Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Topological $ΔG$ homology of rings with twisted $G$-action

Gabriel Angelini-Knoll, Mona Merling, Maximilien Péroux

Abstract

We construct topological $ΔG$-homology for rings with twisted $G$-action. Here a ring with twisted $G$-action is a common generalization of a ring with anti-involution and a ring with $G$-action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its $S^1$-action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its $O(2)$-action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted $C_4$-action, which carries a $Pin(2)$-action. We prove that THQ of a loop space with twisted $C_4$-action can be $Pin(2)$-equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted $G$-action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted $G$-action.

Topological $ΔG$ homology of rings with twisted $G$-action

Abstract

We construct topological -homology for rings with twisted -action. Here a ring with twisted -action is a common generalization of a ring with anti-involution and a ring with -action. This construction recovers as special cases topological Hochschild homology (THH) of rings, with its -action, and Real topological Hochschild homology (THR) of rings with anti-involution, with its -action. A new example of this construction is quaternionic topological Hochschild homology (THQ) of rings with twisted -action, which carries a -action. We prove that THQ of a loop space with twisted -action can be -equivariantly identified with a twisted free loop space. Other new examples of interest are topological symmetric homology and topological hyperoctrahedral homology and more generally topological twisted symmetric homology. We prove a homotopical version of results of Fiedorowicz, Ault, and Graves computing these new topological homology theories on loop spaces with twisted -action. A key step of independent interest in this program is the construction of a new family of crossed simplicial groups, which correspond to operads that encode the structure of rings with twisted -action.
Paper Structure (26 sections, 42 theorems, 222 equations, 5 figures)

This paper contains 26 sections, 42 theorems, 222 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Outline
  3. Generalizations of Hochschild homology
  4. Quaternionic topological Hochschild homology
  5. Equivariant topological Hochschild homology
  6. Equivariant Real topological Hochschild homology
  7. Topological twisted symmetric homology and topological braid homology
  8. Acknowledgements
  9. Crossed simplicial groups
  10. Survey of crossed simplicial groups
  11. The twisted symmetric crossed simplicial group
  12. Twisted TEXT-actions
  13. Twisted TEXT-objects
  14. Rings with twisted TEXT-action as algebras over an operad
  15. Twisted TEXT-objects in TEXT-categories
  16. ...and 11 more sections

Key Result

Theorem A

For a self-dual crossed simplicial group $\Delta \mathbf{G}$, there is an associated topological homology $\mathop{\mathrm{\mathrm{TH}\mathbf{G}}}\nolimits(R)$ which carries a $|\mathbf{G}_{\,\begin{picture}(-1,1)(0.5,-1)\circle*{1.8}\end{picture}\space}|$-action, whose input is a ring spectrum $R$

Figures (5)

  • Figure 1: Table of common crossed simplicial groups
  • Figure 2: Example in $\Delta \mathbf{\Sigma}$ for some $\phi\colon[5]\to [3]$ and $\gamma\in \Sigma_4$.
  • Figure 3: Example in $\Delta \varphi\wr\mathbf{\Sigma}$ for some $\phi\colon[5]\to [3]$ and $\gamma\in \Sigma_4$.
  • Figure 4: The permutation $\sigma_4$ induced by $\gamma_1$ and $\sigma_2$ in $\Delta \varphi\wr \mathbf{\Sigma}$
  • Figure 5: Various crossed simplicial groups and their topological homology

Theorems & Definitions (153)

  • Theorem A
  • Theorem B: \ref{['thm: main computation']}
  • Theorem C
  • Theorem D: \ref{['thm: topological positive hyperoctahedral homology']}
  • Definition 2.1
  • Remark 2.2
  • Proposition 2.3: FL91
  • Example 2.4
  • Lemma 2.5
  • proof
  • ...and 143 more