Homological stability of topological moduli spaces

Krannich, Manuel

doi:10.2140/gt.2019.23.2397

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Homological stability of topological moduli spaces

Manuel Krannich

Geometry & Topology 23 (2019) 2397–2474

DOI: 10.2140/gt.2019.23.2397

Abstract

Given a graded $E_{1}$ –module over an $E_{2}$ –algebra in spaces, we construct an augmented semi-simplicial space up to higher coherent homotopy over it, called its canonical resolution, whose graded connectivity yields homological stability for the graded pieces of the module with respect to constant and abelian coefficients. We furthermore introduce a notion of coefficient systems of finite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilise as well. This generalises a framework of Randal-Williams and Wahl for families of discrete groups.

In many examples, the canonical resolution recovers geometric resolutions with known connectivity bounds. As a consequence, we derive new twisted homological stability results for various examples including moduli spaces of high-dimensional manifolds, unordered configuration spaces of manifolds with labels in a fibration, and moduli spaces of manifolds equipped with unordered embedded discs. This in turn implies representation stability for the ordered variants of the latter examples.

Keywords

homological stability, $E_n$–algebras, operads, configuration spaces, moduli spaces of manifolds, automorphism groups, representation stability

Mathematical Subject Classification 2010

Primary: 55P48, 55R40, 55R80, 57R19, 57R50

References

Publication

Received: 29 January 2018

Revised: 18 September 2018

Accepted: 26 December 2018

Published: 13 October 2019

Proposed: Ralph Cohen

Seconded: Benson Farb, Stefan Schwede

Authors

Manuel Krannich
Centre for Mathematical Sciences University of Cambridge Cambridge United Kingdom

Volume 23, issue 5 (2019)