Homological Stability of Automorphism Groups of Quadratic Modules and Manifolds
Nina FriedrichCentre for Mathematical Sciences, Wilberforce Road, Cambridge CB3 0WB, UK
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 , we get an extension of their result to the case of virtually polycyclic fundamental groups.
Cite this article
Nina Friedrich, Homological Stability of Automorphism Groups of Quadratic Modules and Manifolds. Doc. Math. 22 (2017), pp. 1729–1774DOI 10.4171/DM/609