Periodicity in the homology of moduli spaces of disconnected submanifolds

TL;DR

This paper proves cohomological periodicity and stability results for moduli spaces of multiple embedded manifolds, extending known configuration space results and providing new bounds and conditions.

Contribution

It generalizes cohomological periodicity to moduli spaces of disconnected submanifolds and improves stability results for open manifolds.

Findings

Cohomological periodicity over __\u211b for large n

Integral stability of cohomology for open manifolds

Stability and periodicity for symmetric diffeomorphism groups

Abstract

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 $\mathbb F_\ell$ 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.