Representation stability for homotopy automorphisms

Abstract

We consider in parallel pointed homotopy automorphisms of iterated wedge sums of topological spaces and boundary relative homotopy automorphisms of iterated connected sums of manifolds minus a disk. Under certain conditions on the spaces and manifolds, we prove that the rational homotopy groups of these homotopy automorphisms form finitely generated FI-modules, and thus satisfy representation stability for symmetric groups, in the sense of Church and Farb.

arXiv commentVersion 2. Major revision. Explicit stability ranges added to Theorem A and Theorem B. Proof added that the FI-Q-module considered in Theorem B is the rationalization of an FI-Z-module, constructed in a geometric way. Accepted for publication in "Algebraic and Geometric Topology"

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