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Mathematische Annalen

, Volume 370, Issue 1–2, pp 209–269 | Cite as

\(E_n\)-cell attachments and a local-to-global principle for homological stability

  • Alexander Kupers
  • Jeremy MillerEmail author
Article

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.

Mathematics Subject Classification

55P48 55R80 55R40 57N65 

Notes

Acknowledgements

The authors would like to thank Ricardo Andrade, Kerstin Baer, Ralph Cohen, Søren Galatius, Martin Palmer and the anonymous referee for helpful conversations and comments.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Institut for Matematiske FagKøbenhavns UniversitetKøbenhavn ØDenmark
  2. 2.Department of MathematicsPurdue UniversityWest LafayetteUSA

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