$E_n$-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 for $n\ge 2$ this property is equivalent to homological stability. 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 ${\int }_{R^n}A$ 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 in a way that also applies to compact manifolds. We also give several applications of our results

57 pages, no figures. Major revision

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