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The James Construction and \(\pi _4(\mathbb {S}^{3})\) in Homotopy Type Theory

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Abstract

In the first part of this paper we present a formalization in Agda of the James construction in homotopy type theory. We include several fragments of code to show what the Agda code looks like, and we explain several techniques that we used in the formalization. In the second part, we use the James construction to give a constructive proof that \(\pi _4(\mathbb {S}^{3})\) is of the form \(\mathbb {Z}/n\mathbb {Z}\) (but we do not compute the n here).

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Notes

  1. The code is available at https://github.com/guillaumebrunerie/JamesConstruction and has been tested with Agda 2.5.2. The code fragments are generated directly from the source code using agda --latex and a custom script to extract the relevant parts.

References

  1. Brunerie, G.: On the homotopy groups of spheres in homotopy type theory. PhD thesis (2016). arxiv:1606.05916

  2. Cockx, J., Abel, A.: Sprinkles of extensionality for your vanilla type theory. TYPES (2016). http://www.types2016.uns.ac.rs/images/abstracts/cockx.pdf

  3. Hou, K.-B., Finster, E., Licata, D.R., Lumsdaine, P.L.: A mechanization of the Blakers-Massey connectivity theorem in homotopy type theory. LICS (2016). https://doi.org/10.1145/2933575.2934545

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  4. Licata, D.R., Brunerie, G.: A cubical approach to synthetic homotopy theory. LICS (2015). https://doi.org/10.1109/LICS.2015.19

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  5. The Univalent Foundations Program: Homotopy Type Theory: Univalent Foundations of Mathematics. Institute for Advanced Study, Princeton, NJ (2013). http://homotopytypetheory.org/book

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Authors and Affiliations

  1. Institute for Advanced Study, Princeton, NJ, USA

    Guillaume Brunerie

Authors
  1. Guillaume Brunerie
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Correspondence to Guillaume Brunerie.

Additional information

This material is based upon work supported by the National Science Foundation under Agreement Nos. DMS-1128155 and CMU 1150129-338510.

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Brunerie, G. The James Construction and \(\pi _4(\mathbb {S}^{3})\) in Homotopy Type Theory. J Autom Reasoning 63, 255–284 (2019). https://doi.org/10.1007/s10817-018-9468-2

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  • DOI: https://doi.org/10.1007/s10817-018-9468-2

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