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Homological stability for moduli spaces of disconnected submanifolds, I

Martin Palmer

Algebraic & Geometric Topology 21 (2021) 1371–1444
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, 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.

Keywords
moduli spaces of submanifolds, homological stability, embedding spaces, configuration spaces
Mathematical Subject Classification 2010
Primary: 55R80, 57S05
Secondary: 57N20, 58B05
References
Publication
Received: 27 March 2019
Revised: 16 January 2020
Accepted: 24 April 2020
Published: 11 August 2021
Authors
Martin Palmer
Institutul de Matematică Simion Stoilow al Academiei Române
Bucharest
Romania
https://mdp.ac