Erik Lindell: Stable invariants of some topological moduli spaces
Time: Fri 2023-05-26 13.00
Location: Albano campus, lärosal 4, house 1
Doctoral student: Erik Lindell , Department of Mathematics, Stockholm University
Opponent: Andrew Putman (University of Notre Dame)
Supervisor: Dan Petersen
This thesis consists of three papers, treating stability phenomena in var ious 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 Fn, 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 Fn 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 Fn, defined similarly as the Torelli group of a surface.