a semifree $G$-Poincar\'e space for a periodic finite group $G$ is highly connected, giving a useful construction tool for manifold structures on equivariant Poincar\'e spaces.