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

TL;DR

This paper establishes a connection between bounded generation of $E_n$-algebras and homological stability, providing a local-to-global principle that extends stability results from local algebraic structures to global topological spaces.

Contribution

It introduces bounded generation for $E_n$-algebras, proves their equivalence to homological stability for $n \u2265 2$, and develops a local-to-global principle for homological stability in topological chiral homology.

Findings

Bounded generation is equivalent to homological stability for $E_n$-algebras when $n \u2265 2$.

Homological stability of $A$ implies stability of $ ext{int}_M A$ for any connected non-compact manifold $M$.

The results apply to both non-compact and compact manifolds through a reformulation using scanning.

Abstract

We define bounded generation for $E_n$-algebras in chain complexes and prove that for $n \geq 2$ 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