In the early 1980s, Johnson defined a homomorphism, where is the Torelli group of a closed, connected, and oriented surface of genus g with a boundary component and is the corresponding surface without a boundary component. This is known as the Johnson homomorphism. We study the map induced by the Johnson homomorphism on rational homology groups and apply it to abelian cycles determined by disjoint bounding-pair maps, in order to compute a large quotient of in the stable range. This also implies an analogous result for the stable rational homology of the Torelli group of a surface with a marked point instead of a boundary component. Further, we investigate how much of the image of this map is generated by images of such cycles and use this to prove that in the pointed case, they generate a proper subrepresentation of for and g large enough.

We describe new graphical models of the framed little disks operad which exhibit large symmetry dg Lie algebras.

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.

The Torelli group of an orientable smooth surface is the group of isotopy classes of orientation preserving diffeomorphisms that act trivially on the first homology group of the surface. In this paper, we study the rational homology groups of Torelli groups of smooth, compact and orientable surfaces. More specifically, 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 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.

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.

We compute the cohomology groups of the automorphism group of the free group F_n, with coefficients in arbitrary tensor products of the standard representation H₁(F_n,Q) and its dual, in a range where n is sufficiently large compared to the cohomological degree and the number of tensor factors.