This licentiate thesis consists of two papers about topics related to representation stability for different automorphisms groups of topological spaces and manifolds.

In Paper I, we study the rational homology groups of \textit{Torelli groups} of smooth, compact and orientable surfaces. The Torelli group of a smooth surface is the group of isotopy classes of orientation preserving diffeomorphisms that act trivially on the first homology group of the surface. In the paper, we study a certain class of stable homology classes, i.e. classes that exist for sufficiently large genus, and explicitly describe the image of these classes under a higher degree version of the \textit{Johnson homomorphism}, as a representation of the symplectic group. This gives a lower bound on the dimension of the stable homology of the group, as well as providing some further evidence that these homology groups satisfy representation stability for symplectic groups, in the sense of Church and Farb.

In Paper II, we study pointed homotopy automorphisms of iterated wedge sums of spaces as well as boundary relative homotopy automorphisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these, for simply connected CW-complexes and closed manifolds respectively,  satisfy representation stability for symmetric groups, in the sense of Church and Farb.

This thesis consists of three papers, treating stability phenomena in various automorphism groups in topology. In Papers I and III, we study the group (co)homology of certain mapping class groups of surfaces and graphs, or their respective Torelli subgroups, while the subject of Paper II is homotopy automorphisms of higher-dimensional spaces and manifolds.

The subject of Paper I is the rational homology of the Torelli group of a smooth, compact and orientable surface, which is the group of isotopy classes of self-homeomorphisms that act trivially on the first homology group of the surface. Using a map known as the Johnson homomorphism, we compute a large quotient of the rational homology of the Torelli group, in a range where the genus of the surface is sufficiently large in comparison to the homological degree.

In Paper II, we study in parallel pointed homotopy automorphisms of iterated wedge sums of topological spaces and boundary relative homotopy automorphisms of iterated connected sums of manifolds with a disk removed. We prove that the rational homotopy groups of these satisfy something called representation stability for representations of symmetric groups, under some assumptions on the spaces and manifolds, respectively.

In Paper III, we study the cohomology of the automorphism group of the free group F_n, which can also be viewed as the mapping class group of a graph of loop order n, with coefficients in tensor products of the first rational homology of F_n and its linear dual. In a range where n is sufficiently large compared to the cohomological degree, these cohomology groups are independent of n and the main result of Paper III provides a description of the stable cohomology groups, confirming a conjecture by Djament. These stable cohomology groups are also closely related to the stable cohomology of the Torelli subgroup of the automorphism group of F_n, defined similarly as the Torelli group of a surface.