Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories

Daniel Graves

doi:10.5802/crmath.698

Article de recherche - Géométrie et Topologie

Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories
[Théorèmes de type Pirashvili–Richter pour les théories d’homologie réflexive et diédrale]

Daniel Graves ¹

¹ Lifelong Learning Centre, University of Leeds, Woodhouse, Leeds, LS2 9JT, UK

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 43-55.

Résumé
Abstract

L’homologie réflexive et l’homologie diédrale sont les théories d’homologie associées respectivement aux groupes simpliciaux croisés réflexifs et diédrale. Il a été démontré récemment que la première contient des informations intéressantes à propos de l’homotopie $C_{2}$ -équivariante et que sa structure est liée à l’étude d’objets « réels » en topologie algébrique. Ce dernier suscite depuis longtemps un intérêt pour ses applications pour la théorie de l’homotopie $O (2)$ -équivariante et ses connexions avec la $K$ -théorie algébrique hermitienne. Dans cet article, nous montrons que les théories de l’homologie réflexive et dièdre peuvent être interprétées comme une homologie de foncteurs sur des catégories d’ensembles non commutatifs, à la manière des résultats de Pirashvili et Richter pour les théories de Hochschild et de l’homologie cyclique.

Reflexive homology and dihedral homology are the homology theories associated to the reflexive and dihedral crossed simplicial groups respectively. The former has recently been shown to capture interesting information about $C_{2}$ -equivariant homotopy theory and its structure is related to the study of “real” objects in algebraic topology. The latter has long been of interest for its applications in $O (2)$ -equivariant homotopy theory and connections to Hermitian algebraic $K$ -theory. In this paper, we show that the reflexive and dihedral homology theories can be interpreted as functor homology over categories of non-commutative sets, after the fashion of Pirashvili and Richter’s result for the Hochschild and cyclic homology theories.

Reçu le : 2024-01-28
Révisé le : 2024-10-22
Accepté le : 2024-10-22
Publié le : 2025-03-06

DOI : 10.5802/crmath.698

Classification : 18G15, 18G90, 16E40, 16E30
Keywords: Reflexive homology, dihedral homology, functor homology, crossed simplicial group, involutive non-commutative sets, involutive algebra
Mots-clés : Homologie réflexive, homologie diédrale, homologie de foncteurs, groupe simplicial croisé, ensembles involutifs non commutatifs, algèbre involutive

Affiliations des auteurs :

Daniel Graves ¹

¹ Lifelong Learning Centre, University of Leeds, Woodhouse, Leeds, LS2 9JT, UK

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Pirashvili{\textendash}Richter-type theorems for the reflexive and dihedral homology theories},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {43--55},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.698},
     language = {en},
}

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Daniel Graves. Pirashvili–Richter-type theorems for the reflexive and dihedral homology theories. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 43-55. doi : 10.5802/crmath.698. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.698/

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