Periodicity in the homology of moduli spaces of disconnected submanifolds

We show that the moduli space of $n$ suitably embedded copies of a closed smooth manifold $P$ inside a closed smooth manifold $M$ satisfies cohomological periodicity over ${\mathbb{F}}_{\mathrm{\ell }}$ when $n$ grows, with an explicit linear bound on the period and the periodicity range. This generalizes a known result about configuration spaces. We also show integral stability of the cohomology when $M$ is open, reproving a result of Palmer and improving the slope when inverting $2$. The main input in the proof is Goodwillie and Klein’s multiple disjunction lemma for embedding spaces. As a corollary we get stability and periodicity results for some classes of symmetric diffeomorphism groups of manifolds.