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

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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.

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Notes

  1. See also the appendix of the arXiv-version of this paper.

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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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Authors and Affiliations

  1. Institut for Matematiske Fag, Københavns Universitet, Universitetsparken 5, 2100, København Ø, Denmark

    Alexander Kupers

  2. Department of Mathematics, Purdue University, 150 N. University Street, West Lafayette, IN, 47907-2067, USA

    Jeremy Miller

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  1. Alexander Kupers
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  2. Jeremy Miller
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Correspondence to Jeremy Miller.

Additional information

Alexander Kupers is supported by a William R. Hewlett Stanford Graduate Fellowship, Department of Mathematics, Stanford University, and was partially supported by NSF Grant DMS-1105058.

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Kupers, A., Miller, J. \(E_n\)-cell attachments and a local-to-global principle for homological stability. Math. Ann. 370, 209–269 (2018). https://doi.org/10.1007/s00208-017-1533-3

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