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Homological stability of topological moduli spaces

Manuel Krannich

Geometry & Topology 23 (2019) 2397–2474

Given a graded E1–module over an E2–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.

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
Received: 29 January 2018
Revised: 18 September 2018
Accepted: 26 December 2018
Published: 13 October 2019
Proposed: Ralph Cohen
Seconded: Benson Farb, Stefan Schwede
Manuel Krannich
Centre for Mathematical Sciences
University of Cambridge
United Kingdom