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Real Hochschild homology as an equivariant Loday construction

Ayelet Lindenstrauss, Birgit Richter, Foling Zou

Abstract

Equivariant Loday constructions are a means for providing geometric interpretations of equivariant homology theories. They are usually constructed for a simplicial $G$-set and a $G$-Tambara functor. We study situations where -- depending on the isotropy subgroups occurring in the simplicial $G$-set -- one can work with $H$-Tambara functors for a suitable subgroup $H$ of $G$. We apply this to give an interpretation of Real Hochschild homology of discrete $E_σ$-rings as equivariant Loday constructions where we consider $2m$-gons with a geometrically defined action of the dihedral groups $D_{2m}$ for all $m \geq 1$. The action of symmetric groups on $1$-skeleta of permutohedra also gives examples with isotropy groups $C_2$.

Real Hochschild homology as an equivariant Loday construction

Abstract

Equivariant Loday constructions are a means for providing geometric interpretations of equivariant homology theories. They are usually constructed for a simplicial -set and a -Tambara functor. We study situations where -- depending on the isotropy subgroups occurring in the simplicial -set -- one can work with -Tambara functors for a suitable subgroup of . We apply this to give an interpretation of Real Hochschild homology of discrete -rings as equivariant Loday constructions where we consider -gons with a geometrically defined action of the dihedral groups for all . The action of symmetric groups on -skeleta of permutohedra also gives examples with isotropy groups .
Paper Structure (10 sections, 17 theorems, 98 equations)

This paper contains 10 sections, 17 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Change of groups for Tambara functors
  3. Group actions on norms
  4. Loday constructions for Tambara functors with respect to isotropy subgroups
  5. Trivial isotropy
  6. One non-trivial isotropy group (and its conjugates)
  7. Finite $\Sigma_n$-simplicial sets with small isotropy for all $n \geqslant 2$
  8. A normal isotropy subgroup and its subgroups
  9. Two non-trivial isotropy groups (and their conjugates)
  10. Real Hochschild homology

Key Result

Lemma 2.1

Assume that $H$ and $H'$ are finite groups and that $\varphi \colon H \rightarrow H'$ is an isomorphism. Let ${\underline{R}}$ be a $H'$-Tambara functor. Then the assignment gives rise to an $H$-Tambara functor $\varphi^*{\underline{R}}$.

Theorems & Definitions (45)

  • Lemma 2.1
  • proof
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 3.3
  • proof
  • ...and 35 more