Representation stability for homotopy automorphisms (Q6614584)

The authors give two examples of representation stability, in the sense of Church and Farb, for representations of the symmetric group on the rational homotopy groups of certain spaces of homotopy automorphisms. Let \(S\) be a finite set, \(X\) a simply connected finite complex, and \(X_S = \bigvee_S X\) the wedge sum. Let \(\mathrm{aut}_*(X_S)\) denote the space of pointed homotopy automorphisms of \(X_S\). The first main result is that the functor \(S \mapsto \pi^\mathbb{Q}_k( \mathrm{aut}_*(X_S))\) from the category of finite sets and injections to vector spaces is a finitely-generated \(FI\)-module as in [\textit{T. Church} et al., Duke Math. J. 164, No. 9, 1833--1910 (2015; Zbl 1339.55004)]. Further, if \(d\) is the coconnectivity of \(H_*(X; \mathbb{Q}),\) then this \(FI\)-module has weight \(\leq k + d - 1\) and stability degree \(\leq k +d.\)\N\NNext, let \(M^d\) be a closed, oriented manifold. Given a finite set \(S\), let \(M_S\) denote the \(S\)-fold connected sum of \(M\) with itself, with an open \(d\)-disk removed. A deformation retract \(M_S \simeq \bigvee_S M_1\) implies an \(FI\)-module on the homotopy groups of \(\mathrm{aut}_*(M_S)\). Let \(\mathrm{aut}_{\partial}(M_S)\) denote the space of homotopy automorphism of \(M_S\) that fix the boundary. The second main result gives a lifting of \(FI\)-modules \(S \mapsto \pi_k( \mathrm{aut}_*(X_S))\) to \(S \mapsto \pi_k( \mathrm{aut}_\partial(X_S))\). After rationalization, the weight and stability-degrees are bounded by the formulas as above. The proof here depends on work in [\textit{A. Berglund} and \textit{I. Madsen}, Acta Math. 224, No. 1, 67--185 (2020; Zbl 1441.57033)].\N\NThe paper contains helpful exposition on \(FI\)-modules and the rational homotopy theory of automorphisms of manifolds. Additional results of independent interest include the development of a theory of \(FI\)-Lie models and a proof that the groups of path-components of the monoids \( \mathrm{aut}_\partial(M_S), \mathrm{Homeo}_\partial(M_S),\) and \( \mathrm{Diffeo}_{\partial}(M_S)\) each contain a copy of the permutation group \(\Sigma S.\)