In this paper, the authors introduce and consider higher order representation stability for the homology of ordered configuration spaces of manifolds, inspired by the notion of secondary homological stability due to [S. Galatius et al., Publ. Math., Inst. Hautes Étud. Sci. 130, 1–61 (2019; Zbl 1461.55011)].
For a smooth, connected, non-compact manifold \(M\) of dimension \(n \geq 2\), adding a new marked point makes the \(i\)th homology \(H_i (FM ) : k \mapsto H_i (F_k M)\), where \(F_k M\) is the configuration space of \(k\) ordered points in \(M\), into a free \(\mathrm{FI}\)-module (an \(\mathrm{FI}\)-module is a functor from the category of finite sets and injections to abelian groups; \(k\) is the arity).
The new ingredient here is the secondary stabilization morphism \[ H_i (F_k M) \rightarrow H_{i+1} (F_{k+2} M) \] induced by a class of \(H_1 (F_2 (\mathbb{R}^n))\); this is zero if \(n >2\). These morphisms assemble to give \(H_* (FM)\) the structure of a graded \(\mathrm{FIM}^+\)-module; the category of \(\mathrm{FIM}^+\)-modules is analogous to that of \(\mathrm{FI}\)-modules, with the inclusion morphisms generated by adding pairs of points and with a skew-symmetry property for compositions.
The authors first give a new proof – and generalization to the non-orientable case – of the representation stability result of [T. Church et al., Duke Math. J. 164, No. 9, 1833–1910 (2015; Zbl 1339.55004)] that the \(\mathrm{FI}\)-module indecomposables \(H_0^{\mathrm{FI}} (H_i (FM))\) vanish in arity \(k\) for \(k>2i\); for \(\dim M >2\) they later strengthen this to \(k>i\).
The proof of this and subsequent results uses the arc resolution spectral sequence \(E^r_{*,*}(S)\) associated to the augmented semi-simplicial space \(\mathrm{Arc}_\bullet (F_S M)\), for a finite set \(S\). The key fact is that \(E^\infty_{p,q}(S)=0\) if \(p+q \le |S|-2\), which is deduced from a result of [A. Kupers and J. Miller, Math. Ann. 370, No. 1–2, 209–269 (2018; Zbl 1384.55009)].
The \(E_1\)-page identifies as \(E^1_{p,q} = \mathrm{Inj}_p (H_q (F M) )\) (\(q \geq 0\), \(p \geq -1\)), where \(\mathrm{Inj}_\bullet \mathcal{V}\) is the twisted injective word complex of an \(\mathrm{FI}\)-module \(\mathcal{V}\). The fact that the \(\mathrm{FI}\)-modules \(H_* (FM)\) are free implies that the \(E^2\)-page can be expressed in terms of \(H_0^{\mathrm{FI}} (H_* (FM))\) and the homology of the injective word complex \(\mathrm{Inj}_\bullet\). In each arity, the reduced homology of \(\mathrm{Inj}_\bullet\) is concentrated in a single degree and the authors give a new characterization of this representation; this is of independent interest.
These ingredients lead to the proof of the representation stability result. Higher representation stability arises through the differentials \(d_r\), for \(r\geq 2\); in particular, \(d_2\) is closely related to the secondary stabilization morphisms. More precisely, the graded \(\mathrm{FIM}^+\)-module structure on \(H_* (FM)\) leads to a decomposition as \(\mathrm{FIM}^+\)-modules \[ H_0^{\mathrm{FI}} (H_*(FM)) \cong \bigoplus_{i\geq 0} \mathcal{W}_i ^M. \] The authors’ secondary representation stability result states that, for coefficients in a field of characteristic zero, each \(\mathcal{W}_i^M\) is finitely-generated as an \(\mathrm{FIM}^+\)-module.