Homological stability of topological moduli spaces

Abstract

A. Given a graded E 1 -module over an E 2 -algebra in spaces, we construct an augmented semisimplicial 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 coe cients. We furthermore introduce a notion of coe cient systems of nite degree in this context and show that, without further assumptions, the corresponding twisted homology groups stabilise as well.is 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 e.g. moduli spaces of high-dimensional manifolds, unordered con guration spaces of manifolds with labels in a bration, and moduli spaces of manifolds equipped with unordered embedded discs. is in turn implies representation stability for the ordered variants of the la er examples.is said to satisfy homological stability if the induced maps in homology are isomorphisms in degrees that are small relative to n. ere is a well-established strategy for proving homological stability that traces back to an argument by illen for the classifying spaces of a sequence of inclusions of groups G n . Given simplicial complexes whose connectivity increases with n and on which the groups G n act simplicially, transitively on simplices, and with stabilisers isomorphic to groups G n−k prior in the sequence, stability can o en be derived by employing a spectral sequence relating the di erent stabilisers. In [RW ], Randal-Williams and Wahl axiomatised this strategy of proof, resulting in a convenient categorical framework for proving homological stability for families of discrete groups that form a braided monoidal groupoid. eir work uni es and improves many classical stability results and has led to a number of applications since its introduction [Fri ; GW ; PW ; Ran ; SW ]. However, homological stability phenomena have been proved to occur not only in the context of discrete groups, but also in numerous non-aspherical situations, many of them of a moduli space avour, such as unordered con guration spaces of manifolds [McD ; Seg ; Seg ], the most classical example, or moduli spaces of high-dimensional manifolds [GR ; GR ] to emphasise a more recent one. e majority of the stability proofs in this context resemble the original line of argument for discrete groups, and one of the objectives of the present work is to provide a conceptualisation of this pa ern.Instead of considering the single spaces M n and the maps M n → M n+1 between them one at a time, it is bene cial to treat them as a single space M = n ≥0 M n together with a grading M : M → N 0 to the nonnegative integers, capturing the decomposition of M into the pieces M n , and a stabilisation map s : M → M that restricts to the maps M n → M n+1 , so it increases the degree by one. From the perspective of homotopy...

Twisted homological stability for configuration spaces

“…This alternative proof is also sketched in Appendix A of [KM14]. Moreover, homological stability for C n (M, π), with twisted as well as constant coefficients, is implied by the recent work [Kra17] of Krannich (cf. Remark 1.5).…”

“…This suggests an underlying connection between representation stability and twisted homological stability. We note that representation stability for H * (F n (M, X); Q) may also be deduced from twisted homological stability for C n (M, X) with respect to a different twisted coefficient system than the one considered in Corollary C: see Corollary 5.17 of [Kra17] for the details. The twisted coefficient system used in that case does not fit into the setting of Theorem A.…”

“…Recently, Krannich [Kra17] has extended the techniques of [RW17] to a topological setting, where one begins with an N-graded E 1 -module over an E 2 -algebra. Considering C(R 2 ) = n C n (R 2 ) as a module over itself, he recovers Theorem D of [RW17] for the braid groups.…”

“…Another family of coefficient systems (different from those appearing in Theorem A) with respect to which twisted homological stability is known, is abelian coefficient systems. See §5.6.2 of [RW17] for the case of the braid groups and Theorem D(i) of [Kra17] for configuration spaces in general. A special case of the latter theorem (homological stability for configuration spaces with the abelian twisted coefficients Z[Z/2]) was proved earlier by the author in [Pal13].…”

Homological stability for Artin monoids

“…Proving the K(π, 1) conjecture for families of Artin groups continues to be an active area of research to this day. the categorical set up of [21], however, the classifying spaces of the sequence can be shown to assemble into an E 1 -module over an E 2 algebra, as in [14]. Combining the results of [14] with the high connectivity results established in this paper, our homological stability result with constant coefficients (Theorem A) can most likely be enhanced to one with abelian coefficients and coefficient systems of finite degree in the sense of [14,Section 4].…”