Homological Stability of Automorphism Groups of Quadratic Modules and Manifolds

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 $2n ≥ 6$, we get an extension of their result to the case of virtually polycyclic fundamental groups.

2010 Mathematics Subject Classification: 19G05, 19B10, 19B14, 57S05

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