On the geometric fixed-points of real topological cyclic homology

We give a formula for the geometric fixed-points spectrum of the real topological cyclic homology of a bounded below ring spectrum, as an equaliser of two maps between tensor products of modules over the norm. We then use this formula to carry out computations in the fundamental examples of spherical group-rings, perfect $\mathbb{F}_p$-algebras, and $2$-torsion free rings with perfect modulo $2$ reduction. Our calculations agree with the normal L-theory spectrum in the cases where the latter is known, as conjectured by Nikolaus.


Introduction
The trace methods were introduced in [BHM93] as an effective way of studying the algebraic K-theory of suitable rings, by mapping it to more computable invariants which are typically constructed from the topological Hochschild homology spectrum THH and its cyclic action. One particularly successful invariant is the topological cyclic homology TC defined from suitably derived fixed-points of the cyclic structure of THH. If one tries to extend these methods to the algebraic K-theory of forms or to cobordisms of forms (that is Grothendieck-Witt and L-theory respectively) one discovers the real topological Hochschild homology THR, a dihedral refinement of THH, and the real topological cyclic homology TCR.
The real topological Hochschild homology THR(A) of a ring (or ring spectrum) with antiinvolution A has been introduced in unpublished notes of Hesselholt and Madsen. It is an O(2)equivariant spectrum whose underlying S 1 -spectrum is THH(A), and where the subgroup Z 2 of O(2) generated by a reflection acts via a combination of a reflection of the circle and the antiinvolution of A. The Z 2-equivariant homotopy type of THR(A) has been studied extensively: In [Høg16] it has been computed for spherical group-rings in terms of free loop spaces. In [DMPR21] we studied some of its fundamental structural properties and we computed it for F p and Z. In [HKY + 20] it has been related to equivariant factorisation homology and calculated for equivariant Thom spectra. In [HW21] the Hopf algebroid structure on the homotopy groups of THR(F 2 ) is described and used to give an independent proof of the Segal conjecture for the group of order 2. A key feature which makes these calculations accessible is the description of THR(A) as a derived tensor product, and in particular of its Z 2-geometric fixed-points spectrum as the derived tensor product of module spectra where A acts on A φ Z 2 on the left and on the right by the "Frobenius actions", described informally respectively by the formulas a ⋅ x = axw(a) and x ⋅ a = w(a)xa, and w is the antiinvolution of A.
The real topological cyclic homology TCR(A; p), for a prime number p, is a Z 2-equivariant spectrum introduced in [Høg16], whose underlying spectrum is the p-typical topological cyclic homology TC(A; p). Its construction is analogous to the classical definition of TC(A; p) of [BHM93], by taking the homotopy limit over certain maps R, F ∶ THR(A) C p n+1 → THR(A) C p n 2020 Mathematics Subject Classification: Primary 19D55, 11E70; Secondary 55P91, 13F35 in the category of Z 2-spectra, thus involving the equivariant structure of THR with respect to the finite dihedral subgroups D p n of O(2) and the Weyl actions of Z 2 ≅ D p n C p n on THR(A) C p n . Alternatively, Quigley and Shah give in [QS19] a construction of TCR for bounded below spectra as an equaliser analogous to Nikolaus and Scholze's definition of TC of [NS18]. The goal of this paper is to describe the geometric fixed-points TCR(A; p) φ Z 2 in terms of derived smash products in the same spirit of the formula for THR(A) φ Z 2 above, and use this description to carry out calculations in some fundamental examples.
A ring spectrum with anti-involution A is canonically a left and a right module over the Hill-Hopkins-Ravenel norm (see [HHR16]) of its underlying spectrum, by means of maps described informally respectively by sending a ⊗ b ⊗ x and x ⊗ a ⊗ b to axw(b) and w(a)xb. By taking Z 2-geometric fixed-points these give the left and right Frobenius A-module structure on A φ Z 2 mentioned above. By applying the monoidal functor N C2 e , we also obtain a left and a right N C2 e A-module structure on N C2 e (A φ Z 2 ). Here we are making the point of distinguishing between the subgroups Z 2 and C 2 of O(2), generated respectively by a reflection and the rotation of order two.
Theorem A. Let A be a ring spectrum with anti-involution, and suppose that the underlying spectrum and the Z 2-fixed-points of A are bounded below. Then for every odd prime p there is a natural equivalence of spectra For the prime 2, there is a natural equivalence with the homotopy equaliser where f forgets the fixed-points and r maps to the C 2 -geometric fixed-points, followed by the We prove this Theorem in §2.1 for odd primes, and in §2.2 for the prime 2. Our proof proceeds by identifying the Z 2-geometric fixed-points of THR(A) C p n inductively over n ≥ 0, together with the structure maps R, F ∶ THR(A) C p n+1 → THR(A) C p n . The key ingredient is a result of [LW12] which gives a certain pushout decomposition of the universal space of the family of reflections of O(2). We suspect that our theorem could also be proved starting from the description of TCR for bounded below spectra given in [QS19] using the same technique. Our proof of Theorem A is given more generally for bounded-below real p-cyclotomic spectra, see Theorem 2.14.
We use the formula of Theorem A to compute the geometric fixed-points of TCR in some fundamental examples, starting with spherical group-rings. Every topological monoid M with anti-involution w∶ M op → M has an underlying Z 2-equivariant homotopy type. The genuine Z 2-equivariant suspension spectrum S[M ] ∶= Σ ∞ + M of the latter gets canonically the structure of a ring-spectrum with anti-involution. The monoid M acts on its fixed-points subspace M Z 2 by m ⋅ x = mxw(m) and x ⋅ m = w(m)xm, and the corresponding 2-sided bar construction admits a "Frobenius endomorphism" defined simplicially by ψ(x, m 1 , . . . , m n , y) = (x, m 1 , . . . , m n , ym n . . . m 1 xw(m 1 ) . . . w(m n )y). It also has an involution which reverses the order of the factors of the bar construction and applies w to each component. The following is analogous to the classical description of TC of spherical group-rings of [BHM93] and [NS18,Theorem IV.3.6].
Theorem B. Let M be a well-pointed topological monoid with anti-involution. Then there is a pullback square where the right vertical map is the transfer. In particular for M = * there is an equivalence where RP ∞ −1 is the homotopy fibre of the transfer trf∶ Σ ∞ + RP ∞ = S hC2 → S.
Let us point out that the pullback square of Theorem B does not require any 2-completion. In particular the calculation of TCR(S; 2) φ Z 2 of Theorem B confirms the expected equivariant homotopy type of TCR(S; 2), that appeared in unpublished work of Høgenhaven [Høg16]. We prove this theorem in section §3.1, and we calculate this pullback in §3.2 in the case where M is a discrete group with various assumptions on the involution and the 2-torsion. In particular we determine it fully for M = Z with the minus involution and with the trivial involution, and for M = C 2 . In Corollary 3.4 for every pointed Z 2-space X, we consider the special case of the equivariant loop space M = Ω σ X = Map * (S σ , X), where S σ is the sign representation sphere, and Z 2 acts on the loop space by conjugation. We use Theorem B to describe the cofibre of an assembly map Σ ∞ + (X Z 2 ) ⊗ (S ⊕ RP ∞ −1 ) → TCR(S[Ω σ X]; 2) φ Z 2 in terms of the cofibre of the diagonal ∆∶ X Z 2 → X Z 2 × X X Z 2 , where the homotopy pullback is along the fixed-points inclusions. In particular if the involution on X is trivial these cofibres vanish and we obtain a splitting This calculation shows that TCR(S[Ω σ X]; 2) φ Z 2 is equivalent to Weiss and Williams' hyperquadratic L-theory of the pointed space X, which satisfies the same decomposition by [WW14,Theorem 4.3, Corollary 4.4].
There is in fact a deeper relationship between TCR and L-theory, especially in view of the following result, which we explain in more details at the end of the introduction. Given a discrete commutative ring A, and we write TCR(A; 2) for the TCR spectrum of the Z 2-equivariant Eilenberg-MacLane commutative ring spectrum of A equipped with the trivial involution.
Theorem C. Let k a perfect field of characteristic 2, and Z the ring of integers. There are equivalences of spectra TCR(k; 2) φ Z 2 ≃ ⊕ n≥0 (Σ 2n−1 Hk ⟨x + x 2 x ∈ k⟩ ⊕ Σ 2n H F 2 ) In the case of perfect fields, we are in fact able to calculate the full Z 2-equivariant homotopy type of TCR(k; 2): In §4.2 we use the description of TRR(k; 2) φ Z 2 from Theorem 2.7 to show that TRR(k; 2) is the Eilenberg-MacLane spectrum of the constant Mackey functor on the ring of 2-typical Witt vectors W (k; 2), where F corresponds to its Frobenius. In particular in Theorem 4.8 we show that π 0 THR(k; 2) D 2 n is isomorphic to the (n + 1)-truncated 2-typical Witt vectors of k (this is true for all commutative rings at odd primes by [DPJM22,Theorem C], but it fails in general at the prime 2, see Remark 4.9). We are then able to conclude the following: Theorem D. For every perfect field k of characteristic 2, there is an equivalence of Z 2equivariant spectra TCR(k; 2) ≃ HZ 2 ⊕ Σ −1 Hcoker(1 − F ), where F ∶ W (k; 2) → W (k; 2) is the 2-typical Witt vector Frobenius and the underline denotes the constant Mackey functor.
A similar decomposition holds for odd primes by a much easier argument, see Proposition 4.2. Finally, we prove a flat base-change result for TCR φ Z 2 , showing that if f ∶ A → B is a flat map of discrete commutative rings such that the geometric fixed-points of B are base-changed along f from those of A, then TCR(B; 2) φ Z 2 is "almost" base-changed from TCR(A; 2) φ Z 2 , up to some care with the different module structures on HA φ Z 2 (see Corollary 5.10 for the precise statement). This allows us to extend the calculations above as follows.
Theorem E. For every perfect F 2 -algebra A, there is an equivalence of spectra For every ring B with no 2-torsion and perfect modulo 2 reduction, TCR(B; 2) φ Z 2 is a wedge of Eilenberg-MacLane spectra with homotopy groups for all l ≥ 0, and zero for n ≤ −2.

Real TC and L-theory
The relationship between TC and L-theory was originally observed by Weiss and Williams and studied by Weiss and Rognes. They were investigating whether, under certain conditions on a ring spectrum with anti-involution A, the quadratic L-theory L q (A) is equivalent to the Z 2-Tate construction of the fibre of the trace map K(A) → TC(A) after 2-completion. Nikolaus then formulated an uncompleted version of this statement, conjecturing that TCR(A; 2) φ Z 2 should be equivalent to the genuine normal L-theory of A, defined as the cofibre of the canonical symmetrisation map. Here Mod ω A is the ∞-category of compact A-modules, and Ϙ A ∶ (Mod ω A ) op → Sp is a certain Poincaré structure in the sense of Lurie's formalism of L-theory, which is defined using the Frobenius module structure of A φ Z 2 (see [CDH + 23, 3.2.6 and 3.2.10] for the details). A proof of this conjecture will appear in work of Harpaz, Nikolaus, and Shah [HNS21].
By construction, is an equivalence. One can then see that Nikolaus' conjecture implies the original conjecture of Weiss and Williams provided the fibre of the trace map becomes Borel-complete after 2-completion.
In the case of spherical group-rings, L n (S[Ω σ X]) is the hyperquadratic L-theory of [WW14] by [CDH + 20a, Corollary 4.6.1], and as mentioned above it is equivalent to TCR(S[Ω σ X]; 2) φ Z 2 by [WW14, Theorem 4.3, Corollary 4.4] and Corollary 3.4. The normal L-theory L n (k) is also well-understood if k is a perfect field of characteristic 2, for example by work of Kato and Ranicki, and its homotopy groups agree with the ones of the geometric fixed-points of TCR of Theorem C (see Remark 4.5 for more details on the description of these L-groups). Finally, L n (Z) is calculated by Taylor and Williams in [TW79] (see also [HLN21, Corollary 3.9 and 6.2]) and agrees with our calculation of TCR(Z; 2) φ Z 2 of Theorem C. We are not aware of a flat base-change type of result analogous to Corollary 5.10 for these normal L-spectra, nor if they have been computed for all the rings of Theorem E.

Equivariant spectra
Let G be a compact Lie group. In this paper we will be interested in the case where G is the orthogonal group O(2), or one of its subgroups. We write Sp G for the stable model category of orthogonal spectra with an action of G, equipped with the flat model structure of [Sto11]. This is a model for the homotopy theory of genuine G-spectra. We recall that the weak equivalences are the π * -isomorphism, where π * is the equivariant homotopy groups Mackey-functor.
We denote by ⊗ the derived smash product of G-spectra and of modules in G-spectra, which can be obtained by applying the smash product to a flat replacement of the orthogonal G-spectra (that is to a cofibrant replacement in the flat model structure). We also denote by ⊗ the tensor of a pointed G-space Z and a G-spectrum X: For every closed subgroup H ≤ G, we denote the genuine fixed-points (which is the strict fixedpoints of a fibrant replacement) and the geometric fixed-points functors respectively by We recall that the geometric fixedpoints functor can be defined from the genuine fixed-points functor as called the isotropy separation sequence. We will be particularly interested in the case where G = O(2) and H = C p is the cyclic subgroup of O(2) of rotations of order p, for some prime p. Then for any O(2)-spectrum X, we have a fibre sequence of O(2) C p -spectra and hence for any n ≥ 0, there is a fibre sequence of O(2) C p n+1 -spectra By choosing the reflection over the real coordinate axis, we can identify O(2) with the semi-direct product Z 2 ⋉ S 1 . Here the nontrivial element τ of Z 2 corresponds to the latter reflection. In particular, one has the dihedral subgroups D p n = Z 2 ⋉ C p n ≤ Z 2 ⋉ S 1 = G. If we restrict the family (⊉ C p ) to the dihedral group D p n+1 for n ≥ 0, then it becomes the family R consisting of the trivial group and of those subgroups generated by the reflections in D p n+1 . Hence by restricting to D p n+1 C p , we get a fibre sequence of D p n+1 C p -spectra and by taking fixed-points a fibre sequence of D p n+1 C p n+1 = Z 2-spectra (1) We will abuse notation and always write R for the family of reflections in D p n+1 , for different p and n. Although these families are different, their classifying spaces ER are always modelled by the restriction to the appropriate dihedral group of the O(2)-space defined by the unit sphere S(C ∞ ).
In what follows we will always consider homotopy limits and homotopy colimits of spaces and spectra and will just refer to them as limits and colimits.

Ring spectra with anti-involution and real topological Hochschild homology
We give a short recollection of the construction and main properties of the dihedral structure on topological Hochschild homology, mainly from [BHM93] and [DMPR21].
We recall that a ring spectrum with anti-involution is an orthogonal ring spectrum A equipped with a morphism of orthogonal ring spectra w∶ A op → A such that w 2 = id. We endow A with the genuine Z 2-equivariant homotopy type defined by w. More precisely, a morphism of ring spectra f ∶ A → B commuting with the involutions w is an equivalence if it is a genuine Z 2-equivariant equivalence in the category Sp Z 2 of orthogonal Z 2-spectra (see [DMPR21,A1] for a model structure on their category).
The cyclic nerve of A in the category of orthogonal spectra inherits a levelwise involution, which acts on A⊗A ⊗n by applying in each factor w, fixes the first tensor factor, and reverses the order of the remaining n factors. This involution, together with the levelwise C n+1 -actions which rotate the tensor factors, defines a dihedral spectrum in the sense of [FL91, S 1.5, Example 5] and [Lod87] that we denote by N di A. Its geometric realisation is then an orthogonal spectrum with O(2)-action ([FL91, Theorem 5.3] and [Lod87, Proposition 3.10]), which we regard as a genuine O(2)-equivariant spectrum.
In [DMPR21] we studied the Z 2-equivariant homotopy type of THR(A), where Z 2 is the subgroup of O(2) generated by the reflection over the x-axis. In particular we provided an equivalence of Z 2-spectra (under the standing assumption that A is flat) where ⊗ N Z 2 e A denotes the derived smash product in the category of modules over the Hill-Hopkins-Ravenel norm construction N Z 2 e A of the underlying ring spectrum A of [HHR16]. The norm acts on A respectively on the left and on the right by A, A) is the two-sided bar construction of these actions, which computes the derived smash product. We then deduced an equivalence of spectra where A acts on A φ Z 2 via the geometric fixed-points of the actions of N Z 2 e A on A, using the diagonal equivalence (N Z 2 e A) φ Z 2 ≃ A. We refer to these actions as the Frobenius module structures of A φ Z 2 .
The present paper will focus on the equivariant homotopy type of THR(A) with respect to the finite dihedral subgroups of O(2). We now give a recollection of materials on dihedral objects and simplicial subdivisions, which we use to model the equivariant homotopy type of THR(A) with respect to the finite dihedral subgroups simplicially. We recall that a dihedral orthogonal spectrum is a simplicial orthogonal spectrum X • ∶ ∆ op → Sp whose n-simplicies X n are equipped with an action of the dihedral group D n+1 = Z 2 ⋉ C n+1 , which is suitably compatible with the simplicial structure [FL91,Proposition 3.4]. The geometric realisation of X • has an induced action of O(2) = Z 2 ⋉ S 1 by [FL91,Theorem 5.3]. The action of the reflection generating Z 2 on X • is induced by the maps where w is the action of the generator of Z 2 ≤ D n+1 on the n-simplicies, and ω n sends (t 0 , . . . , t n ) ∈ ∆ n to (t n , . . . , t 0 ) [FL91, Lemma 5.6(ii)].
be the r-fold subdivision of X. Let g n be the generator of C n+1 and w the generator of Z 2. The action of g n+1 rn+r−1 on the n-simplicies (sd r X • ) n = X rn+r−1 defines a simplicial action of C r on sd r X • , and there is a C r -equivariant isomorphism where δ r sends t ∈ ∆ n to (t, t, . . . , t) r ∈ ∆ rn+r−1 [BHM93,§1]. This isomorphism is moreover Z 2-equivariant, where the action of Z 2 on the left-hand side is defined from the maps w ⊗ ω n above as for X • . Let us finally make this Z 2-action simplicial.
Let sd e ∶ ∆ op → ∆ op be the functor that sends For example, Y • could be a dihedral object X • where w n acts by the action of the generator of Z 2 as a subgroup of D n+1 , or Y • = sd r X • where w n acts as the generator of Z 2 as a subgroup of D r(n+1) . The geometric realisation of Y • has a Z 2-action defined as above from the maps w n ⊗ ω n . We now let be the corresponding subdivision. The action of w 2n+1 on the n-simplicies (sd e Y • ) n = Y 2n+1 defines a simplicial Z 2-action on sd e Y • . There is an isomorphism where δ e sends t ∈ ∆ n to (t, ω n (t)) 2 ∈ ∆ 2n+1 , and this isomorphism is clearly Z 2-equivariant. Combining these subdivisions we obtain a D r -equivariant isomorphism sd e sd r X • ≅ X • for every dihedral orthogonal spectrum X • , and in particular an isomorphism of genuine orthogonal D r -spectra for every r ≥ 1.
Remark 1.1. By the latest isomorphism it follows that THR sends equivalences of ring spectra with anti-involution to equivalences of genuine D r -spectra, for every r ≥ 1. Indeed, the nsimplicies of sd e sd r N di A are the orthogonal spectrum where C r acts cyclically on r and the generator of Z 2 acts as described above (and we recall that the tensor product indicates the smash product of a flat replacement). This indexed smash product sends an equivalence of orthogonal spectra to a genuine D r -equivalence by [BDS16, Theorem 3.2.16] (see also [HHR16,Proposition B.209]), and thus its realisation is also a genuine D r -equivalence (since sd e sd r N di A is a good simplicial spectrum by the argument of [DMPR21, Lemma 2.14]).
We can now refine this equivalence to an equivariant equivalence with respect to the action of the Weyl group. The normaliser of Z 2 inside of O(2) is Z 2 × C 2 , where C 2 is generated by the rotation of order 2, and therefore the Weyl group of Z 2 inside of O(2) is isomorphic to C 2 . In particular THR(A) φ Z 2 is a genuine C 2 -spectrum, that we now describe in terms of derived smash products.
Lemma 1.2. There is an equivalence of C 2 -spectra where A is regarded as a C 2 -spectrum via the isomorphism Z 2 ≅ C 2 , the norm N C2 e A acts on A by the right action defined above, and on N C2 e (A φ Z 2 ) by applying the monoidal functor N C2 e to the left Frobenius action of A on A φ Z 2 .
Proof. The equivalence of genuine D 4 = C 2 × Z 2-equivariant spectra THR(A) ≅ sd e sd 2 N di A defined above gives rise to an equivalence of C 2 -spectra where C 2 acts on the third term by reversing the order of the smash products, and the isomorphism rearranges the factors by pairing the factors which are swapped.

Real cyclotomic spectra and real topological cyclic homology
We now review the definitions of the main objects of study of the paper. These are completely analogous to the classical definitions surrounding topological cyclic homology of [BHM93], and are carried out by carefully lifting all the constructions to the category of Z 2-equivariant spectra. These constructions were laid out in [Høg16] using Bökstedt's model for real topological Hochschild homology, and we recast them here for the model of THR above. The two approaches are equivalent by the comparison results of [DMPR21] and [DMP + 19]. Definition 1.3. Let p be a prime. A real p-cyclotomic spectrum is an O(2)-spectrum T ∈ Sp O(2) equipped with a map of O(2)-spectra where O(2) acts on the left-hand side by restriction along the root isomorphism O(2) → O(2) C p , and which is a D p n -equivalence for all n ≥ 0.
The prime example of a real p-cyclotomic spectrum (for all prime p) is the real topological Hochschild homology spectrum THR(A) of a ring-spectrum with anti-involution A. The cyclotomic structure maps are in fact isomorphisms, defined on the dihedral bar construction from the diagonal isomorphisms A ≅ (A ⊗p ) φCp (see e.g. [ABG + 18], or [DMP + 19, §5], and we remind that this is an isomorphism since A is assumed to be flat). In particular, they induce a S 1 C p -equivalence and a Z 2-equivalence on realisations, and thus an O(2)-equivalence (see [DPJM22,§3.3]). For every real p-cyclotomic spectrum T , the isotropy separation sequence (1) defines fibre sequences of Z 2-spectra for every n ≥ 0, and the composite of the right-hand arrow and the equivalence is denoted by R∶ T C p n+1 → T C p n . Since the cyclotomic structure map is O(2)-equivariant, using appropriate root isomorphisms, we see that R is O(2)-equivariant.
Definition 1.4. Let T be a real p-cyclotomic spectrum. For every integer n ≥ 0, we let TRR n+1 (T ; p) be the Z 2-spectrum where Z 2 is identified with the subgroup of O(2) C p n generated by the reflection of the x-axes, and If A is a ring spectrum with anti-involution, we write The inclusion of subgroups C p n−1 ≤ C p n defines a map F ∶ TRR n+1 (T ; p) → TRR n (T ; p), which is equivariant for the Weyl actions and thus in particular Z 2-equivariant. It also commutes with the map R since R is O(2)-equivariant and therefore induces a map of Z 2-spectra F ∶ TRR(T ; p) → TRR(T ; p) by passing to the limit, whose underlying map is the Frobenius of [BHM93].
Definition 1.5. Let T be a real p-cyclotomic spectrum. The real topological cyclic homology of T is the Z 2-spectrum defined as the equaliser If A is a ring spectrum with anti-involution, we let the real topological cyclic homology of A be the Z 2-spectrum TCR(A; p) = eq TRR(A; p) As in the classical definition of TC, since R and F commute one can alternatively define The underlying spectrum of TCR(T ; p) is by construction the topological cyclic homology spectrum TC(T ; p) of [BHM93] (see also [DMP + 19, Theorem 1.3]). The focus of the paper is to understand the Z 2-equivariant homotopy type of TCR(T ; p), and in particular its geometric fixed-points spectrum.

The Geometric fixed-points of TCR
The aim of this section is to give a simple formula for the Z 2-geometric fixed-points of TCR(A; p), when A and its Z 2-fixed-points are bounded below. This object turns out to be interesting only for the prime p = 2, but we will start with the easier case of odd primes.

The odd primary case
In the odd primary case the geometric fixed-points of TRR admit a very simple description, as they split as a product.
Theorem 2.1. For any odd prime p, real p-cyclotomic spectrum T , and n ≥ 1, there is a natural equivalence under which the maps F, R∶ TRR n+1 (T ; p) φ Z 2 → TRR n (T ; p) φ Z 2 respectively project off the first and the (n + 1)-st summand.
Before diving into the proof, we observe that if T = THR(A) is the real topological Hochschild homology of a ring spectrum with anti-involution A, we have an explicit description of the geometric fixed-points In particular if A is the Eilenberg-MacLane ring spectrum of a discrete ring with anti-involution and 1 2 ∈ A, then we have that THR(A) φ Z 2 = 0, and we obtain the following. Corollary 2.2. If A is a discrete ring with anti-involution and 1 2 ∈ A, then for every odd prime p.
The crucial combinatorial ingredient that makes the odd-primary case so simple compared to the prime 2 is that for p odd, any two reflections in D p n are conjugate, and the Weyl group of a reflection is trivial. Applying [LW12, Corollary 2.8] to the trivial family {1} ⊂ R, one gets a pushout square of Z 2-spaces This pushout is the main ingredient for establishing the following result, of which Theorem 2.1 is an immediate consequence.
Proposition 2.3. Let T be a real p-cyclotomic spectrum with p odd. Then for every n ≥ 1, the square is a pullback whose horizontal fibres are equivalent to T φ Z 2 . Here we interpret TRR 0 (T ; p) = 0.
Proof. Let us start by identifying the fibre of the map R. For any group G and G-space E, we writeẼ for the cofibre of the based map E + → S 0 which takes E to the non-basepoint of S 0 . By definition, the geometric fixed-points of the map R fits into the commutative square where the left vertical map and the bottom horizontal map are induced by the canonical maps S 0 → ED p n → ER. The left vertical map is an equivalence since its fibre is where ε∶ D p n → D p n C p n = Z 2 is the quotient map, and ε * E Z 2 ∧ ED p n + is a contractible D p n -space. By mapping the pushout square (2) with additional disjoint base points to the pushout of denotes the induction) and taking cofibres, we get a pushout of pointed D p n -spaces The fibre of R is therefore given by the spectrum By restricting the map ED p n → ER to D p n−1 , we recover the map ED p n−1 → ER. Using this and that the Frobenius map F ∶ TRR n+1 (T ; p) → TRR n (T ; p) is induced by the subgroup inclusion C p n−1 ⊂ C p n , under the equivalences above the map between the horizontal fibres identifies with the map induced by the subgroup inclusion D p n−1 ⊂ D p n . By applying the double coset formula on the source and target this map corresponds to the identity of T φ Z 2 , showing that the Frobenius on horizontal fibres is an equivalence.
We want to conclude the section with a similar splitting for TRR(T ; p) φ Z 2 , by commuting geometric fixed-points with an infinite limit. This can be done by means of the following wellknown result originally observed by Adams (see e.g. [Ada95, Section III.15.2]), and we sketch an argument for completeness. We say that a G-spectrum is bounded below if all of its fixed-points are bounded below, and in case G is infinite we also require that there is a uniform bound over all the closed subgroups of G.
Proof. Since equivariant homotopy groups commute with infinite products it suffices to show that EG ⊗ holim n X n ≃ holim n ( EG ⊗ X n ). This reduces to showing that For a fixed homotopy group only the finite skeleta of EG matter since the X i are uniformly bounded below, and using that EG is of finite type, the statement reduces to showing that S ⊗ − and G + ⊗ − commute with infinite products. The first is obvious and the second follows from the Wirthmüller isomorphism.
Corollary 2.5. Let T be a bounded below real p-cyclotomic spectrum, with p odd. Then there are natural equivalences Proof. The first equivalence follows from Lemma 2.4 and Theorem 2.1, since the tower defining TRR(T ; p) is uniformly bounded below since the spectra T C p n and T D p n are by assumption bounded below with a uniform bound over n. For the second equivalence we observe that by the same results the Frobenius of TRR(T ; p) φ Z 2 is equivalent to the map which projects off the first component. The equaliser of F and the identity is thus T φ Z 2 mapping diagonally to the infinite product.

The geometric fixed-points of TRR
In this section we give a formula for TRR n (T ; 2) φ Z 2 for any bounded below real 2-cyclotomic spectrum T . As mentioned earlier, the subgroups structure of the dihedral groups D 2 n is more involved than the one of D p n for odd p, and this makes our formula for TRR n (T ; 2) φ Z 2 more interesting. The idea of the proof is again to compute TRR n (T ; 2) φ Z 2 inductively, by finding a suitable replacement for the square of Proposition 2.3. Recall that we have chosen the reflection along the x-axis τ inside D 2 n = Z 2 ⋉ C 2 n , where Z 2 is the subgroup generated by τ . If we denote by σ n the generator of C 2 n , then σ n τ is a reflection which is not conjugate to τ . We denote the normaliser and Weyl group of Z 2 inside D 2 n C 2 n−1 respectively by The Weyl group C 2 is generated by the image of σ n , which we denote again by σ n . This group acts on the spectrum TRR n (T ; 2) φ Z 2 , for all n ≥ 1 and any real 2-cyclotomic spectrum T .
In the case n = 1 we are interested in two maps analogous to the restriction and the Frobenius. The map r is induced by the canonical map from the fixed-points to the geometric fixed-points, followed by the equivalence given by the cyclotomic structure of T , and f is the canonical map induced by the subgroup inclusion 1 ⊂ C 2 .
Example 2.6. Suppose that T = THR(A) is the real topological Hochschild homology spectrum of a ring spectrum with anti-involution A. Under the equivalence of Lemma 1.2, the identification (T φ Z 2 ) φC2 ≃ T φ Z 2 coming from the cyclotomic structure corresponds to the equivalence where the first equivalence is the monoidality of the geometric fixed-points combined with the diagonal isomorphism (N C2 e A) φC2 ≅ A, and the second is the general canonical equivalence for respectively right and left A-modules X and Y , where X is regarded as a left A-module via the anti-involution.
The maps f and r are related to F and R respectively, in the following manner. Let c be the map where ε∶ C 2 × Z 2 → Z 2 is the projection, and the arrow is induced by including families of subgroups, by noticing that ε * E Z 2 = E{1, C 2 } as universal spaces of subgroups of C 2 × Z 2. The naturality of the canonical map from fixed-points to geometric fixed-points gives canonical homotopies r ○ c ≃ R and f ○ c ≃ F . In particular, for every n ≥ 1 the iterated Frobenius map factors as Finally, we recall from Section 1.3 that the maps R and F commute since R is O(2)-equivariant.
Theorem 2.7. For every real 2-cyclotomic spectrum T and n ≥ 1, the square is a pullback of spectra, where the square commutes by the homotopies RF = F R, σ n R = Rσ n+1 , and r ○ c ≃ R. The Weyl action of σ n+1 on TRR n+1 (T ; 2) φ Z 2 is given inductively by the strictly commutative diagram Remark 2.8. By inductively unravelling the formula of 2.7 we obtain an equivalence for all n ≥ 1, where we wrote T φ ∶= T φ Z 2 for short and all the products denote fibre products over T φ , subscripts indicating along which maps we are taking the pullbacks. We can then further unravel the structure maps. The restriction map R∶ TRR n+1 (T ; 2) φ Z 2 → TRR n (T ; 2) φ Z 2 corresponds to projecting away the outer two factors for n ≥ 2, and for n = 1 to the composite The Weyl action σ n+1 ∶ TRR n+1 (T ; 2) φ Z 2 → TRR n+1 (T ; 2) φ Z 2 is induced by the map which reverses the order of the product factors. The Frobenius F ∶ TRR n+1 (T ; 2) φ Z 2 → TRR n (T ; 2) φ Z 2 is slightly more delicate to describe. For n = 1 we have already mentioned the description. For n ≥ 2 it is induced by the map defined schematically on the product by interpreted as follows. In order to map into the pullback TRR n (T ; 2) φ Z 2 we need to exhibit homotopies where the second is the canonical homotopy provided by the equivariance of f with respect to the Weyl action. Similarly, γ 0 is given by The rest of the section will be devoted to the proof of Theorem 2.7. Our proof relies on a pushout relating ED 2 n and ER which we now deduce from [LW12, Corollary 2.8].
Recall that τ and σ n are the respective generators of Z 2 and C 2 n inside D 2 n ≅ Z 2 ⋉ C 2 n , and that σ n τ is a reflection which is not conjugate to τ . We let H n be the subgroup generated by σ n τ . We denote the respective normalisers and Weyl groups by We observe that N (H n ) is abstractly isomorphic to Z 2 × C 2 but one has to be careful with this identification, since Z 2 and H n represent different conjugacy classes. By [LW12, Corollary 2.8], there is a pushout of D 2 n -spaces We observe that the classifying spaces that show up in [LW12, Corollary 2.8] at the lower left corner are indeed equivalent to EW (H n ) and EW (Z 2). This pushout square leads to the following analogue of Proposition 2.3.
Proposition 2.9. For every n ≥ 2, the Frobenius induces a commutative diagram where the rows are fibre sequences, and Remark 2.10. We note that the spectra (T φ Z 2 ) hW (Z 2) , (T φHn ) hW (Hn) and (T φHn−1 ) hW (Hn−1) are all equivalent. This is because H n−1 and Z 2 become conjugated in D 2 n , and consequently H n and H n−1 are conjugated in D 2 n+1 . It is however important to point out that H n−1 and Z 2 are not conjugated in D 2 n−1 , and this plays a role while identifying the map F on the fibres of R.
Proof of 2.9. Let us start by calculating the fibres of the horizontal maps. The pushout square (3) induces a pushout square of pointed D 2 n -spaces The map R is, just as in the proof of Proposition 2.3, given by the map induced by the right vertical map of square (4) (this part of 2.3 does not use that p is odd).
Since square (4) is a pushout, the cofibre of R is equivalent to where E ⟨Z 2⟩ and E ⟨H n ⟩ are pointed N (Z 2) and N (H n )-spaces which fixed-points S 0 only at Z 2 and H n , respectively, and have contractible fixed-points at all other subgroups. Let us now identify the first summand, the identification of the second one being similar. By the projection formula and untwisting the action on T , we see that and therefore that the first summand of the fibre of R is equivalent to The other summand is identified similarly. Now the is the projection, again by observing that this smash product also has isotropy S 0 at Z 2 and contractible isotropy on all other subgroups (using Elmendorf's theorem [Elm83]). By definition of the geometric fixed points with respect to a normal subgroup, we get that one summand of the fibre of R is where the last equivalence uses the Adams isomorphism.
Let us now identify the map F on the fibres of R. As we have just seen R∶ TRR n+1 (T ; 2) φ Z 2 → TRR n (T ; 2) φ Z 2 is equivalent to the appropriate fixed-points of the map T ⊗ ED 2 n → T ⊗ ER. For simplicity let us denote the summands of its fibre by Since ED 2 n and ED 2 n−1 are equivalent as D 2 n−1 -spaces, the diagram of Proposition 2.9 is then equivalent to the outer diagram of where the maps F are the restriction maps on fixed-points, and the vertical arrows are induced by the inclusion D 2 n−1 ⊂ D 2 n . Let us analyse the bottom left vertical equivalence. The summand (B where we used that N (H n ) ∩ D 2 n−1 = C 2 and that E ⟨H n ⟩ is trivial when restricted to C 2 . The first equivalence follows from the projection formula as in the identification of the summands of the fibre of R above. This shows that F is trivial on the second summand of the fibres. Let us now apply the double coset formula to the first summand (A where σ n is the generator of C 2 n , with N (Z 2) ∩ D 2 n−1 = N (Z 2) and σ n conjugating Z 2 and H n−1 . We therefore find that The previous proposition holds as stated only for n ≥ 2. The correct analogue for n = 1 is the following: Proposition 2.11. The map c induces id ⊕0 of horizontal fibres in the following diagram Proof. The top horizontal fibre sequence is from 2.9, and the bottom one is immediate from the isotropy separation sequence of the C 2 -spectrum T φ Z 2 and the definition of r.
In order to describe F on the fibre we observe that there is a commutative diagram of pointed where ε∶ D 2 2 = C 2 × Z 2 → Z 2 is the projection, (⊉ Z 2) and (⊉ D 2 2 ) are the families of subgroups of D 2 2 which do note contain Z 2 and D 2 2 respectively, and the arrows in the diagram are induced by the inclusions of families. By Elmendorf's theorem [Elm83], the induced map on horizontal cofibres is the projection where we have used the identifications of Proposition 2.9. By the calculations of Proposition 2.9, by smashing the square above with T and taking C 2 × Z 2-fixed-points, we obtain the diagram of the statement.
The identification of the Weyl action follows immediately from the proof of Proposition 2.9: Lemma 2.12. For any n ≥ 1, the Weyl action on fibres is described by the diagram Proof. As seen in the proof of Proposition 2.9, the fibre of R consists of two summands The first summand (A C 2 n n+1 ) φ Z 2 decomposes into the wedge of two summands according to the double cosets D 2 n D 2 n+1 N (Z 2) = {1, σ n+1 }, and the action of σ n+1 permutes these two summands.
Proof of 2.7. By iterating Proposition 2.9 and Proposition 2.11 the map F n−1 induces an equivalence between the first summands of the horizontal fibres of the diagram of Theorem 2.7. Similarly by 2.12 the maps F n−1 σ n+1 and F n−1 σ n induce an equivalence between the second summands of the horizontal fibres.
Let us now identify the Frobenius F ∶ TRR n+1 (T ; 2) φ Z 2 → TRR n (T ; 2) φ Z 2 for n ≥ 1. The identification for n = 1 follows from the pullback description and the canonical homotopy c ○ f ≃ F . For n ≥ 2, we consider the following diagram whose front and back faces are pullbacks: This diagram is a homotopy commutative cube, since its front and back faces and its arrows are equivalent to those of the outer part of the strictly commutative diagram where the maps φ are induced by the canonical map S 0 → ER. This identifies the Frobenius map. The Weyl action can be identified with a similar argument.

The geometric fixed-points of TFR and TCR
In this section, we use Theorem 2.7 to identify the Z 2-geometric fixed-points of TCR(T ; 2) for every bounded-below real 2-cyclotomic spectrum T .
It turns out that it is simpler to describe the endomorphism R on the limit taken over the Frobenius, rather than describing the Frobenius on TRR(T ; 2). For simplicity, we will again write T φ for T φ Z 2 .
Theorem 2.13. Let T be a bounded below real 2-cyclotomic spectrum. Then TFR(T ; 2) φ Z 2 is equivalent to the homotopy inverse limit n which project away the last factor, i.e. given by (x 1 , x 2 , . . . , x n+1 ) ↦ (x 1 , . . . , x n ). Under this identification, the endomorphism R∶ TFR(T ; 2) φ Z 2 → TFR(T ; 2) φ Z 2 corresponds to the map induced on limits by the projection Proof. Let us first observe that since T is bounded-below, by Lemma 2.4, TFR(T ; 2) φ Z 2 is equivalent to the homotopy inverse limit of For convenience we introduce the notation: Projecting onto the first n factors gives maps A n → B n . These maps commute with the Frobenius F on A n and the projection proj l on B n by the description of F in Remark 2.8, thus defining a morphism of towers. We will now show that this morphism is a pro-equivalence and thus induces an equivalence on homotopy inverse limits. Hence by Theorem 2.7 and Remark 2.8 we obtain the description of TFR(T ; 2) φ Z 2 . Let us define a homotopy pro-inverse B n+1 → A n , for n ≥ 2, by the map induced by (x 1 , x 2 , . . . , x n , x n+1 ) → (x 1 , x 2 , . . . , x n , x n , . . . , x 2 , x 1 ).
The identifications between the components in the pullback A n are defined exactly as in Remark 2.8 for the Frobenius, in particular where the middle homotopy is the canonical one and the last path is just σ 1 applied to the first homotopy. That this map is indeed a pro-inverse [BK72, Section III- §2], follows immediately from the description of F in Remark 2.8.
Let us now identify R on TFR(T ; 2) φ Z 2 . By the description of the map R in Remark 2.8 we see that for any n ≥ 1, the diagram Since the horizontal maps are entries of a pro-equivalence, passing to limits along F gives the desired result.
Finally we are ready to prove the main result of this section: Theorem 2.14. Let T be a bounded below real 2-cyclotomic spectrum. Then there is a natural equivalence Proof. Recall that TCR(T ; 2) φ Z 2 is equivalent to the equaliser

Now consider the commutative diagram
By Theorem 2.13 if we pass to the inverse limits horizontally and then take the equaliser we get TCR(T ; 2) φ Z 2 . Equivalently we can take equalisers in each degree vertically and then pass to the inverse limit. In general, given maps a, b∶ X → Y , the equaliser of the projections off the first and last component (and where all the products denote fibre products over Y ), is equivalent to the equaliser of a and b. Thus each vertical equaliser above is equivalent to and the induced maps are equivalences.

TCR of spherical monoid-rings
We apply the formulas of the previous section to calculate the geometric fixed points of the real topological cyclic homology of spherical monoid-rings, and in particular for the sphere spectrum. In §3.1 we give the general formula and analyse a certain assembly map, and in §3.2 we carry out some calculations for discrete groups.

TCR of spherical monoid-rings and assembly
Let M be a topological monoid with anti-involution, that is a map of monoids w∶ M op → M such that w 2 = id (e.g. M is a group and w is inversion). The Z 2-equivariant suspension spectrum of the underlying Z 2-equivariant space is then a ring spectrum with anti-involution, where the multiplication is inherited from the multiplication of M . We recall that, since S is the monoidal unit of the tensor product of spectra, there is an equivalence of O(2)-spectra where B di M is the dihedral bar construct of the monoid M with respect to the product of spaces (see [DMPR21,Proposition 5.12]). Thus from Corollary 2.5 and Lemma 1.2 we immediately obtain that for every odd prime p that is to say there is a commutative diagram where the diagonal map is the fixed-points inclusion.
Example 3.1. The typical example of a monoid with anti-involution is the signed loop space where X is a pointed Z 2-space, S σ is the sign representation sphere, and Z 2 acts on the loop space by conjugation. In this case the dihedral bar construction is equivalent to the free loop space again with the conjugation action of Z 2 (see [DMPR21,Remark 5.13]). Let us spell out the map ψ under this identification. By passing to the upper half-circle, the Z 2-fixed points of Λ σ X can be identified with those paths in X which start and end at a fixed-point, or in other words the homotopy pullback of the fixed-points inclusion along itself. Since the C 2 -action on Λ σ X is given by the 180 ○ degrees rotation followed by the involution of X pointwise, the residual C 2 -action on X Z 2 × X X Z 2 flips the direction of the path and applies the involution of X pointwise. Hence, there is an isomorphism Under this identification the fixed-points inclusion corresponds to the map which sends a path γ to the concatenation γ ⋆ w(γ), where γ is the inverse path. This is some sort of squaring operation reminiscent of the Frobenius.
We are finally able to describe the geometric fixed-points of TCR(S[M ]; 2) (notice the analogy with [BHM93] and [NS18, Theorem IV.3.6]): Theorem 3.2. Let M be a well-pointed topological monoid with anti-involution. Then there is a pullback square where the right vertical map is the transfer from orbits to fixed-points, followed by the forgetful map to the underlying spectrum. In particular for M = * there is an equivalence Proof. From the identification of THR(S[M ]) with the dihedral bar construction of M we obtain an equivalence of C 2 -spectra Thus the tom-Dieck splitting (see e.g. [Sch20, Section 6]) gives an equivalence where the maps r and f correspond respectively to the maps where r is the projection onto the first summand, and f is the map ψ on the first summand and the transfer on the second. Thus the equaliser of r and f is computed by the pullback above, and it is equivalent to TCR(S[M ]; 2) φ Z 2 by Theorem 2.1. If M = * the bottom horizontal map of the pullback square is zero, and the pullback splits as the fibre of trf and S. and by Let us now turn our attention to the case where G is a group-like topological monoid with involution, that is a topological monoid with involution G such that π 0 G is a group. In this case the canonical map is an equivalence (see [Moi20]), where B σ G is the subdivision of BG with the simplicial involution that sends (g 1 , . . . , g 2n+1 ) to (w(g 2n+1 ), . . . , w(g 1 )). The fixed-points space of X = B σ G is then the one-sided bar construction of G acting on its fixed-points set by x ⋅ g = w(g)xg. We will therefore phrase the next results in terms of signed loop spaces G = Ω σ X, where X is any pointed Z 2-space. We also note that the fixed-points subspace of G = Ω σ X is the fibre of the inclusion where α ∈ ΩX acts on a path γ from the base-point to a fixed-point by concatenation γ ⋅ α = ω(α) ⋆ γ.
Corollary 3.4. For every well-pointed Z 2-space X there is a fibre sequence If the involution of X is trivial, Q is zero and there is a natural equivalence Proof. The diagonal ∆∶ X Z 2 → X Z 2 × X X Z 2 is clearly equivariant for the Weyl C 2 -action on the pullback and the trivial action on X Z 2 . It therefore induces a commutative diagram where the bottom left map is zero since ψ is the identity on constant paths. The limit of the top row is TCR(S[Ω σ X]; 2) φ Z 2 by Theorem 3.2 and Example 3.1 the limit of the bottom row is (Σ ∞ + X Z 2 ) ⊗ (S ⊕ RP ∞ −1 ). By taking cofibres vertically we obtain the fibre sequence of the statement.
If the involution on X is trivial the diagonal map ∆∶ X → X × X X is an equivalence and thus Q is trivial.
If the involution of X is not trivial the cofibre Q need not be zero, as illustrated in the following example.
Example 3.5. Suppose that X is a pointed space, and let us consider the pointed free Z 2space X b = X ∧ E Z 2 + . Since its fixed-points are contractible, the fixed-points of G since this is the space of paths from the base-point to a fixed-point of X b . In this case B(G Z 2 , G, G Z 2 ) ≃ B(ΩX, ΩX, ΩX) ≃ ΩX, and the map ψ∶ ΩX → ΩX sends a loop γ to γ ⋆ γ and it is therefore null. By Theorem 3.2 there is a pullback where C 2 acts on Ω σ X by the loop inversion. There is therefore a splitting In this case the map ∆ of Corollary 3.4 is easily seen to split, and the homotopy orbits summand corresponds to the summand Q.
Example 3.6. Suppose that the involution of G is inversion w = (−) −1 ∶ G op → G. Then the fixed-points space of G consists of the set of elements of order 2. If G has no 2-torsion we are in the situation of Corollary 3.4 where G Z 2 = 1 and (B σ G) Z 2 ≃ BG, and For example let us consider the spherical Laurent polynomial ring S[t, t −1 ] ∶= S[Z], where the involution acts by inversion in Z, i.e. swaps t and t −1 . Then Now suppose that G is a discrete group with a general anti-involution w∶ G op → G. The bar construction B(G Z 2 , G, G Z 2 ) is the nerve of a groupoid, and therefore after a choice of representatives for its isomorphism classes it decomposes as where the equivalence relation identifies (w(g)xg, y) with (x, gyw(g)), and the automorphism group of (x, y) ∈ G Z 2 × G Z 2 is the subgroup Aut(x, y) = {g ∈ G w(g)xg = x , gyw(g) = y}.
Let us now determine the map ψ and the Weyl action, so that all the ingredients of Theorem 3.2 are in place. In particular the homotopy orbits of B(G Z 2 , G, G Z 2 ) for the Weyl action can be computed using this strict action of τ .
Proof. The description of ψ is immediate from the formula before Example 3.1. The description of the action of the generator follows from the well-known fact that if G is a groupoid with strict duality, that is a functor w∶ G op → G such that w 2 = id, then the Z 2-actions on BG defined respectively by the levelwise duality together with inverting the order of the simplex coordinates, as in §1.2, and the one defined by the endofunctor are homotopy coherently equivalent. We were not able to track down a proof, so we include an argument for the reader's convenience. After applying the subdivision functor sd e from §1.2 to the nerve of G, the two actions are respectively equivalent to the simplicial actions defined levelwise on sd e N G by w and w ○ (−) −1 . The subdivided nerve sd e N G is isomorphic to the nerve of the twisted arrow category of G, and the two actions correspond respectively to the ones induced by the (covariant) endofunctors w and w ○ (−) −1 , defined on the objects of the twisted arrow category respectively by → w(y)).
Let us regard these actions as functors Z 2 → Gpd to the category of groupoids, where both send the unique object of Z 2 to the twisted arrow category of G. Then the diagram w(y) exhibits the 2-cells of a pseudonatural isomorphism on the identity transformation between the functors Z 2 → Gpd. Thus the two actions on the geometric realisations are homotopy coherently equivalent.
With these formulas at hand one should in principle be able to determine the pullback of Theorem 3.2, as its maps consist of products of diagonals and transfers Σ ∞ + RP ∞ → S. The combinatorics of which components are hit by the diagonals are complicated in this generality, but we compute them fully in the following special cases.
Example 3.8. Suppose that the order-two elements of G are included in the centre of G, and that the involution on G is inversion. This is exactly the situation where the action of G on G Z 2 is trivial. It follows that G Z 2 = G 2 consists of the elements of order 2, and The map ψ sends the component (x, y) to the component is (x, x) via the identity of BG, and the involution freely permutes the components indexed by pairs (x, y) with x ≠ y, and is trivial on the components (x, x). There is therefore a splitting where P is the pullback where ∆ c ⊂ G 2 × G 2 is the complement of the diagonal with the involution that flips the factors, and q is the sum of the maps that send the component [x ≠ y] respectively to the components x and y.
For example for G = Z with the minus involution we recover the calculation for S[t, t −1 ] of the example above, since in this case G 2 = 1. On the other hand for G = C 2 the map q is the diagonal, and P is the pullback of the transfer along itself, and Here we also see that TCR(S[C 2 ]; 2) φ Z 2 splits off an RP ∞ −1 ⊗ Σ ∞ + BC 2 -summand, since the pullback of the two transfers splits as RP ∞ −1 ⊕ Σ ∞ + RP ∞ , but this splitting is however non-canonical.
Example 3.9. Now suppose that G is abelian and endowed with the trivial involution. Then G Z 2 = G with left and right G-actions g ⋅ x ∶= 2g + x. The components of the two-sided bar construction are described by a bijection which sends [x, y] to (x+y, [y]). Under this equivalence the C 2 -action sends (x, z) to (x, [x]+z), and ψ to ψ(x, z) = (2x, [x] + z).
The C 2 -fixed-points set of G × G 2 is therefore the set of pairs of the form (2g, x), and G × G 2 decomposes C 2 -equivariantly as If we assume additionally that G has no 2-torsion, then the fundamental groups of the two-sided bar construction vanish since the corresponding groupoid has only trivial automorphisms. The pullback diagram describing TCR(S[G]; 2) φ Z 2 then takes the form where ∆ sends the component of an orbit [g, x] with g ∉ 2G diagonally to the components (g, x) and (g, [g] + x).
Let us now identify this pullback under the additional assumption that G does not have elements infinitely divisible by 2, that is for any 0 ≠ g ∈ G there exists n ∈ N, such that g = 2 n x does not have a solution. Under this assumption we can easily compute the cofibre of id −Σ ∞ + ψ. Indeed, from the commutative diagram we see that the lower horizontal map induced on cofibres is an equivalence. Hence the cofibre of id −Σ ∞ + ψ is equivalent to the cofibre of its restriction Since ψ(0, z) = (0, z), we see that the zero map 0∶ (G 2) + ⊗ S → (G 2) + ⊗ S splits off from the given map and hence the cofibre contains the summand (G 2) + ⊗ (S ⊕ S 1 ). Let us now compute the cofibre of By the non-divisibility condition, this morphism induces injection on homotopy groups. By inspecting its cokernel on homotopy groups, we see that the cofibre is ((2G ∖ 4G) × G 2) + ⊗ S. All in all we get a cofibre sequence where ζ includes (0 × (G 2)) + ⊗ S into (G 2) + ⊗ S, sends the (2 n h × G 2) + ⊗ S-summand via the identity to the summand (2h × G 2) + ⊗ S for any h ∈ G ∖ 2G and n ≥ 1, and sends the summand .
From the pullback square above we find that TCR(S[G]; 2) φ Z 2 is the fibre of the map which is given by the wedge where P is the pullback By using that any non-zero element g ∈ G can be uniquely written as 2 n γ, where n is a nonnegative integer and γ ∈ G ∖ 2G, we can write P as We note that To summarise, for every abelian G with trivial involution, no 2-torsion, and no elements infinitely divisible by 2 In particular the group G = Z with the trivial involution gives rise to the spherical Laurent polynomials S[t, t −1 ] ∶= S[Z] with the involution which acts trivially on the generators, and where we took the liberty of enumerating the summands non-canonically.

TCR of perfect fields
In [HM97], Hesselholt and Madsen identified the p-typical topological cyclic homology spectrum TC(k; p) of a perfect field k of characteristic p as the sum where F ∶ W (k; p) → W (k; p) is the Frobenius homomorphism of the ring W (k; p) of p-typical Witt vectors. Their calculation relies on the fact that the ring π 0 THR(A) C p n is isomorphic to the ring W n+1 (A; p) of (n + 1)-truncated p-typical Witt vectors, which holds for every commutative ring A (see [HM97,Theorem F]). The situation for π 0 THR(A) D p n is not completely analogous, and requires particular care. We start by recalling from [DMPR21, Corollary 5.2] that, for every commutative ring with involution A, there is an isomorphism of rings where A Z 2 is the subring of invariants of A, and the quotient is by the ideal generated by the elements of the form 1 ⊗ aa − aa ⊗ 1 for some a ∈ A (here we use that a + a = (a + 1)(a + 1) − aa − 1 to simplify the second relation of [DMPR21,5.2], so that in particular 2b The restriction map π 0 (THR(A) Z 2 ) → π 0 THR(A) then corresponds to the multiplication map where the second map is the inclusion. For perfect fields, the map µ induces an isomorphism A Z 2 ⊗ N A Z 2 ≅ A Z 2 , and the same is true in the following cases: Remark 4.1. i) Let us start by noticing that for every additive generator and therefore all the elements of the kernel of µ are 2-torsion (where the second equality follows from the identity above). Thus µ is an isomorphism when A is 2-torsion free, for example for fields of odd characteristic. ii) There is a section for µ∶ A Z 2 ⊗ N A Z 2 → A Z 2 , that sends a to a ⊗ 1. Therefore µ is always surjective, and it is an isomorphism if and only if this section is itself surjective. iii) If the multiplication map A Z 2 ⊗ A Z 2 → A Z 2 is an isomorphism, for example for A = Z n for any integer n, then so is µ. iv) If the involution of A is trivial and the modulo 2 reduction of A is semi-perfect (that is the mod 2 Frobenius is surjective), then every element a ∈ A can be written as a = c 2 + 2d for some c, d ∈ A. Then we can write a generator of A ⊗ N A as which shows that the section A → A ⊗ N A is surjective. This example covers the case of perfect fields of characteristic 2. v) If the involution of A is not trivial, a similar argument shows that the section A Z 2 → A Z 2 ⊗ N A Z 2 is surjective if every element a ∈ A Z 2 can be written as a = cc + d + d for some c, d ∈ A, or in other words if the composite is surjective, where N (a) = aa and tran(a) = a + a. vi) Suppose that there exists an element e ∈ A with the property that e + w(e) = 1, for example if 2 ∈ A is a unit. By Frobenius reciprocity, this is equivalent to the surjectivity of tran∶ A → A Z 2 , since any element x ∈ A Z 2 can be written as (explicitly, x = ex + w(ex)). Thus this condition is equivalent to the vanishing of HA φ Z 2 . By the previous item µ is an isomorphism.
An example where the multiplication map is not an isomorphism is provided by the groupring Z[C 2 ] with the trivial involution, where is not isomorphic to Z[C 2 ] Z 2 (see [DMPR21,Section 5.2]).
If the multiplication µ∶ A Z 2 ⊗ N A Z 2 → A Z 2 is an isomorphism, it follows from [DMPR21, Theorem 5.1] that the Z 2-Mackey functor π 0 THR(A) is the fixed-points Mackey functor of the ring with involution A. On the other hand, if the prime p odd we show in [DPJM22,Theorem 3.7] that π 0 THR(A) D p n is also a ring of Witt vectors, and combining these results we obtain a ring isomorphism for every odd prime p and commutative ring A satisfying any of the assumptions of Remark 4.1.
In the next section we use this last isomorphism to determine TCR of perfect fields of odd characteristic. In the subsequent ones we examine the relationship between π 0 THR(A) D 2 n and the Witt vectors for the prime 2, and determine TCR of perfect fields of characteristic 2.  This map is an equivalence on underlying spectra by [HM97, Theorem 5.5], and it is therefore sufficient to prove that it is an equivalence on geometric fixed-points. The spectrum TRR(k; p) Z 2 has the structure of a ring spectrum. Moreover there is an isomorphism π 0 (TRR(k; p) Z 2 ) ≅ W (k; p) and therefore 2 = tran(1) is a unit in π 0 (TRR(k; p) Z 2 ), see [DPJM22, Corollary 3.14].

TCR of perfect fields of odd characteristic
Since the transfers vanish in the geometric fixed points, we have that 2 is both a unit and zero in π 0 (TRR(k; p) φ Z 2 ), and therefore π 0 (TRR(k; p) φ Z 2 ) is the zero ring. Since TRR(k; p) φ Z 2 is a ring spectrum its homotopy groups are a module over the zero ring, and therefore it must be contractible.
According to Definition 1.5 and the previous paragraph, the Z 2-spectrum TCR(k; p) is equivalent to the equaliser of Z 2-spectra eq HW (k; p) The kernel of id −F ∶ W (k; p) → W (k; p) is equal to W (F p ; p) which is isomorphic to Z p , and this completes the proof.

TCR of perfect fields of characteristic 2
The calculation of TCR(k; 2) for a perfect field of characteristic 2 is more involved than the odd primary case. This is because the geometric fixed-points spectrum of TRR(k; 2) is not trivial, and thus we cannot directly apply the argument of Proposition 4.2. The first step is to understand the geometric fixed-points of TRR n (k; 2), using the formula of Theorem 2.7.

The geometric fixed points of TRR n for perfect fields of characteristic 2
Let us fix a perfect field k of characteristic 2, and let us compute additively TRR n (k; 2) φ Z 2 , TRR(k; 2) φ Z 2 and TCR(k; 2) φ Z 2 . We let k denote the constant Mackey functor of k.
Lemma 4.3. There is an equivalence of genuine C 2 -equivariant spectra where ρ is the regular representation of C 2 . In particular there is an equivalence of spectra Proof. By splitting Hk φ Z 2 using the Frobenius Hk-module structure we obtain a decomposition This uses that k is perfect and hence k considered as a k-module via the Frobenius is again a 1-dimensional k-vector space. By using inductively that, for every R-modules X and Y , there is an equivalence of N C2 (R)-modules we find that the C 2 -norm of Hk φ Z 2 decomposes as a N C2 e Hk-module as Σ n+m (C 2 ) + ⊗ N C2 e (Hk)).
By the description of THR(k; 2) φ Z 2 of Lemma 1.2, there is an equivalence of C 2 -spectra where the last identification is the equivalence given by tensoring with the unit of the norm of Hk (and similarly for the identification on the induced summands). Now let us identify the fixed-points. Notice that Σ nρ Hk is a module over Hk and therefore its fixed-points spectrum is a wedge of Eilenberg-MacLane spectra. Moreover a straightforward calculation in Bredon homology shows that when n ≤ i ≤ 2n, and π C2 i (Σ nρ Hk) = 0 otherwise. In the following proposition the summands are arranged exactly as in Lemma 4.3. In particular, the summands indexed on (n, m) with n < m in the source come from the induced summands. Similarly, the summands indexed on (n, m) with n ≠ m in the target corresponds to the induced summands.
Proposition 4.4. For any perfect field k of characteristic 2, the maps r, f ∶ (THR(k) where r kills the (n, m)-summands with n < m and maps the (n, m)-summands with m ≤ n to the (n, m)-summand via the inverse Frobenius of k, and f kills the (n, m)-summands with m < n, includes the summand (n, n), and embeds diagonally the (n, m)-summands with n < m into the sum of the summands (n, m) and (m, n).
Proof. Recall that the map r in general is given by where first map is the canonical map from the genuine to the geometric fixed-points and the equivalence is given by the cyclotomic structure. If we take T = THR(k), then the last equivalence is described in Example 2.6. As genuine and geometric fixed-points commute with coproducts we need to identify r on each of the summands of Lemma 4.3. Since geometric fixed-points kill induced spectra, r vanishes on the summands (n, m) with n < m. The identification on the other summands follows from observing that the canonical map π C2 * (Σ nρ Hk) → π * ((Σ nρ Hk) φC2 ) ≅ π * (Σ n (Hk) φC2 ) ≅ π * (⊕ l≥0 Σ n+l Hk) induces the inverse Frobenius of k in degrees n ≤ * ≤ 2n (cf. with [NS18, Example IV.1.2]). Similarly, we need to compute f on each summand, and its identification follows from the fact that the restriction map is the identity only in degree * = 2n and zero otherwise, whereas is the diagonal for all n and m.
Remark 4.5. From Proposition 4.4 and Theorem 2.14 we obtain that Indeed by 4.4, the map r − f is an isomorphism in π * when restricted and corestricted to the summands with n ≠ m. It is therefore an isomorphism in odd degrees, and its long exact sequence decomposes into exact sequences for every l ≥ 0. Again by 4.4, the kernel of r − f is the kernel of id − (−)∶ k → k. Since k is a field this is F 2 . Similarly the cokernel of r − f is the cokernel of id − (−), which since k is perfect it is also the cokernel of id +(−) 2 . We also remark that these groups agree with the homotopy groups of the cofibre L n (k) of the canonical map L q (k) → L(Mod ω A , Ϙ gs k ). induced by the symmetrisation map from the quadratic to the genuine Poincaré structure, as defined in [CDH + 23, CDH + 20a, CDH + 20b]. Indeed the even homotopy groups of L q (k) are the Witt groups of quadratic forms over k, and since k is a field the odd groups vanish [Ran92,Proposition 22.7]. The map above is an isomorphism in degrees lass than or equal to −3 and surjective in degree −2 by [CDH + 20b, Theorem 5], and therefore the cofibre L n (k) is −1-connected. In degrees greater or equal to −1 the homotopy groups of the target are the symmetric Witt groups of k in even degrees and zero in odd degrees, by [CDH + 20b, Corollary 1.3.5]. The map is the symmetrisation map from quadratic to symmetric Witt groups, which is zero since k has characteristic 2. Thus the homotopy groups of L n (k) are the symmetric Witt groups of k in even non-negative degrees, and the quadratic ones in odd non-negative degrees. The (−1)-st homotopy group of L n (k) is the kernel of the symmetrisation map, and therefore again the quadratic Witt group. The quadratic and symmetric Witt groups of a perfect field of characteristic 2 are respectively k (x + x 2 ) and F 2 , see e.g. [Kat82, Theorem (1)].
In order to understand the full equivariant homotopy type of TCR(k; 2) will need to calculate the homotopy groups of TRR(k; 2) φ Z 2 .
Theorem 4.6. Let k be a perfect field of characteristic 2. For any l ≥ 1, there is an isomorphism n,m≥0,n+m= * k.
The maps R, F ∶ TRR l+1 (k; 2) φ Z 2 → TRR l (k; 2) φ Z 2 and the Weyl action are described on homotopy groups as follows. The map R kills the (n, m)-summands with n ≠ m and is the inverse Frobenius of k on the summands (n, n). The map F kills the (n, m)-summands with m < n, is the identity of k on the summands (n, n), and embeds the (n, m)-summands with n < m diagonally into the sum of the (n, m) and (m, n)-summands. The Weyl action of σ l swaps the (n, m)-summand and the (m, n)-summand for all n, m ≥ 0. In particular the homotopy groups and the maps are all independent of l.
Proof. We prove the theorem by induction on l, using the pullbacks of Theorem 2.7. For n = 1 the pullback of 2.7 implies that TRR 2 (k; 2) φ Z 2 is equivalent to the pullback (since the right vertical map is the diagonal for n = 1 in 2.7). Consider the Mayer-Vietoris sequence associated to (T φ ) C2 × T φ (T φ ) C2 : . . .
By the inductive assumption the iterated map F l−1 is in fact equal to a single map F . The right horizontal map then sends a triple (x, y, z) of finite sequences indexed on the pairs of integers n, m ≥ 0 to the pair of sequences This map is clearly surjective for all * , and therefore π * TRR l+1 (k; 2) φ Z 2 is isomorphic to its kernel. This consists of those triples (x, y, z) such that x (n,m) = y (n,m) = 0 for n > m and z (n,m) = 0, for n ≠ m, and √ x (n,n) = √ y (n,n) = z (n,n) , which is isomorphic to the direct sum on all pairs of natural numbers by setting w (n,m) = y (m,n) for n > m, and w (n,m) = x (n,m) , for n < m, and w (n,n) = x (n,n) . Let us now describe R and F under these isomorphisms. By 2.7 the map R sends (x, y, z) to z, and therefore under the isomorphism above R(w) (n,m) = 0 , n ≠ m √ w (n,n) , n = m.
Again by 2.7 the map F sends (x, y, z) to (x, x, F (z)). Thus under the identification above ,n) , n > m x (n,m) = w (n,m) , n < m x (n,n) = w (n,n) , n = m.
Corollary 4.7. Let k be a perfect field of characteristic 2. There is a natural isomorphism π * TRR(k; 2) φ Z 2 ≅ k if * is even 0 otherwise, and the Frobenius endomorphism F ∶ TRR(k; 2) φ Z 2 → TRR(k; 2) φ Z 2 is the Frobenius of k on homotopy groups.
Proof. By Theorem 4.6 the map R on homotopy groups is the map n+m= * k (where n, m ≥ 0) which is the inverse Frobenius on the summands (n, n), and zero everywhere else. It is an idempotent up to isomorphism, and therefore it satisfies the Mittag-Leffler condition. It follows that where the last isomorphism is induced by the projection onto the first component of the limit and onto the summand 2n = * when * is even, and it is zero otherwise. After composing with the shift automorphism of the limit, R becomes by definition the identity, and F the Frobenius of k.

The components of TRR and the ring of Witt vectors of perfect fields
As we will show in Remark 4.9 below, the ring π 0 THR(A) D 2 n is not necessarily the ring of Witt vectors of π 0 THR(A) Z 2 , not even when the latter is isomorphic to A. However, this is still the case for perfect fields, as we show now.
Theorem 4.8. Let k be a perfect field of characteristic 2, equipped with the trivial involution. Then for every n ≥ 0, the restriction map is an isomorphism, and the Verschiebung, Frobenius, and restriction maps of the Witt vectors correspond respectively to tran D 2 n D 2 n−1 , res D 2 n D 2 n−1 , and R.
Proof. Let us start with a commutative ring with involution A, and follow the strategy of [HM97] and [DPJM22] of analysing the long exact sequence induced on homotopy groups by the fibre sequence ER + ⊗ C 2 n THR(A) → THR(A) C 2 n → THR(A) C 2 n−1 .
The components of the fixed-points of the fibre are then calculated by the colimit where O R is the full subcategory of the orbit category of D 2 n generated by the reflections and the trivial group (this follows for example from the fact that ER is the homotopy colimit over O R of the funtor that takes D 2 n H to the discrete space D 2 n H, see e.g. [LO01, Lemma 2.2]).
The crucial difference between 2 and the odd primes is that for the prime 2 the category O R has two components, generated by the distinct conjugacy classes of the reflections τ and στ , where σ is the generator of the cyclic group C 2 n . Therefore the colimit above is isomorphic to the pushout of abelian groups along the transfer maps tran Z 2 e ∶ A → (π 0 THR(A) Z 2 ) C2 , where the coinvariants are taken with respect to the action of the Weyl group C 2 . Under this identification, the transfer map to π 0 THR(A) D 2 n is the transfer tran D 2 n Z 2 on the first summand, and σ n tran D 2 n Z 2 on the second summand, where σ n is the action of the generator of the Weyl group. The corresponding long exact sequence on homotopy groups is then Let us now compute the boundary map of this sequence in the case where k is a perfect field of characteristic 2 with the trivial involution. Since k is a field of characteristic 2, the isomorphism of [DMPR21, Corollary 5.2] is π 0 THR(k) Z 2 ≅ k ⊗ S k where S ⊂ k is the subfield of squares. Moreover since k is perfect, the restriction map which is induced by the multiplication map of k, is an isomorphism, and the Weyl action on the source must be trivial. The transfer map k ≅ π 0 THH(k) is multiplication by 2 by the double-coset formula, and therefore zero. Thus the long exact sequence above becomes Now suppose inductively that the restriction map res∶ π 0 THR(k) D 2 n−1 → π 0 THR(k) C 2 n−1 is an isomorphism, and identify the target with W n (k; 2) by the isomorphism of [HM97, Theorem F]. Then the restriction map defines a commutative diagram with exact rows By the inductive assumption the Z 2-action on π 0 THR(k) C 2 n−1 is trivial (since res is surjective) and the transfer tran D 2 n−1 C 2 n−1 identifies with the multiplication by 2 map on the Witt vectors W n (k; 2), whose kernel is k. It follows that the cokernel of φ, and therefore that of ∂, is isomorphic to k. Thus, if k is finite, the image of ∂ must have as many elements as k does, and therefore since it is included in ∆ it must be equal to it. This concludes the proof in the case where k is finite.
If k is infinite, we cannot yet conclude that the image of ∂ is the whole diagonal, since we only know that (k ⊕ k) Im ∂ is abstractly isomorphic to k. We do however know that the image of ∂ is ∆ for the finite field F 2 , and the naturality of ∂ with respect to the morphism of fields F 2 → k shows that at least (1, 1) must belong to Im ∂. Since R∶ THR(k) D 2 n → THR(k) D 2 n−1 is a map of ring spectra, ∂ is a map of π 0 THR(k) D 2 n -modules. Moreover the isomorphism is an isomorphism of π 0 THR(k) D 2 n -modules, where π 0 THR(k) D 2 n acts on each π H 0 THR(k) via the restriction map, and the transfers are linear over these restrictions by the Frobenius reciprocity formula of the D 2 n -Mackey functor π 0 THR(k). In particular π 0 THR(k) D 2 n acts diagonally on k ⊕ k, via the restriction map res D 2 n e ∶ π 0 THR(k) D 2 n → π 0 THH(k) = k. This map factors as π 0 THR(k) D 2 n res → π 0 THR(k) C 2 n ≅ W n−1 (k; 2) where the first map is surjective by the argument above. Since k is of characteristic 2, the iterated Frobenius is given by F n−2 (a 1 , . . . , a n−1 ) = a 2 n−2 1 , which is surjective since k is perfect. Thus given any x ∈ k, we can choose an element z of π 0 THR(k) D 2 n which maps to x by the restriction res D 2 n e . Then since the image of ∂ is a submodule of k ⊕ k containing (1, 1), we have that (x, x) = z ⋅ (1, 1) is also in the image of ∂, and thus Im ∂ = ∆.
The identification of the Witt vectors operators V , F and R follows from the commutative diagrams and the fact that the maps of the bottom row correspond respectively to V, F and R by [HM97,Theorem F]. Note that to show that the first diagram commutes one needs to use the doublecoset formula and the fact that the quotient D 2 n−1 D 2 n C 2 n is trivial.
Remark 4.9. The restriction map of Theorem 4.8 is not generally an isomorphism. For example for the ring of integers, there is a map of short exact sequences where Z ⊕ 2 Z is the pushout of the transfer 2∶ Z → Z along itself, which is isomorphic to Z × Z 2, and the left-hand map is the identity on each summand. Thus the middle restriction is not an isomorphism, and moreover π 0 THR(Z) D2 has 2-torsion. The top row of the diagram comes from the long exact sequence on homotopy groups for the map R of the proof of Theorem 4.8, upon showing that its connective homomorphism ∂ is in this case zero. To see this, we map the sequence to the analogous sequence for F 2 via the canonical quotient map Z → F 2 , and obtain a commutative diagram with exact rows where the second vertical map from the left is induced by the projection on each summand. Thus if we can show that the left vertical map is zero, we will have that the upper ∂ maps into the kernel of the projection Z ⊕ 2 Z → F 2 ⊕ F 2 , which is the subgroup of elements (2n, 0), and isomorphic to Z. The group π 1 THR(Z) Z 2 is however isomorphic to Z 2 by [DMPR21,Proposition 5.22], and therefore ∂ is 0.
We still need to verify that the left vertical map is zero. We look at its effect on the isotropy separation sequences for the Z 2-spectrum THR, and obtain a diagram One can in fact show that the connecting homomorphism is zero also for the larger dihedral groups, by calculating the first part of the long exact sequence for R on geometric fixed points using the calculations of section 5.1. One then obtains short exact sequences for every n ≥ 1. We will address this in future work.
Proposition 4.10. Let k be a perfect field of characteristic 2. The tower of abelian groups satisfies the Mittag-Leffler condition, and therefore there is an isomorphism of rings π 0 TRR(k; 2) Z 2 ≅ W (k; 2).
Proof. We need to analyse the images in π 1 of the composite maps R j . Let (⊉ C 2 j ) be the family of subgroups of D 2 n+j that do not contain C 2 j (it is the family R when j = 1). By taking the D 2 n+j C 2 j -fixed points of the isotropy separation sequence for the subgroup C 2 j ⊂ D 2 n+j we obtain a fibre sequence of spectra and after identifying the third term with THR(k) D 2 n using the real cyclotomic structure we obtain a fibre sequence The group of components of the fibre can be calculated as the colimit where O (⊉C 2 j ) is the full subcategory of the orbit category of D 2 n+j spanned by the subgroups

TCR of perfect fields of characteristic 2
We now combine the results of the previous two sections to prove the following theorem.
Theorem 4.11. For any perfect field k of characteristic 2, there is an equivalence of Z 2equivariant ring spectra TRR(k; 2) ≃ HW (k; 2), where W (k; 2) is the constant Green functor of the abelian group with trivial involution W (k; 2).
The map TRR(k; 2) → HW (k; 2) is an underlying equivalence by [HM97, Theorem 4.5]. Hence it suffices to show that it is an equivalence after applying the geometric fixed points. By the calculation of Theorem 4.6 (and in particular using the formula for R) we see that the map TRR(k; 2) φ Z 2 → THR(k; 2) φ Z 2 induces injections on homotopy groups. Hence it suffices to show that after applying the geometric fixed points the lower horizontal map induces an injection on the image of the left vertical map. Indeed, this will imply that the upper horizontal map induces an injection on the homotopy groups of the geometric fixed points, and since these are either 0 or 1 dimensional k-vector spaces (the target has homotopy groups W (k; 2) 2 ≅ k in even non-negative degrees) it must also be surjective. The lower map is, on geometric fixed-points, the multiplication map Hk φ Z 2 ⊗ Hk Hk φ Z 2 → Hk φ Z 2 .
By [DMPR21, Proposition 5.19] the induced map on homotopy groups where w 1 = 1, w 2 = 1 and v = 1, sends both w 1 and w 2 to v. This implies that its restriction is injective, and by the description of R of Theorem 4.6 k[w 1 w 2 ] is exactly the image of the left vertical map on homotopy groups after applying geometric fixed points.
Corollary 4.12. For any perfect field k of characteristic 2, one has an equivalence of genuine where F ∶ W (k; 2) → W (k; 2) is the Witt vector Frobenius.
Proof. It follows from Theorem 4.8 that F ∶ TRR(k; 2) → TRR(k; 2) corresponds to the Witt vector Frobenius HF ∶ HW (k; 2) → HW (k; 2) under the equivalence of Theorem 4.11. It is an easy exercise in Witt vectors to see that ker(1 − F ) ≅ W (F 2 ; 2) ≅ Z 2 . Hence we get π 0 TCR(k; 2) ≅ Z 2 and π −1 TCR(k; 2) ≅ coker(1 − F ), and all the other homotopy Mackey functors of TCR(k; 2) vanish. Since coker(1 − F ) is a Z 2module coming from a Z 2 -module, its homological dimension over the Green functor Z 2 is less than or equal to 1. The universal coefficient theorem in the category of modules over the Green functor Z 2 now implies that in fact TCR(k; 2) splits as claimed (in case k = F 2 this is obvious since F = id).

TCR of the integers and perfect rings
In this section we will calculate the homotopy type of TCR(A; 2) φ Z 2 where A is either a perfect F 2 -algebra or 2-torsion free ring with a perfect mod 2 reduction (for example the Witt vectors of a perfect F 2 -algebra). We will first calculate TCR(Z; 2) φ Z 2 , and then deduce TCR(A; 2) φ Z 2 by a base-change formula from F 2 and Z.

The geometric fixed-points of TCR(Z; 2)
Let us denote by N A ∶= N C2 e HA the C 2 -equivariant norm of the Eilenberg-MacLane ring spectrum of a commutative ring A. We regard HZ (the C 2 -equivariant Eilenberg-MacLane spectrum for the constant Mackey functor Z) as an N Z-module via the multiplication map N Z → HZ. We then consider HZ φ Z 2 as an H Z-module via the induced map on geometric fixed-points H Z ≃ (N Z) φ Z 2 → HZ φ Z 2 , and obtain a splitting of H Z-modules Again using the description of THR(Z) φ Z 2 as the derived smash product of Lemma 1.2 and the splitting above just as in Lemma 4.3 , we obtain an equivalence of genuine C 2 -spectra In order to apply Theorem 2.14 to compute TCR(Z; 2) φ Z 2 , we need to understand the genuine C 2 -fixed points of this spectrum. By the Wirthmüller isomorphism, the genuine C 2 -fixed point spectrum of the induced summands are The genuine fixed points of the terms Σ 2nρ ((N F 2 ) ⊗ N Z HZ) are more laborious.
Lemma 5.1. There is a fibre sequence of C 2 -spectra where the left-hand spectrum is the Eilenberg MacLane spectrum of the abelian group Z ⊕ Z 2 with involution w(a, x) = (a, [a] + x). In particular the homotopy Mackey functors of the cofibre are where the restriction is the canonical projection and the transfer is injective, π 1 ((N F 2 ) ⊗ N Z HZ) ≅ Z 2 which is the constant Mackey functor of Z 2, and the other homotopy groups vanish.
Proof. Let us first calculate the fibre of the canonical map N Z → N F 2 given by the norm of the reduction modulo 2. There is a commutative C 2 -equivariant diagram of spectra (see [DM16, Section 1.1] for details on G-diagrams) N Z 1⊗2 Lemma 5.2. The C 2 -equivariant homotopy groups of Σ kρ ((N F 2 ) ⊗ N Z HZ) for even k are Proof. We calculate the equivariant homotopy groups from the fibre sequence of C 2 -spectra from Lemma 5.1. We start by calculating the equivariant homotopy groups of the first two spectra. These are respectively the Bredon homology groups of S kρ with coefficients in the Mackey functors of the abelian groups Z with the the trivial involution, and Z ⊕ Z 2 with the involution w(a, x) = (a, [a] + x). These are respectively the homology of the chain complexes where the non-zero groups are sitting between degree k and 2k (and k is even), and with the non-zero groups sitting in the same degrees. The first complex has homology groups Z 2 in even degrees between k and 2k − 2, a Z in degree 2k, and zero everywhere else. The differentials of the second complex are respectively for all (a, x) ∈ Z × Z 2. Its homology is then concentrated in even degrees between k and 2k, with in even degrees between k and 2k − 2 and (2 Z) × Z 2 in degree 2k. The long exact sequence of the above fibre sequence therefore splits and gives rise to exact sequences give the groups of the statement.
Since HZ → (N F 2 ) ⊗ N Z HZ is a map of C 2 -equivariant algebras, the fixed point spectrum (Σ kρ (N F 2 ) ⊗ N Z HZ) C2 is a module over HZ and therefore splits as a wedge of Eilenberg-MacLane spectra. As a consequence of Lemma 5.2 we obtain an equivalence The identification of the map r∶ (THR(Z) φ Z 2 ) C2 → THR(Z) φ Z 2 in terms of the above splittings will contain higher stable cohomology operations, and this complicates the calculation of the equaliser of r and f . However, like in the case of fields, it is possible to compute r on homotopy groups and after identifying only a portion of the matrix describing r we will be able to compute TCR(Z; 2) φ Z 2 using Theorem 2.14.
Proposition 5.4. Under the above splittings, the map r∶ (THR(Z) φ Z 2 ) C2 → THR(Z) φ Z 2 corresponds to the map with the following properties. It is zero on the (n < m)-summands. On the summands (n > m) it has components which are zero if n ≠ n ′ or m ′ < m, and the identity if n = n ′ and m = m ′ . The entry is given by the matrix The remaining entries are zero on homotopy groups, but generally contain higher stable cohomology operations (cf. [NS18, Section IV.1]).
Proof. The map r vanishes on the summands (n < m) since it factors through the geometric fixed-points of an induced spectrum which are zero. From the description of r of Example 2.6 we see that r preserves the wedge decomposition over n, and thus its components vanish for n ≠ n ′ . It remains to identify for every fixed n ≥ 0. The fibre sequence of Lemma 5.1 induces a commutative diagram (2,0) For any underlying connective C 2 -spectrum X, the canonical map π C2 * (Σ lρ X) → π * ((Σ lρ X) φC2 ) induces an isomorphism in degrees * < 2l and a surjection in degree * = 2l, since the homotopy orbits of Σ lρ X are (2l − 1)-connected. By applying this fact to the vertical maps of the commutative diagram above we obtain the description of r on the summands (n > m). Let us finally compute the map Using that the fibre sequence of Lemma 5.1 is HZ-linear, by considering the relevant summands in the diagram above we get a morphism of exact triangles where r is the map we are trying to compute, and β Z is the Bockstein of Composing the Bockstein β Z with the projection H2 Z → H F 2 gives the Bockstein for 0 → Z 2 → Z 8 → Z 4 → 0 which gives the desired result.
Theorem 5.5. There is an equivalence of spectra Proof. For simplicity we use the symbol ⊕ n>m≥0 to denote ⊕ (n,m),n>m≥0 (Σ 2n+2m (H F 2 ⊕ΣH F 2 )) and a similar symbol for the summands indexed by the pairs (n, m) with 0 ≤ n < m. Consider the commutative diagram in the stable homotopy category, where the vertical sequences are fibre sequences: We explain the maps in the diagram: The top map ω is the composite pr ○(r − f ) ○ incl and is an equivalence since by Propositions 5.3 and 5.4 it is an isomorphism on homotopy groups. The map α is then defined to be (r − f ) ○ incl ○ω −1 and by construction is of the form (id, ϕ, id), for some map The lower right vertical map is −ϕ on the outer summands and the identity on the middle summand. The map M is the induced map on the cofibres. Propositions 5.3 and 5.4 imply that the map ϕ is zero on the summand ⊕ 0≤n<m . On the other hand, the restriction of r − f to the summand Σ 4n (H Z 4 ⊕ ΣH F 2 ) cannot hit ⊕ n>m≥0 , since the cohomology operations do not decrease degrees and r preserves the n-coordinate. Thus M is given in matrix form by the wedge ⊕ The fibres of r − f and M are equivalent since ω is an equivalence. This completes the proof by Theorem 2.14.

Flat base-change and perfect rings
We recall that we always regard the geometric Z 2-fixed points of a ring spectrum with antiinvolution A as a left A-module via the geometric fixed-points of the map of Z 2-spectra N Z 2 e A⊗ A → A, and similarly as a right A-module via A ⊗ N Z 2 e A → A. We call these respectively the left and right Frobenius module structures on A φ Z 2 . We will always denote by ⊗ A the derived tensor product of A-modules.
Definition 5.6. A map α∶ A → B of ring-spectra with anti-involution is called φ-flat if the map induced by the map of left A-modules α∶ A φ Z 2 → α * B φ Z 2 , is an equivalence of spectra.
Example 5.7. Let α∶ A → B be a map of commutative rings with trivial involution which is flat 2-locally. Then the induced map on Eilenberg-MacLane spectra is φ-flat precisely if the maps adjoint to α 2 are isomorphisms, where ϕ (−) denotes the module structure r ⋅ x ∶= r 2 x, and (−) 2 the two-torsion. Indeed since α is flat 2-locally, α is φ-flat precisely if is an isomorphism. Since HA φ Z 2 is the connective cover of HA t Z 2 which is 2-periodic, and similarly for HB φ Z 2 , this is equivalent to showing that B ⊗ A ϕĤ i (Z 2, A) → ϕĤ i (Z 2, B) are isomorphisms for i = 0, 1, and this is exactly the assumption above. In particular: i) If B is a perfect F 2 -algebra with trivial involution, then the map F 2 → B is φ-flat. Indeed the maps above are both isomorphic to the Frobenius (−) 2 ∶ B → B. ii) If B is a commutative ring with trivial involution with no 2-torsion, and B 2 is perfect, then Z → B is φ-flat. Indeed the maps above are in this case respectively the Frobenius of B 2 and the map 0 → B 2 .
Recall that, as a C 2 -spectrum, THR(A) φ Z 2 is equivalent to B(A; N C2 e A; N C2 e (A φ Z 2 )), where A is regarded as a C 2 -spectrum via the identification C 2 ≅ Z 2 (see Lemma 1.2). In particular THR(A) φ Z 2 is canonically a module over A in the category of C 2 -spectra, by acting on the left copy of A in the bar construction. induced by α∶ THR(A) φ Z 2 → α * THR(B) φ Z 2 is an equivalence of C 2 -spectra. Here B is considered as a C 2 -spectrum via the isomorphism C 2 ≅ Z 2 (see Lemma 1.2).
Proof. Let us first show that the map is an equivalence on underlying spectra. This is the where the left A-module structure on the source is the left Frobenius structure on the right A φ Z 2 -factor (or equivalently the right one on the left factor), and similarly for the B-module structure on the target. Since A is commutative this A-module structure agrees with the left Frobenius structure on the first A φ Z 2 -factor, and therefore the map factors as where the two right pointing arrows are equivalences since α is φ-flat.
Let us now verify that this map is an equivalence on C 2 -geometric fixed-points. From the bar construction we see that, this is the map induced by the map α∶ THR(A) φ Z 2 → α * THR(B) φ Z 2 , where THR(A) φ Z 2 = A φ Z 2 ⊗ A A φ Z 2 is a left A φ Z 2 -module via left multiplication on the left factor (notice that A φ Z 2 is a ring spectrum since A is commutative), and similarly for the left B φ Z 2 -module structure on the target. This map then factors as where for the last equivalence we used that α is φ-flat.
Proposition 5.9. Under the assumptions of 5.8, suppose moreover that the restriction maps A Z 2 → A and B Z 2 → B are equivalences (for example if A and B are the Eilenberg-MacLane spectra of commutative rings with trivial involutions). Then there is an equivalence and the maps f, r∶ (THR(B) φ Z 2 ) C2 → THR(B) φ Z 2 correspond under the equivalences of Proposition 5.8 respectively to the tensor of the restriction maps and to the tensor of the canonical map to the geometric fixed-points and the map r of THR(A) φ Z 2 where the first equivalence is from the fact that α is φ-flat, and the second is the canonical one.
Proof. The first statement follows from the fact that if the restriction maps of A and B are equivalences, the canonical map is an equivalence for every A-module X (which is cofibrant under our standing assumption). Indeed since the source and target of this map commute with homotopy colimits in X, it is sufficient to check it on the generators A and A ⊗ (C 2 ) + of the category of A-modules. For A this is the canonical equivalence For A ⊗ (C 2 ) + this is the map where the arrow is induced by the map of A C2 -modules α∶ A → α * B, where A is an A C2 -module via the restriction A C2 → A, and similarly for B. This is an equivalence since the restrictions of A and B are. The identifications of f and r follow by naturality and unravelling the definitions, using Example 2.6 for the cyclotomic structure.
Corollary 5.10. Let α∶ A → B be a φ-flat map of commutative flat Z 2-equivariant ring spectra, and suppose that the restriction maps A Z 2 → A and B Z 2 → B are equivalences. Then there is an equaliser diagram where ν∶ B ⊗ A A φ Z 2 → B φ Z 2 is the equivalence from the φ-flatness condition, and can∶ B C2 → B φC2 is the canonical map.
Remark 5.11. One cannot conclude from Corollary 5.10 that TCR(B; 2) φ Z 2 is the base-change of TCR(A; 2) φ Z 2 , nor that it is a B-module. This is because the maps f and r computing TCR(A; 2) φ Z 2 are A-linear with respect to two different A-module structures.
Proof. By Example 5.7 and Corollary 5.10 the maps r and f computing TCR(B; 2) φ Z 2 are maps r, f ∶ ⊕ On homotopy groups they are described by the same projections and diagonals as in the case for Z of Propositions 5.3 and 5.4, except that r is postcomposed with the root isomorphism of the perfect F 2 -algebra B 2. The same argument of the proof of 5.5 gives a fibre sequence