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, 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.
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