$$E_n$$ E n -cell attachments and a local-to-global principle for homological stability

Abstract

We define bounded generation for \(E_n\)-algebras in chain complexes and prove that this property is equivalent to homological stability for \(n \ge 2\). Using this we prove a local-to-global principle for homological stability, which says that if an \(E_n\)-algebra A has homological stability (or equivalently the topological chiral homology \(\int _{\mathbb {R}^n} A\) has homology stability), then so has the topological chiral homology \(\int _M A\) of any connected non-compact manifold M. Using scanning, we reformulate the local-to-global homological stability principle so that it applies to compact manifolds. We also give several applications of our results.