Conventions

Throughout the paper we will let $k$ be a commutative ring. An unadorned tensor product symbol, $\otimes$, will denote the tensor product of $k$-modules. The category of $k$-modules will be denoted by $\mathbf {Mod}_k$. We will denote by $\mathbf {Top}$ the category of compactly generated weak Hausdorff topological spaces. Let $\mathbf {Top}_{\star }$ denote the category of based compactly generated weak Hausdorff topological spaces. We will usually refer to a ‘topological space’ or a ‘based topological space’. When referring to a weak equivalence of topological spaces we mean a $\pi _{\star }$-isomorphism. Several of our results relate to free loop spaces so we will introduce notation for this here. Let $\mathcal {L}$ denote the functor $\mathrm {Maps}(S^1,\, -)$, which sends a based topological space $X$ to the space of unbased continuous maps $S^1\rightarrow X$.

1. Reflexive homology

In this section we recall the definition of the reflexive crossed simplicial group and define its associated homology theory.

1.1 Functor homology and Hochschild homology

We start by recalling some constructions from functor homology, using Hochschild homology as an example.

For a small category $\mathbf {C}$ there are abelian categories $\mathbf {CMod}=\mathrm {Fun}(\mathbf {C},\, \mathbf {Mod}_k)$ and $\mathbf {ModC}=\mathrm {Fun}(\mathbf {C}^{op},\, \mathbf {Mod}_k)$. There is a tensor product

\[ -\otimes_{\mathbf{C}} - \colon \mathbf{ModC} \times \mathbf{CMod} \rightarrow \mathbf{Mod}_k \]

defined as the coend

\[ G\otimes_{\mathbf{C}} F = \int^{C\in \mathrm{Ob}(\mathbf{C})} G(C) \otimes_k F(C) \]

as in [Reference Lane55, Section 3]. It is well-known that this tensor product is right exact with respect to both variables and preserves direct sums [Reference Pirashvili and Richter62, Section 1.6]. The left derived functors of this tensor product are denoted by $\mathrm {Tor}_{\star }^{\mathbf {C}}(-,\,-)$. When $G=k^{\ast }$, the constant functor at $k$, we write $H_{\star }(\mathbf {C},\,F)= \mathrm {Tor}_{\star }^{\mathbf {C}}(k^{\ast },\, F)$.

As an example, we can recover Hochschild homology of a simplicial $k$-module. Recall the category $\Delta$, whose objects are the sets $[n]=\left \lbrace 0,\,\dotsc,\, n\right \rbrace$ for $n\geqslant 0$ and whose morphisms are order-preserving maps [Reference Loday51, B.1]. If we take $\mathbf {C}=\Delta ^{op}$ and a simplicial $k$-module $F\colon \Delta ^{op}\rightarrow \mathbf {Mod}_k$ we recover Hochschild homology. One example that we will be particularly interested in is studying $k$-algebras so we recall the Loday functor.

Let $A$ be an associative $k$-algebra and let $M$ be an $A$-bimodule. There is a functor

\[ \mathcal{L}(A,M)\colon \Delta^{op}\rightarrow \mathbf{Mod}_k \]

given on objects by $[n]\mapsto M\otimes A^{\otimes n}$ and determined on morphisms by

\[ \partial_i\left(m\otimes a_1\otimes \cdots \otimes a_n\right) = \begin{cases} \left( ma_1\otimes a_2\otimes \cdots \otimes a_n\right) & i=0\\ \left(m\otimes a_1\otimes \cdots \otimes a_ia_{i+1}\otimes \cdots \otimes a_n\right) & 1\leqslant i \leqslant n-1\\ \left(a_nm\otimes a_1\otimes \cdots \otimes a_{n-1}\right) & i=n \end{cases} \]

and

\[ s_j\left(m\otimes a_1\otimes \cdots \otimes a_n\right) = \begin{cases} \left(m\otimes 1_A \otimes a_1\otimes \cdots \otimes a_n\right) & j=0\\ \left(m\otimes a_1\otimes \cdots\otimes a_j \otimes 1_A \otimes a_{j+1} \otimes \cdots \otimes a_n\right) & j\geqslant 1. \end{cases} \]

The chain complex associated to this simplicial $k$-module is the Hochschild complex $C_{\star }(A,\,M)$ and its homology is Hochschild homology, $HH_{\star }(A,\,M)$. In particular we have

\[ HH_{{\star}}(A,M)= H_{{\star}}\left(\Delta^{op} , \mathcal{L}(A,M)\right) = \mathrm{Tor}_{{\star}}^{\Delta^{op}}\left(k^{{\ast}}, \mathcal{L}\left(A,M\right)\right). \]

1.2 Reflexive homology

An important source of functor homology theories come from crossed simplicial groups, introduced independently by Fiedorowicz and Loday [Reference Fiedorowicz and Loday31] and Krasauskas [Reference Krasauskas47].

In this paper we will study the homology theory associated to the reflexive crossed simplicial group.

Definition 1.1 Let $R_n=\left \langle r_n\mid r_n^2=1\right \rangle \cong C_2$ for $n\geqslant 0$.

The family of groups $\left \lbrace R_{\star }\right \rbrace$ forms a crossed simplicial group [Reference Fiedorowicz and Loday31, Example 2], whose geometric realization is $C_2$.

Definition 1.2 The category $\Delta R$ has the sets $[n]=\left \lbrace 0,\,\dotsc,\, n\right \rbrace$ for $n\geqslant 0$ as objects. An element of $\mathrm {Hom}_{\Delta R}([n],\,[m])$ is a pair $(\varphi,\, g)$ where $g\in R_n$ and $\varphi \in \mathrm {Hom}_{\Delta }([n],\,[m])$. The composition is determined as follows. For a face map $\delta _i\in \mathrm {Hom}_{\Delta }([n],\,[n+1])$ we define $r_{n+1}\circ \delta _i=\delta _{n-i}\circ r_n$ and for a degeneracy map $\sigma _j\in \mathrm {Hom}_{\Delta }([n],\,[n-1])$ we define $r_{n}\circ \sigma _j=\sigma _{n-j}\circ r_{n-1}$.

Remark 1.3 The category $\Delta R$ also appears in [Reference Krasauskas, Lapin and Solov'ev48, Section 1] with the notation $\Delta \ltimes \mathbb {Z}/2$.

Definition 1.4 Let $F\in \mathrm {Fun}(\Delta R^{op},\,\mathbf {Mod}_k)$. We define the reflexive homology of $F$ to be $HR_{\star }(F)=\mathrm {Tor}_{\star }^{\Delta R^{op}}(k^{\ast },\, F)$, where $k^{\ast }\in \mathrm {Fun}(\Delta R,\, \mathbf {Mod}_k)$ is the constant functor at $k$.

We can extend the Loday functor to a functor $\mathcal {L}(A,\,M)\colon \Delta R^{op}\rightarrow \mathbf {Mod}_k$ in two different ways but first we require some definitions.

Definition 1.5 A $k$-algebra is said to be involutive if it is equipped with an anti-homomorphism of algebras $A\rightarrow A$ of order two, which we will denote by $a\mapsto \overline {a}$. In other words, we equip $A$ with a $k$-linear endomorphism which reverses the order of multiplication and squares to the identity.

Following Loday [Reference Loday51, 5.2.1] we recall the notion of an involutive $A$-bimodule.

Definition 1.6 Let $A$ be an involutive $k$-algebra. An involutive $A$-bimodule is an $A$-bimodule $M$ equipped with a map $m\mapsto \overline {m}$ such that $\overline {a_1ma_2}= \overline {a_2}\, \overline {m}\, \overline {a_1}$ for $a_1$, $a_2\in A$.

Example 1.7 Taking $M=A$ gives one example of an involutive $A$-bimodule. If $A$ is equipped with an augmentation $\varepsilon \colon A\rightarrow k$ we can also take $M=k$ with the trivial involution.

With these definitions we can extend the Loday functor to $\Delta R^{op}$ in two ways.

Definition 1.8 Let $A$ be an involutive $k$-algebra and let $M$ be an involutive $A$-bimodule. We extend the Loday functor $\mathcal {L}(A,\,M)$ to a functor

\[ \mathcal{L}^+\left(A,M\right)\colon \Delta R^{op}\rightarrow \mathbf{Mod}_k \]

by defining

\[ r_n\left(m\otimes a_1\otimes \cdots \otimes a_n\right) = \left(\overline{m}\otimes \overline{a_n} \otimes \cdots \otimes \overline{a_1}\right). \]

Definition 1.9 Let $A$ be an involutive $k$-algebra and let $M$ be an involutive $A$-bimodule. We define

\[ HR_{{\star}}^+(A,M)= \mathrm{Tor}_{{\star}}^{\Delta R^{op}}\left(k^{{\ast}},\mathcal{L}^+(A,M)\right). \]

Definition 1.10 Let $A$ be an involutive $k$-algebra and let $M$ be an involutive $A$-bimodule. We extend the Loday functor $\mathcal {L}(A,\,M)$ to a functor

\[ \mathcal{L}^{-}\left(A,M\right)\colon \Delta R^{op}\rightarrow \mathbf{Mod}_k \]

by defining

\[ r_n\left(m\otimes a_1\otimes \cdots \otimes a_n\right) ={-}\left(\overline{m}\otimes \overline{a_n} \otimes \cdots \otimes \overline{a_1}\right). \]

Definition 1.11 Let $A$ be an involutive $k$-algebra and let $M$ be an involutive $A$-bimodule. We define

\[ HR_{{\star}}^{-}(A,M)= \mathrm{Tor}_{{\star}}^{\Delta R^{op}}\left(k^{{\ast}},\mathcal{L}^{-}(A,M)\right). \]

Remark 1.12 When we take $M=A$ we will omit the coefficients from the notation and write $\mathcal {L}^{\pm }(A)$ and $HR_{\star }^{\pm }(A)$.

Remark 1.13 These reflexive homology groups fit into long exact sequences with the dihedral homology groups by [Reference Krasauskas, Lapin and Solov'ev48, Proposition 3.1]. There exist long exact sequences

\[ \cdots \rightarrow HR_n^+(A)\rightarrow HD_n^+(A)\rightarrow HD_{n-2}^{-}(A)\rightarrow HR_{n-1}^+(A)\rightarrow \cdots \]

and

\[ \cdots \rightarrow HR_n^{-}(A)\rightarrow HD_n^{-}(A)\rightarrow HD_{n-2}^+(A)\rightarrow HR_{n-1}^{-}(A)\rightarrow \cdots \]

where the dihedral homology groups $HD_{\star }^+(A)$ and $HD_{\star }^{-}(A)$ are defined in [Reference Krasauskas, Lapin and Solov'ev48, Section 1]. Furthermore, as discussed in [Reference Loday51, 5.2], if the ground ring $k$ contains $\mathbb {Q}$ then the direct sum of these long exact sequences is Connes’ long exact sequence connecting Hochschild homology and cyclic homology [Reference Loday51, Theorem 2.2.1].

Remark 1.14 For an involutive $k$-algebra $A$, there is natural map $HR_{\star }^+(A) \rightarrow HO_{\star }(A)$, where $HO_{\star }$ denotes the hyperoctahedral homology theory of [Reference Graves39] and [Reference Fiedorowicz30, Section 2]. One can obtain this map by composing the map $HR_{\star }^+(A) \rightarrow HD_{\star }^+(A)$ from the previous remark with the map $HD_{\star }(A) \rightarrow HO_{\star }(A)$ of [Reference Graves39, Subsection 3.3].

2. Biresolution

We begin this section by defining a biresolution of the $k$-constant right $\Delta R^{op}$-module $k^{\ast }$. Fiedorowicz and Loday [Reference Fiedorowicz and Loday31, 6.7] give a general construction for such a biresolution using the homogeneous bar resolution for a group [Reference Lane56, VII.6]. The bicomplex defined here is smaller, making use of the periodic resolution of the group $C_2$. It is also worth noting that our bicomplex includes into the tricomplex of Krasauskas, Lapin and Solov'ev [Reference Krasauskas, Lapin and Solov'ev48, Section 2] for computing dihedral homology.

Definition 2.1 We define a bicomplex $C_{\star,\star }$ of right $\Delta R^{op}$-modules as follows. Firstly we set

\[ C_{p,q}=k\left[\mathrm{Hom}_{\Delta R}\left([q],-\right)\right] \]

for all $p,\,q\geqslant 0$.

The horizontal differential $d\colon C_{p,q}\rightarrow C_{p-1,q}$ for $q\geqslant 0$ and $p\geqslant 1$ is given by

\[ d=\begin{cases} 1-r_q & q\equiv 0,\, 3\,(\mathrm{mod}\, 4), \, p\equiv 1\, (\mathrm{mod}\, 2)\\ 1+r_q & q\equiv 1,\, 2\,(\mathrm{mod}\, 4), \, p\equiv 1\, (\mathrm{mod}\, 2)\\ 1+r_q & q\equiv 0,\, 3\,(\mathrm{mod}\, 4), \, p\equiv 0\, (\mathrm{mod}\, 2)\\ 1-r_q & q\equiv 1,\, 2\,(\mathrm{mod}\, 4), \, p\equiv 0\, (\mathrm{mod}\, 2) \end{cases} \]

where the map $r_q$ is defined by pre-composition.

The vertical differential $b\colon C_{p,q}\rightarrow C_{p,q-1}$ for $p\geqslant 0$ and $q\geqslant 1$ is given by

\[ b=\sum_{i=0}^q ({-}1)^i \delta_i \]

where the maps $\delta _i$ are defined by pre-composition.

Proof. It suffices to show that the bicomplex of $k$-modules obtained by evaluating $C_{\star,\star }$ on each object $[n]$ of $\Delta R^{op}$ is a resolution of $k$.

We fix an object $[n]$. We observe that $C_{p,q}([n])$ is a free $k[C_2]$-module on the set of generators $\mathrm {Hom}_{\Delta }([q],\,[n])$ for each $p\geqslant 0$.

Consider the $E^1$-page of the spectral sequence obtained by taking the horizontal homology of the bicomplex $C_{\star,\star }([n])$. The rows are exact complexes of finitely generated $k[C_2]$-modules. Therefore, the $E^1$-page is isomorphic to the complex

\[ k\left[\mathrm{Hom}_{\Delta}\left([{\star}],[n]\right)\right] \]

with differential $b$ concentrated in the column $p=0$. This is the complex associated to the standard simplicial model of the $n$-simplex, which is acyclic. We deduce that the $E^2$-page of the spectral sequence is isomorphic to a copy of $k$ concentrated in bidegree $(0,\,0)$ as required.

In the introduction we said that Krasauskas, Lapin and Solov'ev [Reference Krasauskas, Lapin and Solov'ev48, Section 3] had given a definition of reflexive homology in terms of hyperhomology. We now demonstrate that the functor homology definition coincides with that definition.

Proposition 2.4 Suppose that $2$ is invertible in the ground ring. Let $A$ be an involutive $k$-algebra and let $M$ be an involutive $A$-bimodule. There exist isomorphisms of graded $k$-modules

\[ HR_{{\star}}^+(A,M)\cong H_{{\star}}\left(\frac{C_{{\star}}(A,M)}{(1-r)} \right)\quad \text{and} \quad HR_{{\star}}^{-}(A,M)\cong H_{{\star}}\left(\frac{C_{{\star}}(A,M)}{(1+r)}\right). \]

3. Reflexive hyperhomology

We can extend the definition of reflexive homology to chain complexes of left $\Delta R^{op}$-modules by defining reflexive hyperhomology. This is a property common to all crossed simplicial groups, as described in [Reference Dunn24, Section 3] for example. We define reflexive hyperhomology in terms of hypertor functors (see [Reference Weibel70, 5.7.8] for instance) over the category $\Delta R^{op}$. In our case, these can be described explicitly as the hyper-derived functors of the tensor product $k^{\ast } \otimes _{\Delta R^{op}}-$ where $k^{\ast }$ is the $k$-constant right $\Delta R^{op}$-module.

Example 3.3 Another important example of a reflexive chain complex arises from involutive DGAs. Recall that a DGA (differential graded algebra, or sometimes chain algebra), $(A,\,d)$, is a graded $k$-algebra, $A$, equipped with a $k$-linear map $d\colon A \rightarrow A$ satisfying the following properties. The map $d$ has degree $-1$, it squares to zero and it satisfies the graded Leibniz rule: $d(ab)=(da)b+(-1)^{\left \lvert a \right \rvert }a(db)$, where $\left \lvert a \right \rvert$ denotes the degree of a homogeneous element $a$.

An involutive DGA, $(A,\,d,\,i)$, is a DGA, $(A,\,d)$, equipped with a chain map

\[ i\colon (A,d)\rightarrow (A,d), \]

written $i(a)=\overline {a}$. The map $i$ must square to the identity and satisfy $\overline {ab}= (-1)^{\left \lvert a \right \rvert \cdot \left \lvert b \right \rvert } \overline {b}\overline {a}$.

Given an involutive DGA $(A,\,d,\,i)$, we can form its reflexive bar construction, $\Gamma (A,\,d,\,i)$, using the simplicial and reflexive structure described in [Reference Dunn24, Section 3]. The $n$-simplices are given by $(A,\,d)^{\otimes (n+1)}$. Let $a=a_0\otimes \cdots \otimes a_n$. The face maps are given by

\[ \partial_i(a)=\begin{cases} a_0\otimes \cdots \otimes a_ia_{i+1}\otimes \cdots \otimes a_n & 0\leqslant i \leqslant n-1\\ ({-}1)^{\left\lvert a_n\right\rvert\left(\left\lvert a_0\right\rvert +\cdots +\left\lvert a_{n-1} \right\rvert\right)} a_na_0\otimes a_1\otimes \cdots \otimes a_{n-1} & i=n. \end{cases} \]

The degeneracy maps are given by

\[ s_j(a)= a_0\otimes \cdots \otimes a_j \otimes 1 \otimes a_{j+1} \otimes \cdots \otimes a_n \]

for $0\leqslant j\leqslant n$.

The reflexive operators are defined by

\[ r(a)= ({-}1)^{\lvert\lvert a \rvert\rvert\cdot n(n-1)/2}\, \overline{a_0}\otimes \overline{a_n}\otimes \cdots \otimes \overline{a_1} \]

where $\lvert \lvert a \rvert \rvert = \sum _{i=1}^{n-1} \sum _{j=i+1}^n \left \lvert a_i\right \rvert \cdot \left \lvert a_j\right \rvert$.

This is another example of a reflexive chain complex.

One example that will be of particular interest in § 8.2 is the following. If $M$ is an involutive topological monoid, the singular chain complex $S_{\star }(M,\,k)$ is an involutive DGA with the involution induced from the one on $M$.

We now define reflexive homology of a chain complex of left $\Delta R^{op}$-modules

Definition 3.4 Let $k_0^{\ast }$ denote the right $\Delta R^{op}$-module $k^{\ast }$ thought of as a chain complex concentrated in degree zero. Let $F_{\star }$ be a non-negatively graded chain complex of left $\Delta R^{op}$-modules. For each $n\geqslant 0$, we define the $n^{th}$ reflexive homology of $F_{\star }$ by

\[ HR_{n}\left(F_{{\star}}\right) = \mathbf{Tor}_{n}^{\Delta R^{op}}\left(k_0^{{\ast}} , F_{{\star}}\right). \]

4. Morita invariance

In this section we prove Morita invariance results for reflexive homology. It is a remarkable property of Hochschild homology [Reference Dennis and Igusa17, Theorem 3.7] (see also [Reference Loday51, Section 1.2]), cyclic homology [Reference Loday and Quillen52, Corollary 1.7] and dihedral homology [Reference Krasauskas, Lapin and Solov'ev48, Theorem 3.4] that the given homology theory applied to the algebra of $(m\times m)$-matrices with entries in a $k$-algebra $A$, involutive in the case of dihedral homology, is isomorphic to the homology of $A$. We prove that reflexive homology shares this property. We also prove a more general result. We show that if two algebras are Hermitian Morita equivalent and satisfy a compatibility condition, then they have the same reflexive homology. It is worth remarking that Morita invariance is not a property shared by every homology theory associated to a crossed simplicial group. For example, it is shown in [Reference Ault2, Remark 88] and [Reference Graves40, Corollary 5.11] that neither symmetric homology nor hyperoctahedral homology satisfy Morita invariance.

4.1 Morita equivalence and Hermitian Morita equivalence

We begin by recalling the definition of Morita equivalence for two (not necessarily involutive) $k$-algebras.

Definition 4.1 Two unital $k$-algebras, $A$ and $B$, are said to be Morita equivalent if there is an $A$-$B$-bimodule $P$ and a $B$-$A$-bimodule $Q$ together with an isomorphism of $A$-bimodules, $u\colon P\otimes _B Q \rightarrow A$, and an isomorphism of $B$-bimodules, ${v\colon Q\otimes _{A} P\rightarrow B}$.

The concept of Morita equivalence can be extended to involutive $k$-algebras. This notion is known as Hermitian Morita equivalence. This was introduced by Frölich and McEvett [Reference Fröhlich and McEvett33, Section 8] (see also [Reference Hahn42] and [Reference Farinati and Solotar27, Section 2]).

Definition 4.2 Let $A$ and $B$ be two unital, involutive $k$-algebras. We say that $A$ and $B$ are Hermitian Morita equivalent if:

1. • $A$ and $B$ are Morita equivalent in the sense of definition 4.1;

2. • the isomorphisms $u$ and $v$ satisfy

   1. – $u(p\otimes q)p^{\prime }= pv(q\otimes p^{\prime })$ and

   2. – $v(q\otimes p)q^{\prime }=qu(p\otimes q^{\prime })$

   for all $p,\,p^{\prime }\in P$ and $q,\,q^{\prime }\in Q$;

3. • there exists an additive bijection $\theta \colon P\rightarrow Q$ satisfying

   1. – $\theta (apb)= \overline {b} \,\theta (p) \,\overline {a}$ for all $a \in A$, $b\in B$ and $p\in P$,

   2. – $u(p\otimes \theta (p^{\prime })) = \overline {u(p^{\prime }\otimes \theta (p))}$ and

   3. – $v(\theta (p)\otimes p^{\prime }) = \overline {v(\theta (p^{\prime })\otimes p)}$.

Remark 4.3 It follows from the fact that $u$ and $v$ are isomorphisms that there exist sets of elements in $P$, say $\left \lbrace p_1,\,\dotsc,\, p_l\right \rbrace$ and $\left \lbrace p_1^{\prime } ,\, \dotsc,\, p_m^{\prime }\right \rbrace$, and sets of elements in $Q$, say $\left \lbrace q_1,\,\dotsc,\, q_l\right \rbrace$ and $\left \lbrace q_1^{\prime } ,\, \dotsc,\, q_m^{\prime }\right \rbrace$, such that

\[ u\left(\sum_{i=1}^l p_j\otimes q_j\right) = 1_A \quad \text{and} \quad v\left(\sum_{j=1}^m q_k^{\prime}\otimes p_k^{\prime}\right) =1_B. \]

Definition 4.4 Let $A$ and $B$ be two unital, involutive $k$-algebras. Suppose that $A$ and $B$ are Hermitian Morita equivalent. Suppose that the additive bijection $\theta$ sends the set $\left \lbrace p_1,\,\dotsc,\, p_l\right \rbrace$ to the set $\left \lbrace q_1,\,\dotsc,\, q_l\right \rbrace$. In this case we say that $A$ and $B$ are compatible.

4.2 The case of involutive matrix algebras

In this subsection we prove that reflexive homology satisfies Morita invariance for matrix algebras.

Let $A$ be an involutive $k$-algebra. The algebra of $(m\times m)$-matrices with entries in $A$, $\mathcal {M}_m(A)$, is an involutive $k$-algebra with involution defined by $\overline {(x_{ij})}=(\overline {x_{ji}})$. In other words, we take the transpose and apply the involution of $A$ entry-wise.

For $n\geqslant 0$, we define a $k$-module morphism

\[ \mathrm{Tr}_n\colon \mathcal{M}_m(A)^{{\otimes} (n+1)}\rightarrow A^{{\otimes} (n+1)} \]

by

\[ \mathrm{Tr}_n\left(X^{(0)}\otimes\cdots \otimes X^{(n)}\right) = \sum_{1\leqslant i_0,\dotsc , i_n\leqslant m} x_{i_0i_1}^{(0)}\otimes x_{i_1i_2}^{(1)}\otimes \cdots \otimes x_{i_ni_0}^{(n)} \]

where $x_{ij}^{(k)}$ is the element in the $i^{th}$ row of the $j^{th}$ column of the matrix $X^{(k)}$. As noted in [Reference Krasauskas, Lapin and Solov'ev48, Section 3] this map is compatible with simplicial structure and the involution operators $r_n$ for $n\geqslant 0$. It follows that there are induced maps

\[ \mathrm{Tr}_{{\star}}\colon HR_n^{{\pm}}\left(\mathcal{M}_m(A)\right) \rightarrow HR_n^{{\pm}}(A) \]

on reflexive homology for all $m\geqslant 1$ and $n\geqslant 0$.

Theorem 4.5 The morphism

\[ \mathrm{Tr}_{{\star}}\colon HR_n^{{\pm}}\left(\mathcal{M}_m(A)\right) \rightarrow HR_n^{{\pm}}(A) \]

on reflexive homology induced from the trace map is an isomorphism for all $m\geqslant 1$ and $n\geqslant 0$.

Proof. By proposition 2.3 we can consider $\mathrm {Tr}_{\star }$ as a morphism on hyperhomology. By the Morita invariance for Hochschild homology the trace maps induce isomorphisms $HH_{\star }(\mathcal {M}_m(A)) \rightarrow HH_{\star }(A)$.

Consider the bicomplexes $C_{\star,\star }\otimes _{\Delta R^{op}} \mathcal {L}^{\pm }(\mathcal {M}_m(A))$ and $C_{\star,\star }\otimes _{\Delta R^{op}} \mathcal {L}^{\pm }(A)$. The isomorphisms $HH_{\star }(\mathcal {M}_m(A)) \rightarrow HH_{\star }(A)$ ensure that the vertical homology spectral sequences for these bicomplexes have isomorphic $E^1$-pages. The result now follows from the comparison theorem [Reference Weibel70, 5.2.12].

4.3 A more general result

In this section we prove a more general Morita invariance statement for reflexive homology. We show that if two algebras are Hermitian Morita equivalent and are compatible in the sense of definition 4.4, then they have the same reflexive homology.

Definition 4.6 Let $A$ and $B$ be two unital, involutive $k$-algebras. Suppose that $A$ and $B$ are Hermitian Morita equivalent, so there exist bimodules $P$ and $Q$ and an additive bijection $\theta \colon P\rightarrow Q$ as in definitions 4.1 and 4.2. Let $M$ be an involutive $A$-bimodule. We define an involution the the $B$-bimodule $Q\otimes _A M\otimes _A P$ by

\[ \overline{q\otimes m \otimes p}=\theta(p)\otimes \overline{m} \otimes \theta^{{-}1}(q). \]

Theorem 4.7 Let $A$ and $B$ be two unital, involutive $k$-algebras. Suppose that they are Hermitian Morita equivalent and that they are compatible in the sense of definition 4.4. Let $M$ be an involutive $A$-bimodule. There exist natural isomorphisms of graded $k$-modules

\[ HR_{{\star}}^{{\pm}}(A,M) \cong HR_{{\star}}^{{\pm}}(B, Q\otimes_A M \otimes_A P), \]

where $Q\otimes _A M \otimes _A P$ is equipped with the involution of definition 4.6.

Proof. In [Reference Loday51, 1.2.7], Loday defines a chain homotopy equivalence

\[ \psi_{{\star}}\colon C_{{\star}}(A,M)\rightarrow C_{{\star}}\left(B, Q\otimes_A M\otimes_A P\right) \]

between the Hochschild complexes. One can check that this is compatible with the involutions on both complexes, yielding a $C_2$-equivariant chain homotopy equivalence, from which the result follows.

5. Calculations

In this section we provide some computations of reflexive homology. We can calculate the reflexive homology of the ground ring in two different ways; using the theory of crossed simplicial groups and direct computation from our bicomplex in the previous section. We demonstrate that our calculations agree with what is already known when working over a field of characteristic zero. We also give explicit descriptions of reflexive homology in degree zero for a commutative algebra.

Proof. Let $T\colon \Delta R^{op}\rightarrow \mathbf {Set}$ denote the trivial reflexive set. Explicitly, $T$ sends every object in $\Delta R^{op}$ to the one point set $\left \lbrace \ast \right \rbrace$ with trivial involution. Using [Reference Fiedorowicz and Loday31, Corollary 6.13] we have

\[ HR_{{\star}}^+(k) \cong HR_{{\star}}^+\left(k[T]\right) \cong H_{{\star}}\left(EC_2\times_{C_2} \left\lvert T\right\rvert, k\right) \cong H_{{\star}}\left(EC_2\times_{C_2} \ast , k\right) \cong H_{{\star}}\left(BC_2 , k\right) \]

as required.

We can also obtain this calculation directly from the bicomplex defined in the previous section.

Proposition 5.2 Let $k$ be a commutative ring with trivial involution. Then

\[ HR_n^+(k) \cong \begin{cases} k & n=0 \\ k/2k & n \text{ odd}\\ {}_2k & n>0 \text{ even} \end{cases} \hskip 6em HR_n^{-}(k)\cong \begin{cases} k/2k & n \text{ even}\\ {}_2k & n \text{ odd} \end{cases} \]

where ${}_2k$ denotes the $2$-torsion of $k$.

Proof. We will prove the result for $HR_n^+(k)$. The result for $HR_n^{-}(k)$ is similar.

Consider the bicomplex $C_{\star, \star }\otimes _{\Delta R^{op}} \mathcal {L}^+(k)$. Since the Hochschild homology of $k$ is isomorphic to $k$ concentrated in degree zero [Reference Loday51, 1.1.6], taking the vertical homology yields the complex

\[ 0\leftarrow k \xleftarrow{0} k \xleftarrow{2} k \xleftarrow{0} k \xleftarrow{2} k \xleftarrow{0} \cdots \]

in row zero. The result now follows by taking homology of this complex.

6. Reflexive homology of a tensor algebra

In this section we calculate the reflexive homology of a tensor algebra. Let $M$ be a $k$-module and consider a module automorphism $m\mapsto \overline {m}$ which squares to the identity. The identity automorphism is an example of such.

Consider the tensor algebra

\[ TM = \bigoplus_{n=0}^{\infty} M^{{\otimes} n} \]

where $M^{\otimes 0}=k$. The product is given by concatenation, see [Reference Loday51, A.1] for instance. The tensor algebra has an involution determined on the $n^{th}$ summand by

\[ r\left(m_1\otimes \cdots \otimes m_n\right) = \left(\overline{m_n} \otimes \overline{m_{n-1}}\otimes \cdots \otimes \overline{m_2}\otimes \overline{m_1}\right). \]

Note that the involution on $k$ is trivial.

Taking $A=TM$ in the definition of the Loday functor $\mathcal {L}^+(A)$, we obtain an induced involution on $\mathcal {L}^+(A)([n])=(TM)^{\otimes {n+1}}$ given by

\[ r(a_0\otimes\cdots \otimes a_n) = \overline{a_0}\otimes \overline{a_n}\otimes \cdots \otimes \overline{a_1}, \]

where each $A_i \in TM$.

Theorem 6.1 Let $M$ be a $k$-module with a module automorphism $m\mapsto \overline {m}$ which squares to the identity. There is a natural isomorphism of $k$-modules

\[ HR_n^+(TM) \cong \begin{cases} H_0\left(C_2, HH_0(TM)\right) & n=0\\ H_{n-1}\left(C_2, HH_1(TM)\right)\oplus H_n\left(C_2, HH_0(TM)\right) & n\geqslant 1. \end{cases} \]

Proof. Consider the reflexive bicomplex for $TM$. By taking the vertical homology we obtain the $E^1$-page of a spectral sequence whose entries are the Hochschild homology groups of $TM$ with horizontal differentials induced from $1\pm r$.

Loday and Quillen [Reference Loday and Quillen52, Lemma 5.2] have calculated the Hochschild homology of a tensor algebra as follows:

\[ HH_n(TM) \cong \begin{cases} \bigoplus_{q\geqslant 0} \frac{M^{{\otimes} q}}{\left\langle 1-t\right\rangle} & n=0\\ \bigoplus_{q \geqslant 1} \left(M^{{\otimes} q}\right)^t & n=1\\ 0 & \text{otherwise}, \end{cases} \]

where $(M^{\otimes q})^t$ are the invariants of the action of the cyclic group $C_q$ on $M^{\otimes q}$ and $M^{\otimes q}/\left \langle 1-t\right \rangle$ are the coinvariants.

Therefore the $E^1$-page of our spectral sequence is concentrated in rows zero and one as follows:

An argument similar to [Reference Lodder53, Lemma 2.2.1] tells us that in row zero we have

\[ r\left(m_1\otimes \cdots \otimes m_q\right) = \left(\overline{m_1} \otimes \overline{m_q}\otimes \cdots \otimes \overline{m_2}\right) \]

and in row one we have

\[ r\left(m_1\otimes \cdots \otimes m_q\right) ={-}\left(\overline{m_1}\otimes \overline{m_q}\otimes \cdots \otimes \overline{m_2}\right). \]

Furthermore, an argument similar to [Reference Lodder53, Lemma 2.3.2] tells us that the differentials on the $E^2$-page are zero and so the spectral sequence collapses.

Taking homology on the $E^1$-page yields

\[ E_{p,0}^2\cong H_p\left(C_2, HH_0(TM)\right) \quad \text{and} \quad E_{p,1}^2\cong H_p\left(C_2, HH_1(TM)\right) \]

with all other $E_{p,q}^2=0$, from which we can read off the result.

Remark 6.2 We deduce from the theorem that the reflexive homology of a tensor algebra has a grading induced from the grading on the Hochschild homology. In homological degree zero, for $q\geqslant 0$, we have

\[ HR_0^+\left(TM\right)_q = H_0\left(C_2,\frac{M^{{\otimes} q}}{\left\langle 1-t\right\rangle}\right). \]

Note that when $q=0$ we have

\[ HR_0^+\left(TM\right)_0=H_0\left(C_2,k\right)\cong H_{0}(BC_2,k) \cong HR_{0}^+(k) \]

and when $q=1$ we have

\[ HR_0^+\left(TM\right)_1=H_0\left(C_2,M\right). \]

In homological degree $p$ we have

\[ HR_p^+(TM)_0 = H_p\left(C_2 , k\right) \cong H_p(BC_2,k) \cong HR_p^+(k) \]

and

\[ HR_p^+(TM)_q = H_p\left(C_2,\frac{M^{{\otimes} q}}{\left\langle 1-t\right\rangle}\right) \oplus H_{p-1}\left(C_2 , \left(M^{{\otimes} q}\right)^t\right) \]

for $q\geqslant 1$.

We note that, as mentioned above, our grading on the reflexive homology of a tensor algebra is induced from the grading on the Hochschild homology of a tensor algebra. For cyclic homology [Reference Loday and Quillen52, Proposition 5.4] and dihedral homology [Reference Lodder53, Theorem 2.1.1] there is a grading induced directly from the grading on a tensor algebra. In both of these cases this follows from analysis of the norm map $N$ (see [Reference Loday51, 2.1.0] for instance) which we do not have in the reflexive case.

7. Reflexive homology of a group algebra

In this section we will study the reflexive homology of the group algebra of a discrete group. We will recall simplicial models for the bar construction and for the classifying space on a group. We will show that these extend to reflexive sets. We will show that the reflexive homology of a group algebra is isomorphic to the $C_2$-equivariant homology of the free loop space on $BG$. As a consequence we will deduce the analogous result for dihedral homology, namely that the dihedral homology of a group algebra is isomorphic to the $O(2)$-equivariant homology of the free loop space on $BG$. We will then show that we can decompose the reflexive homology of a group algebra in terms of the conjugacy classes of the group.

These results fit into a broader story of using the homology theories associated to crossed simplicial groups to calculate interesting information about loop spaces. Hochschild homology and cyclic homology of $k[G]$ are known to coincide with the homologies of $\mathcal {L}BG$ and the $S^1$-equivariant Borel construction on $\mathcal {L}BG$ respectively [Reference Loday51, 7.3.13]. The symmetric and hyperoctahedral theories are known to compute the homology and $C_2$-equivariant homology of certain infinite loop spaces on $BG$, see [Reference Ault2, Corollary 40] and [Reference Graves39, Theorem 8.8].

The decomposition we provide fits into a bigger picture of decomposing homology theories associated to crossed simplicial groups in the case of a group algebra. Results of this form were proved by Burghelea [Reference Burghelea7, Theorem 1] for Hochschild and cyclic homology (see also [Reference Loday51, Theorem 7.4.6]) and by Loday [Reference Loday50, Proposition 4.9] for dihedral homology.

7.1 Bar construction and classifying spaces

We recall some simplicial models for the bar construction [Reference Loday51, 7.3.10] and classifying space [Reference Loday51, B.12] of a group and extend them to reflexive sets.

Definition 7.1 Let $G$ be a discrete group. For $n\geqslant 0$, let $\Gamma _nG=G^{n+1}$, the $(n+1)$-fold Cartesian product. The face maps are defined by

\[ \partial_i\left(g_0,\dotsc , g_n\right)=\begin{cases} \left(g_0,\dotsc ,g_{i-1}, g_ig_{i+1}, g_{i+2},\dotsc , g_n\right) & 0\leqslant i \leqslant n-1\\ \left(g_ng_0, g_1, \dotsc , g_{n-1}\right) & i=n. \end{cases} \]

The degeneracy maps insert the identity element of $G$ into the tuple. We extend this to a reflexive set by defining

\[ r_n\left(g_0,\dotsc , g_n\right)= \left(g_0^{{-}1}, g_n^{{-}1},\dotsc , g_1^{{-}1}\right). \]

Remark 7.2 Loday has already shown that $\Gamma _{\star }G$ is a cyclic set, by defining

\[ t_n\left(g_0,\dotsc , g_n\right) = \left(g_n, g_0,\dotsc , g_{n-1}\right). \]

One can easily check that the reflexive structure and the cyclic structure are compatible, giving $\Gamma _{\star }G$ the structure of a dihedral set. The fact that the reflexive structure that we have defined is compatible with the cyclic structure of $\Gamma _{\star }G$ is key to proving Theorem 7.7.

Definition 7.3 Let $B_nG=G^{n}$ for $n\geqslant 0$. The face maps are given by

\[ \partial_i\left(g_1,\dotsc , g_n\right) = \begin{cases} \left(g_2, \dotsc , g_n\right) & i=0\\ \left(g_1 , \dotsc , g_ig_{i+1}, \dotsc , g_n\right) & 1\leqslant i \leqslant n\\ \left(g_1,\dotsc , g_{n-1}\right) & i=n. \end{cases} \]

The degeneracy maps insert the identity element into the tuple. We can extend this to a reflexive set by defining

\[ r_n\left(g_1,\dotsc , g_n\right) = \left(g_n^{{-}1},\dotsc , g_1^{{-}1}\right). \]

Definition 7.4 Let $G$ be a discrete group. We define the reflexive homology of $G$ to be

\[ HR_{{\star}}^+\left(G,k\right)= HR_{{\star}}^+\left(k\left[B_{{\ast}}G\right]\right). \]

Remark 7.5 Note that we have a projection map, which is a map of reflexive sets,

\[ p\colon \Gamma_{{\star}}G \rightarrow B_{{\star}}G, \]

determined in degree $n$ by $(g_0,\,\dotsc,\, g_n) \mapsto (g_1,\,\dotsc,\, g_n)$.

Remark 7.6 As for $\Gamma _{\star }G$, $B_{\star }G$ has a cyclic structure. This is given by

\[ t_n\left(g_1,\dotsc ,g_n\right)= \left(\left(g_1\cdots g_n\right)^{{-}1}, g_1,\dotsc , g_{n-1}\right), \]

as described in [Reference Loday51, 7.3.3] (with $z=1$) for example. This is compatible with the reflexive structure on $B_{\star }G$, as described in [Reference Dunn24, 2.3], giving $B_{\star }G$ the structure of a dihedral set.

7.2 Reflexive homology of a group algebra

Let $G$ be a discrete group. In this subsection we prove that the reflexive homology of a group algebra $k[G]$ is isomorphic to the homology of the $C_2$-equivariant Borel construction on the free loop space of $BG$.

Theorem 7.7 Let $G$ be a discrete group. Let $k[G]$ denote its group algebra and let $BG$ be its classifying space. Let $\mathcal {L}BG$ denote the free loop space on $BG$. There is an isomorphism of graded $k$-modules

\[ HR_{{\star}}^+\left(k[G]\right) \cong H_{{\star}}\left(EC_2 \times_{C_2} \mathcal{L}BG , k \right), \]

where the $C_2$-action on $\mathcal {L}BG$ is induced from the reflexive structure of $B_{\star }G$ and reversing the direction of loops.

Proof. In [Reference Loday51, 7.3.11], Loday shows that there is a homotopy equivalence $\gamma \colon \left \lvert \Gamma _{\star }G\right \rvert \rightarrow \mathcal {L}BG$. This is done by considering the adjoint functors $S^1\times -$ and $\mathcal {L}(-)$. In particular, there is a map

\[ S^1 \times \left\lvert \Gamma_{{\star}}G\right\rvert \rightarrow \left\lvert \Gamma_{{\star}}G\right\rvert \rightarrow \left\lvert B_{{\star}}G\right\rvert \]

where the first map is induced from the cyclic structure of $\Gamma _{\star }G$, as described in [Reference Fiedorowicz and Loday31, Section 5], and the second is induced from the projection map $p\colon \Gamma _{\star }G \rightarrow B_{\star }G$ described in remark 7.5. The adjoint map is $\gamma$, the necessary homotopy equivalence.

We extend this to a $C_2$-equivariant homotopy equivalence. As noted in remark 7.2, the reflexive structure of $\Gamma _{\star }G$ is compatible with the cyclic structure. Explicitly, the circle $S^1$ is the geometric realization of the cyclic crossed simplicial group, $\left \lbrace C_n \right \rbrace$, by [Reference Loday51, 6.3.6]. The level-wise involution determined by sending the generator of the cyclic group to its inverse induces a $C_2$-action on the geometric realization. Similarly, the reflexive structures on $\Gamma _{\star }G$ and $B_{\star }G$ induce $C_2$-actions on the realizations. Note that the involution on geometric realizations also flips the simplex. One easily checks that with these definitions the map

\[ S^1 \times \left\lvert \Gamma_{{\star}}G\right\rvert \rightarrow \left\lvert B_{{\star}}G\right\rvert \]

is $C_2$-equivariant. Under the adjunction between $S^1 \times -$ and $\mathcal {L}(-)$ this yields that

\[ \gamma \colon \left\lvert \Gamma_{{\star}}G\right\rvert \rightarrow \mathcal{L}BG \]

is a $C_2$-equivariant map, with the involution on $\mathcal {L}BG$ given by applying the involution on $BG$ and reversing the direction of loops. Combining this with [Reference Loday51, 7.3.11], we see that $\gamma$ is a $C_2$-equivariant homotopy equivalence.

Using [Reference Loday51, 7.3.13] and [Reference Fiedorowicz and Loday31, 6.13], we have

\[ HR_{{\star}}^+\left(k[G]\right) \cong H_{{\star}}\left(EC_2 \times_{C_2} \left\lvert\Gamma_{{\star}}G\right\rvert \right) \cong H_{{\star}}\left(EC_2\times_{C_2} \mathcal{L}BG\right) \]

as required.

Using the fact that the reflexive structure that we have described is compatible with the cyclic structure of [Reference Loday51, 7.3.11] we can deduce the analogous result for dihedral homology.

Theorem 7.8 Let $G$ be a discrete group. Let $k[G]$ denote its group algebra and let $BG$ be its classifying space. Let $\mathcal {L}BG$ denote the free loop space on $BG$. There is an isomorphism of graded $k$-modules

\[ HD_{{\star}}^+\left(k[G]\right) \cong H_{{\star}}\left(EO(2) \times_{O(2)} \mathcal{L}BG , k \right), \]

where the $O(2)$-action on $\mathcal {L}BG$ is induced from the dihedral structure of $B_{\star }G$ and the action of $O(2)$ on loops.

Proof. Since the reflexive structure of $\Gamma _{\star }G$ and $B_{\star }G$ used in Theorem 7.7 is compatible with the cyclic structures used in [Reference Loday51, 7.3.11], we can deduce that $\gamma$ is an $O(2)$-equivariant map, from which the result follows.

7.3 Decomposition by conjugacy classes

In this subsection we prove that the reflexive homology of the group algebra of a discrete group can be decomposed into a direct sum indexed by the conjugacy classes of the group, where the summands are defined in terms of a reflexive analogue of group homology. We prove that under nice conditions, for example when the group is abelian, we can give a simple description of the summands.

Definition 7.9 Let $G$ be a discrete group.

• • Let $\overline {k[G]}$ denote $k[G]$ considered as a right $k[G]$-module with the action $h\ast g = g^{-1}hg$.

• • Let $\left \langle G\right \rangle$ denote the set of conjugacy classes of $G$. We will choose a representative $z$ in each class and denote the class by $\left \langle z \right \rangle$.

• • For $z\in G$ we denote by $G_z$ the centralizer of $z$ in $G$, that is, $G_z=\left \lbrace g\in G : gz=zg\right \rbrace$.

Definition 7.10 For a group $G$ and a right $k[G]$-module $M$ we denote by $C_{\star }(G,\,M)$ the Eilenberg-Mac Lane complex [Reference Loday51, C.2, C.3]. We will denote an element in degree $n$ by $h\otimes [g_1,\, \dotsc,\,g_n]$.

In order to prove our decomposition of the reflexive homology of a group algebra we need to recall certain reflexive sets. The involutions defined here can be found in the proof of [Reference Loday50, Proposition 4.9].

Definition 7.11 Let $G$ be a discrete group.

• • We define the action of $r_{n}$ on $k[G]^{\otimes n+1}$ to be determined by

  \[ r_{n}\left(g_0,\dotsc , g_n\right) = ({-}1)^{n(n+1)/2} \left(g_0^{{-}1}, g_n^{{-}1}, \dotsc , g_1^{{-}1}\right). \]

• • Let $M=\overline {k[G]}$. The action of $r_n$ on $C_{n}(G,\,M)$ is given by

  \[ r_{n} \left(h \otimes \left[g_1,\dotsc , g_n\right]\right) = ({-}1)^{n(n+1)/2}\left( g^{{-}1} h^{{-}1} g \otimes \left[g_n^{{-}1}, \dotsc , g_1^{{-}1}\right]\right) \]
  where $g=g_1\cdots g_n$.

• • We define the action of $r_n$ on

  \[ \bigoplus_{\left\langle z \right\rangle \in \left\langle G\right\rangle} C_n\left(G, k\left[ G/G_z\right]\right) \]
  by
  \[ r_{n} \left(h \otimes \left[g_1,\dotsc , g_n\right]\right) = ({-}1)^{n(n+1)/2}\left( g^{{-}1} h^{{-}1} g \otimes \left[g_n^{{-}1}, \dotsc , g_1^{{-}1}\right]\right) \]
  where $g=g_1\cdots g_n$.

Theorem 7.12 Let $G$ be a discrete group. For each $n\geqslant 0$ there is an isomorphism of $k$-modules

\[ HR_n^+\left(k[G]\right) \cong \bigoplus_{\left\langle z \right\rangle \in \left\langle G\right\rangle} HR_n^+\left(G , k[G/G_z]\right). \]

Proof. Let $M=\overline {k[G]}$. Recall the Mac Lane isomorphism

\[ \Phi \colon k[G]^{n+1}\rightarrow C_n\left(G, M\right) \]

from [Reference Loday51, Section 7.4] for example. The Mac Lane isomorphism is compatible with the actions described in definition 7.11. Recall from the proof of [Reference Loday50, Proposition 4.7] that there is an isomorphism of right $k[G]$-modules

\[ \xi \colon \overline{k[G]}\rightarrow \bigoplus_{\left\langle z \right\rangle \in \left\langle G\right\rangle} k\left[ G/G_z\right]. \]

This isomorphism is also compatible with the actions described in definition 7.11

We therefore have isomorphisms

\[ k[G]^{n+1} \xrightarrow{\Phi} C_n\left(G, M\right) \xrightarrow{\xi} \bigoplus_{\left\langle z \right\rangle \in \left\langle G\right\rangle} C_{n}\left(G,k[G/G_z]\right) \]

compatible with the reflexive $k$-module structures and the result follows upon taking reflexive homology.

Corollary 7.13 If $G$ is an abelian group then

\[ HR_{{\star}}^+\left(k[G]\right) \cong \bigoplus_{\left\lvert G\right\rvert} HR_{{\star}}^+\left(G,k\right). \]

Proof. If $G$ is abelian, then each element $z\in G$ has its own conjugacy class and the centralizer $G_z$ is isomorphic to $G$.

For non-abelian groups we can also identify some of the summands in the decomposition.

Proposition 7.14 Let $G$ be a discrete group. The $\left \langle 1\right \rangle$-component of $HR_{\star }^+(k[G])$ is isomorphic to $HR_{\star }^+(G,\,k)$.

Proof. The centralizer $G_1$ is equal to $G$ so the reflexive homology of $C_{\star }(G ,\, k[G/G_1])$ is the reflexive homology of $C_{\star }(G,\,k)$.

Proposition 7.15 Let $G$ be a discrete group. Let $z \in G$ be a central element of order two. The $\left \langle z\right \rangle$-component of $HR_{\star }^+(k[G])$ is isomorphic to $HR_{\star }^+(G,\,k)$.

Proof. Recall from the proof of [Reference Loday50, Proposition 4.7] that there is a quasi-isomorphism

\[ k\otimes C_n\left(G_z\right)\rightarrow k\left[G/G_z\right] \otimes C_n(G) \]

for any $z\in G$. When $z$ is a central element of order two this quasi-isomorphism is compatible with the reflexive structure described in definition 7.11, whence the result.

8. Reflexive homology of singular chains on a Moore loop space

Goodwillie [Reference Goodwillie35, Section V] proved that for a sufficiently nice space $X$, the cyclic homology of the singular chain complex of the Moore loop space of $X$ is isomorphic to the $S^1$-equivariant homology of the free loop space on $X$. Dunn [Reference Dunn24, 3.6] proved that this result can be extended to a result for the dihedral homology theory and $O(2)$-equivariant homology. In this section we will prove the analogous theorem for the reflexive homology theory and $C_2$-equivariant homology. We will also provide a $C_2$-equivariant analogue of Lodder's result for free-loop suspension spaces [Reference Lodder53, 3.3.3].

8.2 Moore loops and free loop spaces

We begin by recalling the definition of the Moore loop space.

Definition 8.3 Let $M\colon \mathbf {Top}_{\star }\rightarrow \mathbf {Top}_{\star }$ denote the Moore loop functor. For $X\in \mathbf {Top_{\star }}$ with basepoint $\star$, the topological space $M(X)$ is the subset of

\[ \mathrm{Hom}_{\mathbf{Top}_{{\star}}}\left([0,\infty),X\right)\times [0,\infty) \]

consisting of all pairs of the form $( f,\, r)$ such that $f(t)=\star$ for all $t\geqslant r$. In general we will omit the $r$ and refer to a Moore loop $f$.

Let $Y$ be a group-like LEC topological monoid with involution. As noted in example 3.3, the singular chain complex $S_{\star }(Y,\, k)$ is an involutive DGA, with the involution induced from $Y$. Recall the reflexive bar construction, $\Gamma (S_{\star }(Y,\, k))$, on $S_{\star }(Y,\, k)$ from example 3.3.

Theorem 8.4 Let $Y$ be a group-like LEC topological monoid with involution. There is an isomorphism of graded $k$-modules

\[ HR_{{\ast}}\left(\Gamma\left(S_{{\star}}\left(Y, k\right)\right)\right)\cong H_{{\ast}}\left(EC_2 \times_{C_2} \mathcal{L}BY,k\right) \]

where the $C_2$-action on $\mathcal {L}BY$ is induced from the involution on $Y$ and reversing the direction of loops.

Proof. This follows directly from [Reference Dunn24, 3.6]. We note that the given theorem relies on Dunn's propositions 2.10, 3.2, 3.3 and 3.5. One can check that these results hold when we only consider the simplicial and reflexive structure from the dihedral objects considered in that paper.

Corollary 8.5 Let $X$ be a connected, LEC topological space. There is an isomorphism of graded $k$-modules

\[ HR_{{\ast}}\left(\Gamma\left(S_{{\star}}\left(MX, k\right)\right)\right)\cong H_{{\ast}}\left(EC_2 \times_{C_2} \mathcal{L}X,k\right), \]

where $X$ is equipped with the trivial involution and $MX$ is equipped with the involution which reverses the direction of loops.

Proof. May [Reference May58, 15.4] shows that there is a map $\xi \colon BMX\rightarrow X$, which is a weak homotopy equivalence if $X$ is connected. As noted in [Reference Dunn24, 2.9], with the given involutions the map $\xi$ is $C_2$-equivariant. The corollary now follows from Theorem 8.4 by setting $Y=MX$.

8.3 Moore loops and free loop-suspension spaces

In this section we prove a reflexive analogue of Lodder's result for free loop-suspension spaces [Reference Lodder53, 3.3.3]. We identify the $C_2$-equivariant homology of a free loop-suspension space $\mathcal {L}\Sigma X$ as the reflexive homology of the singular chains on a reflexive space constructed from the Moore loop space of $X$. In this case we can use a smaller construction than the reflexive bar construction on a DGA since the suspension of a topological space is always path-connected.

We begin by recalling the definition of the suspension functor.

Definition 8.6 Let $\Sigma \colon \mathbf {Top}_{\star } \rightarrow \mathbf {Top}_{\star }$ denote the suspension functor $S^1 \wedge -$.

Definition 8.7 Let $X$ be a based topological space. Let $\Sigma X=S^1 \wedge X$ be equipped with the involution $\overline {(t,\,x)}=(1-t,\,x)$, given by reversing the suspension co-ordinate. We define a reflexive topological space

\begin{align*} M\Sigma X(-)\colon \Delta R^{op} \rightarrow \mathbf{Top}_{{\star}} \end{align*}

as follows.

For each $n\geqslant 0$, we have $M\Sigma X([n])=(M\Sigma X)^{n+1}$, the $(n+1)$-fold Cartesian product of the Moore loop space on $\Sigma X$.

The face maps $\partial _i\colon M\Sigma X([n])\rightarrow M\Sigma X([n-1])$ are given by concatenation of loops:

\[ \partial_i\left(f_0, \dotsc , f_n\right) = \begin{cases} \left( f_0,\dotsc , f_{i-1}, f_i\cdot f_{i+1} , f_{i+2},\dotsc , f_n\right) & 0\leqslant i \leqslant n-1\\ \left(f_n\cdot f_0, f_1,\dotsc , f_{n-1}\right) & i=n. \end{cases} \]

The degeneracy maps $s_j\colon M\Sigma X([n])\rightarrow M\Sigma X([n+1])$ are given by inserting the trivial loop:

\[ s_j\left(f_0, \dotsc , f_n\right)= \left(f_0, \dotsc , f_j , 1 , f_{j+1},\dotsc , f_n\right) \]

for $0\leqslant j \leqslant n$.

The reflexive operators $r_n\colon M\Sigma X([n]) \rightarrow M\Sigma X([n])$ for $n\geqslant 0$ are given by reversing the direction of loops and applying the involution on $\Sigma X$:

\[ r_n\left(f_0, \dotsc , f_n\right) = \left(\overline{f}_0^{{-}1}, \overline{f}_n^{{-}1}, \dotsc , \overline{f}_1^{{-}1}\right). \]

Definition 8.8 Let $\mathcal {S}$ denote the reflexive chain complex obtained by taking the singular chain complex $S_{\star }(M\Sigma X(-),\, k)$ on the reflexive topological space $M\Sigma X(-)$.

Theorem 8.9 Let $X$ be a based topological space. Let $\mathcal {S}$ denote the singular chain complex on the reflexive topological space $M\Sigma X(-)$. There is an isomorphism of graded $k$-modules

\[ HR_{{\ast}}\left(\mathcal{S}\right) \cong H_{{\ast}}\left( EC_2 \times_{C_2} \mathcal{L}\Sigma X,k\right). \]

Proof. By combining the constructions in [Reference Loday51, Appendix B] for singular homology with results on homotopy colimits for crossed simplicial groups [Reference Fiedorowicz and Loday31, Section 6] we see that given a reflexive space $Y_{\star }$, there are isomorphisms

\[ HR_{n}\left(\mathcal{S}(Y_{{\star}})\right) \cong H_n\left(\mathrm{hocolim}_{\Delta R^{op}}\left(Y_{{\star}}\right)\right)\cong H_n\left(EC_2\times_{C_2} \left\lvert Y_{{\star}}\right\rvert\right), \]

where the $C_2$-action on $\left \lvert Y_{\star }\right \rvert$ is induced from the reflexive structure.

Goodwillie [Reference Goodwillie35, Section V] has constructed a weak equivalence $\left \lvert M\Sigma X(-)\right \rvert \rightarrow \mathcal {L}\Sigma X$. By incorporating the reflexive action defined in [Reference Lodder53, Section 3.4] we extend this to a $C_2$-equivariant weak equivalence. The proof of this fact follows by mimicking [Reference Lodder53, 3.4.1], replacing the dihedral crossed simplicial group with the reflexive crossed simplicial group.

We therefore have a weak equivalence

\[ EC_2\times_{C_2} \left\lvert M\Sigma X(-)\right\rvert \rightarrow EC_2 \times_{C_2}\mathcal{L}\Sigma X \]

which yields

\[ HR_{{\ast}}\left(\mathcal{S}\right)\cong H_{{\ast}}\left(EC_2\times_{C_2} \left\lvert M\Sigma X(-)\right\rvert\right) \cong H_{{\ast}}\left( EC_2 \times_{C_2} \mathcal{L}\Sigma X,k\right) \]

as required.

9. Relationship to involutive Hochschild homology

Involutive Hochschild homology was introduced by Braun [Reference Braun4] to study involutive algebras and involutive $A_{\infty }$-algebras. Fernàndez-València and Giansiracusa [Reference Fernàndez-València and Giansiracusa29] developed the homological algebra of involutive Hochschild homology. In particular, they showed that involutive Hochschild homology can be expressed as $\mathrm {Tor}$ over an involutive version of the enveloping algebra. The original motivation for constructing involutive Hochschild homology was to extend work of Costello [Reference Costello15] to study the connection between unoriented two-dimensional topological conformal field theories and involutive Calabi-Yau $A_{\infty }$-algebras [Reference Braun4], [Reference Fernàndez-València28]. Involutive Hochschild homology also arises in the study of the (co)homology of involutive dendriform algebras [Reference Das and Saha16]. We show that under certain conditions involutive Hochschild homology coincides with reflexive homology.

Let $k$ be a field. Let $A$ be an involutive $k$-algebra and let $A^e=A\otimes A^{op}$ denote the enveloping algebra. For this section only, let $C_2=\left \langle t\mid t^2=1\right \rangle$. The group $C_2$ acts on $A^e$ by the rule $t(a_1\otimes a_2)=\overline {a_2} \otimes \overline {a_1}$. Fernàndez-València and Giansiracusa define the involutive enveloping algebra $A^{ie}$ to be $A^e\otimes k[C_2]$ with the product determined by

\[ \left(a_1\otimes t^i\right) \left(a_2\otimes t^j\right)= \left(a_1 \cdot t^i(a_2)\right)\otimes t^{i+j}. \]

The involutive Hochschild homology of $A$ with coefficients in an involutive $A$-bimodule $M$, denoted by $iHH_{\star }(A,\,M)$, is defined to be $\mathrm {Tor}_{\star }^{A^{ie}}(A,\,M)$ [Reference Fernàndez-València and Giansiracusa29, 3.3.1]. Recall that an involutive vector space is projective if, when viewed as a $k[C_2]$-module, it is a direct summand of a free module.

Theorem 9.1 Let $k$ be a field of characteristic zero. Let $A$ be a projective involutive $k$-algebra and let $M$ be an involutive $A$-bimodule. There exists an isomorphism of graded $k$-modules

\[ HR_{{\star}}^+(A,M)\cong \mathrm{Tor}_{{\star}}^{A^{ie}}\left(A,M\right). \]

10. Reflexive structure in real topological Hochschild homology

The structure of the reflexive crossed simplicial group arises implicitly in the literature on real topological Hochschild homology, although the category $\Delta R$ very rarely explicitly arises. In this expository section we explain how certain constructions in real topological Hochschild homology can be stated in terms of the reflexive crossed simplicial group.

We begin by recalling the definition of a real simplicial object from [Reference Dotto18, 1.4.1], [Reference Dotto, Moi, Patchkoria and Reeh21, 1.2], [Reference Høgenhaven45, 1.1], [Reference Angelini-Knoll, Gerhardt and Hill1, 2.3].