<p>Secondary homological stability is a recently discovered stability pattern for the homology of a sequence of spaces exhibiting homological stability in a range where homological stability does not hold. We prove secondary homological stability for the homology of the unordered configuration spaces of a connected manifold. The main difficulty is the case that the manifold is compact because there are no obvious maps inducing stability and the homology eventually is periodic instead of stable. We resolve this issue by constructing a new chain-level stabilization map for configuration spaces.</p>

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.

In this thesis we study the homological behaviour of configuration spaces as the number of objects in the configuration goes to infinity. For unordered configurations of distinct points (possibly equipped with some internal parameters) in a connected, open manifold it is a well-known result, going back to G. Segal and D. McDuff in the 1970s, that these spaces enjoy the property of homological stability. In Chapter 2 we prove that this property also holds for so-called oriented configuration spaces, in which the points of a configuration are equipped with an ordering up to even permutations. There are two important differences from the unordered setting: the rate (or slope) of stabilisation is strictly slower, and the stabilisation maps are not in general split-injective on homology. This can be seen by some explicit calculations of Guest-Kozlowski-Yamaguchi in the case of surfaces. In Chapter 3 we refine their calculations to show that, for an odd prime p, the difference between the mod-p homology of the oriented and the unordered configuration spaces on a surface is zero in a stable range whose slope converges to 1 as p goes to infinity. In Chapter 4 we prove that unordered configuration spaces satisfy homological stability with respect to finite-degree twisted coefficient systems, generalising the corresponding result of S. Betley for the symmetric groups. We deduce this from a general “twisted stability from untwisted stability” principle, which also applies to the configuration spaces studied in the next chapter. In Chapter 5 we study configuration spaces of submanifolds of a background manifold M. Roughly, these are spaces of pairwise unlinked, mutually isotopic copies of a fixed closed, connected manifold P in M. We prove that if the dimension of P is at most (dim(M)−3)/2 then these configuration spaces satisfy homological stability w.r.t. the number of copies of P in the configuration. If P is a sphere this upper bound on its dimension can be increased to dim(M)−3.