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Equivariant Khovanov homotopy types

Jakub Paliga

Abstract

We investigate group actions on homotopy coherent diagrams. This is used to prove an equivalence between realizations of equivariant cubical flow categories and external actions on Burnside functors. In particular, the results imply that the equivariant Khovanov homotopy types defined by Borodzik, Politarczyk, Silvero (arXiv:1807.08795) and Stoffregen, Zhang (arXiv:1810.04769) are equivariantly stably homotopy equivalent.

Equivariant Khovanov homotopy types

Abstract

We investigate group actions on homotopy coherent diagrams. This is used to prove an equivalence between realizations of equivariant cubical flow categories and external actions on Burnside functors. In particular, the results imply that the equivariant Khovanov homotopy types defined by Borodzik, Politarczyk, Silvero (arXiv:1807.08795) and Stoffregen, Zhang (arXiv:1810.04769) are equivariantly stably homotopy equivalent.
Paper Structure (31 sections, 32 theorems, 95 equations)

This paper contains 31 sections, 32 theorems, 95 equations.

Table of Contents

  1. Prerequisites
  2. The cube category
  3. $G$-cell complexes and the equivariant Spanier-Whitehead category
  4. Homotopy coherent diagrams
  5. Homotopy coherent diagrams
  6. Homotopy coherent diagrams as topological diagrams
  7. External actions on homotopy coherent diagrams
  8. Fixed point diagrams
  9. Flow categories
  10. $\langle n\rangle$-manifolds and flow categories
  11. Permutohedra and group actions
  12. The cube flow category
  13. Equivariant cubical neat embeddings
  14. Geometric realization of an equivariant cubical flow category
  15. Burnside functors
  16. ...and 16 more sections

Key Result

Theorem 1

There is an equivariant stable homotopy equivalence

Theorems & Definitions (76)

  • Theorem 1
  • Theorem 1
  • Theorem 1
  • Definition 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Proposition 2.5
  • Definition 2.6
  • ...and 66 more