Homological stability for moduli spaces of disconnected submanifolds, I

TL;DR

This paper extends the concept of homological stability from configuration spaces of points to moduli spaces of higher-dimensional submanifolds, including parametrised and labelled cases, revealing new stability phenomena in geometric topology.

Contribution

It generalises homological stability results to moduli spaces of disconnected submanifolds, including parametrised and labelled variants, broadening understanding of their topological properties.

Findings

Homology stabilises as the number of submanifold components increases.

Stability results hold for both unparametrised and partially-parametrised submanifolds.

Corollaries include stability of diffeomorphism groups under connected sum and singularity addition.

Abstract

A well-known property of unordered configuration spaces of points (in an open, connected manifold) is that their homology stabilises as the number of points increases. We generalise this result to moduli spaces of submanifolds of higher dimension, where stability is with respect to the number of components having a fixed diffeomorphism type and isotopy class. As well as for unparametrised submanifolds, we prove this also for partially-parametrised submanifolds -- where a partial parametrisation may be thought of as a superposition of parametrisations related by a fixed subgroup of the mapping class group. In a companion paper (arXiv:1807.07558), this is further generalised to submanifolds equipped with labels in a bundle over the embedding space, from which we deduce corollaries for the stability of diffeomorphism groups of manifolds with respect to parametrised connected sum and addition of singularities.