Homological stability for moduli spaces of disconnected submanifolds, I

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.

arXiv comment47 pages; incorporated changes suggested by the referee. Final version, to appear in Algebraic & Geometric Topology

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Cited by in corpus (5)

• Topological representations of motion groups and mapping class groups -- a unified functorial construction
• The Burau representations of loop braid groups
• Configuration-mapping spaces and homology stability
• Constructing the Virasoro groups using differential cohomology
• Vertical configuration spaces and their homology