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

We define bounded generation for En-algebras in chain complexes and prove that this property is equivalent to homological stability for n≥2. Using this we prove a local-to-global principle for homological stability, which says that if an En-algebra A has homological stability (or equivalently the topological chiral homology ∫RnA has homology stability), then so has the topological chiral homology ∫MA 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.