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 well-behaved in any sense: for example, the quadratic form may be singular. This extends results by van der Kallen and Mirzaii--van der Kallen respectively. Combining these results with the machinery introduced by Galatius--Randal-Williams to prove homological stability for moduli spaces of simply-connected 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 Galatius--Randal-Williams 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 Atiyah--Hirzebruch 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.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:744925 |
| Date | January 2018 |
| Creators | Friedrich, Nina |
| Contributors | Randal-Williams, Oscar |
| Publisher | University of Cambridge |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | https://www.repository.cam.ac.uk/handle/1810/276986 |
Page generated in 0.0044 seconds