| Popis: |
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. |
