Homological stability for moduli spaces of disconnected submanifolds, I

Palmer, Martin

doi:10.2140/agt.2021.21.1371

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Homological stability for moduli spaces of disconnected submanifolds, I

Martin Palmer

Algebraic & Geometric Topology 21 (2021) 1371–1444

DOI: 10.2140/agt.2021.21.1371

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

Volume 21, issue 3 (2021)