Title:

Automorphism groups of quadratic modules and manifolds

In this thesis we prove homological stability for both general linear groups of modules over a ring with finite stable rank and unitary groups of quadratic modules over a ring with finite unitary stable rank. In particular, we do not assume the modules and quadratic modules to be wellbehaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaiivan der Kallen respectively. Combining these results with the machinery introduced by GalatiusRandalWilliams to prove homological stability for moduli spaces of simplyconnected manifolds of dimension $2n \geq 6$, we get an extension of their result to the case of virtually polycyclic fundamental groups. We also prove the corresponding result for manifolds equipped with tangential structures. A result on the stable homology groups of moduli spaces of manifolds by GalatiusRandalWilliams enables us to make new computations using our homological stability results. In particular, we compute the abelianisation of the mapping class groups of certain $6$dimensional manifolds. The first computation considers a manifold built from $\mathbb{R} P^6$ which involves a partial computation of the Adams spectral sequence of the spectrum ${MT}$Pin$^{}(6)$. For the second computation we consider Spin $6$manifolds with $\pi_1 \cong \mathbb{Z} / 2^k \mathbb{Z}$ and $\pi_2 = 0$, where the main new ingredient is an~analysis of the AtiyahHirzebruch spectral sequence for $MT\mathrm{Spin}(6) \wedge \Sigma^{\infty} B\mathbb{Z}/2^k\mathbb{Z}_+$. Finally, we consider the similar manifolds with more general fundamental groups $G$, where $K_1(\mathbb{Q}[G^{\mathrm{ab}}])$ plays a role.
