Isovariant homotopy theory and fixed point invariants

An isovariant map is an equivariant map between $G$-spaces which strictly preserves isotropy groups. We consider an isovariant analogue of Klein-Williams equivariant intersection theory for a finite group $G$. We prove that under certain reasonable dimension and codimension conditions on $H$-fixed subspaces (for $H\leq G$), the fixed points of a self-map of a compact $G$-manifold can be removed isovariantly if and only if the equivariant Reidemeister trace of the map vanishes. In doing so, we build a new Quillen model structure on the category of isovariant spaces such that $G$-manifolds are both fibrant and cofibrant, yielding an isovariant Whitehead's theorem. In addition, we speculate on the role of isovariant homotopy theory in distinguishing manifolds up to homeomorphism.Comment: 22 pages, comments welcome, v2 fixed typo in first paragrap