Reflexive homology

Abstract

Reflexive homology is the homology theory associated to the reflexive crossed simplicial group; one of the fundamental crossed simplicial groups. It is the most general way to extend Hochschild homology to detect an order-reversing involution. In this paper we study the relationship between reflexive homology and the $C_2$ -equivariant homology of free loop spaces. We define reflexive homology in terms of functor homology. We give a bicomplex for computing reflexive homology together with some calculations, including the reflexive homology of a tensor algebra. We prove that the reflexive homology of a group algebra is isomorphic to the homology of the $C_2$ -equivariant Borel construction on the free loop space of the classifying space. We give a direct sum decomposition of the reflexive homology of a group algebra indexed by conjugacy classes of group elements, where the summands are defined in terms of a reflexive analogue of group homology. We define a hyperhomology version of reflexive homology and use it to study the $C_2$ -equivariant homology of certain free loop and free loop-suspension spaces. We show that reflexive homology satisfies Morita invariance. We prove that under nice conditions the involutive Hochschild homology studied by Braun and by Fernàndez-València and Giansiracusa coincides with reflexive homology.

Stability of Loday constructions

“…In [Gra, Definition 2.1], he defines a bicomplex C * , * which is a bi-resolution of k * . With its help he shows in [Gra,Proposition 2.4] that HR +,k * (A; M ) is the homology of the complex CH k * (A; M )/(1 − r), where r is obtained from the maps r n of (6.1) by…”

“…Theorem 7.2 is a special case of Proposition 7.7 working with k = Z (although we are working over the C 2 -Burnside Tambara functor, not over Z c ) and with M = R. Theorem 7.3 is the relative version.Remark 7.8. Graves states a comparison result in[Gra, Theorem 9.1] between reflexive homology, HR+,k * (A; M ), and involutive Hochschild homology, iHH k * (A; M ). The assumptions are slightly too restrictive there: Fernàndez-València and Giansiracusa prove in [FVG18, Proposition 3.3.3] that iHH k * (A; M ) ∼ = HH k * (A; M ) C 2 if the characteristic of the ground field is different from 2 and Graves shows in [Gra, Proposition 2.4], that HH k * (A; M ) C 2 ∼ = HR +,k * (A; M ) if 2 is invertible in the ground ring.…”