Limits of polyhedra in Gromov–Hausdorff space

The main theorem of this paper is that compact metric spaces which are locally n-connected and which have cohomological dimension ⩽n for some n are precisely the spaces which are cell-like images of finite polyhedra. We show that this leads to a well-defined simple homotopy theory for such spaces. We also show that these spaces are precisely the compact metric spaces which are limits of polyhedra in Gromov’s topological moduli spaces M(n, ρ) for some choice of ρ and n. In addition, we prove that every precompact subset of M(n, ρ) contains only finitely many simple homotopy types. In the final section, we discuss the problem of determining which metric spaces are limits of closed manifolds in M(n, ρ) for some n and ρ.