Section 5.1, which treats integral homotopy theory for relative homotopy automorphisms, has developed greatly since the first preprint version of this paper, thanks to many other people. Our decision to consider the integral homotopy groups of the homotopy automorphisms of iterated connected sums is inspired by an answer by Ryan Budney to a question by Saleh at MathOverflow. The method used to prove Theorem B(a) was suggested by Manuel Krannich, who has also provided several other very helpful comments. In addition, he was in the committee for the PhD defense of Lindell, where he, together with Fabian Hebestreit, pointed out several minor errors in the paper and had some very helpful suggestions for improvements.Saleh was supported by the Knut and Alice Wallenberg Foundation through grant 2019.0521.2 Representation stability, FI-modules and Lie models of FI-spaces
ConventionsThroughout the paper, we will use R to denote a commutative ring, which we will assume to be Noetherian for convenience. We will mainly work over the field Q, so unless otherwise specifically stated, all vector spaces are over Q. We will use "dg" to abbreviate the term differential graded. FI denotes the category of finite sets with injective maps as morphisms.If S is a finite set, we will use jS j to denote its cardinality, and we will write †.S / WD Aut FI .S / for the symmetric group on S . If S D n WD f1; 2; : : : ; ng, we will simply write †.S / D † n for brevity.Recall that the irreducible Q-representations of † n are indexed by partitions of weight n, ie sequences of nonnegative integers D .