Homological stability for moduli spaces of disconnected submanifolds, II - Symmetric diffeomorphism groups and parametrised connected sum

Martin Palmer

Published 2018-07-19Version 1

In Part I (arXiv:1805.03917) we showed that the moduli spaces of n mutually isotopic, embedded copies of a closed manifold P in a connected, open manifold M parametrised modulo an open subgroup G of the diffeomorphism group of P are homologically stable as n goes to infinity, as long as a certain condition on the relative dimensions of M and P is satisfied. In Part II (this paper) we use this to deduce homological stability results for symmetric diffeomorphism groups of manifolds, with respect to the operation of parametrised connected sum (for example, surgery). As a special case of this, we see that the diffeomorphism groups of manifolds with certain conical singularities are homologically stable with respect to the number of singularities of a fixed type. To do this, we first extend the main result of Part I to moduli spaces of disconnected, labelled submanifolds - in which the components of the submanifolds are equipped with labels in a fibration over an embedding space. This in turn is a corollary of twisted homological stability for moduli spaces of disconnected, unlabelled submanifolds with finite-degree coefficients, which we prove using the main (untwisted) result of Part I as an input.