Math. Z. manuscript No. (will be inserted by the editor)

On cyclic fixed points of spectra

Marcel Bökstedt · Robert R. Bruner · Sverre Lunøe–Nielsen · John Rognes

Received: date / Accepted: date

Abstract For a finite p-group G and a bounded below G-spectrum X of finite type mod p, the G-equivariant Segal conjecture for X asserts that the canonical map X^{G} → X^{hG}, from G-fixed points to G-homotopy fixed points, is a p-adic equivalence. Let C_{p^{n}} be the cyclic group of order p^{n}. We show that if the C_{p}-equivariant Segal conjecture holds for a C_{p^{n}}-spectrum X, as well as for each of its geometric fixed point spectra Φ^{C_{p^{e}}} (X) for 0 < e < n, then the C_{p^{n}}-equivariant Segal conjecture holds for X. Similar results also hold for weaker forms of the Segal conjecture, asking only that the canonical map induces an equivalence in sufficiently high degrees, on homotopy groups with suitable finite coefficients.

Keywords Segal conjecture · cyclic p-group · fixed points · Tate construction · smash power · topological Hochschild homology

Mathematics Subject Classification (2000) 55P42 · 55P91 · 55T15

1 Introduction

Let p be any prime number. Graeme Segal’s Burnside ring conjecture [1] for a finite p-group G asserts that if X = S_{G} is the genuinely G-equivariant sphere spectrum, then the canonical map X^{G} → X^{hG} = F (EG_{+}, X)^{G} is a p-adic equivalence. For cyclic groups G = C_{p} of prime order the conjecture was proved by Lin [15] and Gunawardena [11], [2]. Thereafter Ravenel [19], [20] gave an inductive proof of Segal’s conjecture for finite cyclic p-groups G = C_{p^{n}} of order p^{n}, starting from Lin and Gunawardena’s theorems. Ravenel’s result was superseded by Carlsson’s proof [7] of the Segal conjecture for all finite p-groups, but as we shall show here, Ravenel’s methods are also of interest in a more general context, where X is a quite general G-spectrum.

M. Bökstedt Dept. of Mathematical Sciences, Aarhus University, Denmark, E-mail: marcel@imf.au.dk R. R. Bruner Department of Mathematics, Wayne State University, USA, E-mail: rrb@math.wayne.edu S. Lunøe–Nielsen Department of Mathematics, University of Oslo, Norway, E-mail: sverreln@math.uio.no J. Rognes Department of Mathematics, University of Oslo, Norway, E-mail: rognes@math.uio.no

As was elucidated by Miller and Wilkerson [18], Ravenel’s methods give two proofs of the Segal conjecture for cyclic groups—one computational using the modified Adams spectral sequence, and one non-computational, using explicit geometric constructions. The object of this paper is to generalize Ravenel’s geometric proof of the Segal conjecture to show that X^{G} → X^{hG} is “close to” a p-adic equivalence for G = C_{p^{n}}, assuming that X^{C} → X^{hC} and similar maps are “close to” such an equivalence for C = C_{p}. Our main technical results are Theorems 2.4 and 2.5. Their statements involve (W, k)-coconnected maps and geometric fixed points, which are discussed in Definitions 2.1 and 2.3, respectively. See Example 2.2 for more on how a (W, k)-coconnected map is close to a p-adic equivalence.

In the special cases X = B^{∧p^{n}} and X = T HH(B), where B^{∧p^{n}} is a specific C_{p^{n}}-equivariant model for the p^{n}-th smash power of a symmetric spectrum B, and T HH(B) is the topological Hochschild homology of a symmetric ring spectrum B, the geometric fixed points are well understood, as explained in Theorems 2.7 and 2.8, respectively. In the special cases W = S^{−1}/p^{∞} and W = F (V, S), where V is a finite p-torsion spectrum, the (W, k)-coconnected maps are well understood in terms of p-completion and homotopy with V - coefficients, as explained in Examples 2.9 and 2.10, respectively. In the doubly special case when X = T HH(B) and W = S^{−1}/p^{∞}, our results recover the main theorem of Tsalidis [24].

2 Statement of results

We first formalize the notion of being close to a p-adic equivalence. Throughout the paper we assume that a pair (W, k) has been chosen as in the following definition. The hypothesis on W ensures that the function spectrum F (W, Y) is contractible whenever the p-adic completion Y_{p}is contractible.

Definition 2.1 Let S^{−1}/p^{∞} be a Moore spectrum with homology ℤ/p^{∞} concentrated in degree −1, so that F (S^{−1}/p^{∞}, Y) = Y_{p}for each spectrum Y. Let W be an object in the localizing ideal [13, Def. 1.4.3(d)] of spectra generated by S^{−1}/p^{∞}, i.e., the smallest thick subcategory of spectra that contains S^{−1}/p^{∞} and is closed under arbitrary wedge sums, as well as under smash products with arbitrary spectra.

Let k be an integer, or the symbol −∞. We say that a spectrum Y is (W, k)-coconnected if π_{∗}F (W, Y) = 0 for all ∗ ≥ k. We say that a map of spectra f: Y_{1} → Y_{2} is (W, k)-coconnected if hofib(f) is (W, k)-coconnected, or equivalently, if π_{∗}F (W, Y_{1}) → π_{∗}F (W, Y_{2}) is injective for ∗ = k and an isomorphism for all ∗ > k.

Example 2.2 The most obvious choice for W is the Moore spectrum S^{−1}/p^{∞} itself, in which case F (W, Y) = Y_{p}, so a map f: Y_{1} → Y_{2} is (W, k)-coconnected if and only if the p-completed map f_{p}: (Y_{1})_{p} → (Y_{2})_{p} induces an injection on π_{∗} for ∗ = k and an isomorphism for ∗ > k. When k = −∞, this is the same as being a p-adic equivalence.

Alternatively, we may take W = F (V, S), where V is a finite CW spectrum whose integral homology is p-torsion, in which case F (W, Y) ≃ V ∧ Y by Spanier–Whitehead duality. In this case f: Y_{1} → Y_{2} is (W, k)-coconnected if and only if the map 1 ∧ f: V ∧ Y_{1} → V ∧ Y_{2} induces an injection V_{∗}(Y_{1}) = π_{∗}(V ∧ Y_{1}) → π_{∗}(V ∧ Y_{2}) = V_{∗}(Y_{2}) for ∗ = k and an isomorphism for ∗ > k. The Smith–Toda complexes V (m) for m ≥ 0, see [22] and [23], are examples of such finite p-torsion spectra.

Next, we recall some comparison maps between fixed points, homotopy fixed points, geometric fixed points and Tate constructions.

Definition 2.3 Let C = C_{p} ⊂ C_{p^{n}} = G and Ḡ = G/C ≅ C_{p^{n−1}}. Let λ = ℂ(1) be the basic faithful G-representation of complex rank one, and S^{λ} its onepoint compactification. Let ∞λ be the direct sum of a countable number of copies of λ. Its unit sphere S(∞λ) = EG is a free contractible G-CW space, and its one-point compactification S^{∞λ} = ẼG sits in a G-homotopy cofiber sequence EG_{+} → S^{0} → ẼG, where the first map collapses EG to the nonbasepoint.

Let X be a G-spectrum, in the sense of [14], and consider the vertical map

\xymatrix{EG_{+} ∧ X \ar[r] \ar[d]^{≃_{G}} & X \ar[r] \ar[d] & ẼG ∧ X \ar[d] \\ EG_{+} ∧ F (EG_{+}, X) \ar[r] & F (EG_{+}, X) \ar[r] & ẼG ∧ F (EG_{+}, X)}

of horizontal G-homotopy cofiber sequences. Passing to G-fixed point spectra we obtain a vertical map

[figure: X_{hG}; N; X^{G}; R; Γ_{n}; Φ^{C}(X)Ḡ; Γ̂_{n}; X_{hG}; N^{h}; X^{hG}; R^{h}; X^{tG}]

of horizontal homotopy cofiber sequences, often called the norm–restriction sequences [10, Diag. (C), (D)]. Here

X_{hG} = EG_{+} ∧_{G} X  (homotopy orbits)
X^{hG} = F (EG_{+}, X)^{G}  (homotopy fixed points)
X^{tG} = [ẼG ∧ F (EG_{+}, X)]^{G}  (Tate construction)

and there is a preferred Ḡ-equivariant equivalence

[ẼG ∧ X]^{C} ⟶^{≃} Φ^{C}(X)  (geometric fixed points)

inducing the upper right hand equivalence [ẼG ∧ X]^{G} ≃ Φ^{C}(X)Ḡ. For more details, see e.g. [12, Prop. 2.1].

The right hand square above is homotopy cartesian, so Γ_{n} is (W, k)-co-connected if and only if Γ̂_{n} is (W, k)-coconnected. This observation can be combined with the conclusions of all of the theorems below.

We briefly write H_{∗}(X) = H_{∗}(X; 𝔽_{p}) for the mod p homology of any spectrum, and say that H_{∗}(X) is of finite type if each group H_{m}(X) is finite.

Theorem 2.4 Let X be a G-spectrum with π_{∗}(X) bounded below and H_{∗}(X) of finite type. Suppose that Γ_{1} : X^{C} → X^{hC} and Γ_{n−1} : Φ^{C}(X)Ḡ → Φ^{C}(X)^{hḠ} are (W, k)-coconnected maps. Then Γ_{n}: X^{G} → X^{hG} is (W, k)-coconnected.

Informally, this theorem asserts that if X^{C} → X^{hC} and Y Ḡ → Y^{hḠ} are close to p-adic equivalences, for Y = Φ^{C}(X), then X^{G} → X^{hG} is close to a p-adic equivalence.

Theorem 2.5 Let X be a C_{p^{n}}-spectrum. Suppose, for each of the geometric fixed point spectra

Y = X , Φ^{C_{p}}(X) , …, Φ^{C_{p^{n−1}}} (X) ,

that π_{∗}(Y) is bounded below, H_{∗}(Y) is of finite type and Γ_{1} : Y^{C_{p}} → Y^{hC_{p}} is (W, k)-coconnected. Then Γ_{n}: X^{C_{p^{n}}} → X^{hC_{p^{n}}} is (W, k)-coconnected.

The proofs of Theorems 2.4 and 2.5 are given near the end of Section 3. The bounded below and finite type mod p hypotheses enter in the proof of Proposition 3.8, where we make use of the convergence of an inverse limit of Adams spectral sequences.

The following construction was introduced by the first author, in the context of functors with smash product (FSPs). See [5, §3] for a published account. We are principally interested in the case r = p^{n}.

Definition 2.6 Let B be any symmetric spectrum. The r-th smash power B^{∧r} can be defined as a C_{r}-spectrum by the construction

B^{∧r} = sd_{r}T HH(B)_{0} = T HH(B)_{r−1}

from [12, §2.4]. Its V -th space is defined by a homotopy colimit

(B^{∧r})_{V} = hocolim_{(i_{1},...,i_{r})∈I^{r}} Map(S^{i_{1}} ∧ ⋯ ∧ S^{i_{r}}, B_{i_{1}} ∧ ⋯ ∧ B_{i_{r}} ∧ S^{V}) ,

and C_{r} cyclically permutes the smash factors, in addition to its natural action on S^{V}.

To ensure that B^{∧r} has the same naively equivariant homotopy type as the ordinary r-fold smash product B ∧ ⋯ ∧ B, it suffices to assume that B is flat and convergent, see e.g. [17, Lem. 5.5]. Hereafter, when referring to B^{∧r} we always assume that B has first been replaced by an equivalent flat and convergent symmetric spectrum. In [17, Thm. 5.13], the third and fourth authors prove that Γ_{1} : (B^{∧p})^{C_{p}} → (B^{∧p})^{hC_{p}} is a p-adic equivalence whenever π_{∗}(B) is bounded below and H_{∗}(B) is of finite type. This provides the inductive beginning for the following application of Theorem 2.5.

Theorem 2.7 Let B be a symmetric spectrum with π_{∗}(B) bounded below and H_{∗}(B) of finite type. Then

Γ_{n}: (B^{∧p^{n}})^{C_{p^{n}}} → (B^{∧p^{n}})^{hC_{p^{n}}}

is a p-adic equivalence, for each n ≥ 1.

When B is a symmetric ring spectrum, its topological Hochschild homology T HH(B) is a 𝕋-spectrum [12, §2.4], where 𝕋 is the circle group. It is not true in general that Γ_{1} : T HH(B)^{C_{p}} → T HH(B)^{hC_{p}} is a p-adic equivalence, see e.g. [12, Prop. 5.3] and [21, Thm. 4.7], but when it is “approximately” true, then the following theorem is useful.

Theorem 2.8 Let B be a connective symmetric ring spectrum with H_{∗}(B) of finite type, and suppose that

Γ_{1} : T HH(B)^{C_{p}} → T HH(B)^{hC_{p}}

is (W, k)-coconnected. Then

Γ_{n}: T HH(B)^{C_{p^{n}}} → T HH(B)^{hC_{p^{n}}}

is (W, k)-coconnected, for each n ≥ 2.

The proofs of Theorems 2.7 and 2.8 are given at the end of Section 3. In the case B = S there is a G-equivariant equivalence T HH(S) ≃ S_{G},

and Γ_{1} is a p-adic equivalence by the classical Segal conjecture. Also in the cases B = MU (the complex cobordism spectrum) and B = BP (the Brown– Peterson spectrum) it turns out that Γ_{1} for T HH(B) is a p-adic equivalence, as the third and fourth authors show in [16, Thm. 1.1]. This provides examples with k = −∞ for the following special case.

Example 2.9 Taking W = S^{−1}/p^{∞}, the assumption in Theorem 2.8 is that the p-completed map Γ_{1} : (T HH(B)^{C_{p}})_{p} → (T HH(B)^{hC_{p}})_{p} is k-coconnected, i.e., that it induces an injection on π_{k} and an isomorphism on π_{∗} for ∗ > k, and the conclusion is that the p-completed map

Γ_{n}: (T HH(B)^{C_{p^{n}}})_{p} → (T HH(B)^{hC_{p^{n}}})_{p}

is also k-coconnected, for all n ≥ 2. This recovers a theorem of Tsalidis [24, Thm. 2.4].

Example 2.10 Taking W = F (V, S) and V = V (1) = S/(p, v_{1}), the Smith– Toda complex of chromatic type 2 (for p odd), the assumption in Theorem 2.8 is that

V (1)_{∗}(Γ_{1}): V (1)_{∗}T HH(B)^{C_{p}} → V (1)_{∗}T HH(B)^{hC_{p}}

is k-coconnected, and the conclusion is that

V (1)_{∗}(Γ_{n}): V (1)_{∗}T HH(B)^{C_{p^{n}}} → V (1)_{∗}T HH(B)^{hC_{p^{n}}}

is also k-coconnected, for all n ≥ 2. This recovers the generalization of Tsalidis’ theorem used by Ausoni and the fourth author [3, Thm. 5.7] in the special case when B = ℓ, the Adams summand of connective p-local complex K-theory, and k = 2p − 2. The generalized result is used again in [4, Cor. 5.9], for B = ℓ/p = k(1), the first connective Morava K-theory.

3 Constructions and proofs

Definition 3.1 Let λ̄ be the basic faithful Ḡ-representation of complex rank one. Like in Definition 2.3, we let EḠ = S(∞λ̄) and ẼḠ = S^{∞λ̄}. The usual map from the homotopy colimit to the categorical colimit is a Ḡ-equivalence hocolim_{j} S^{jλ̄} ⟶^{≃} S^{∞λ̄} = ẼḠ. The pullback of λ̄ along the canonical projection G → Ḡ is the p-th tensor power λ^{p} = ℂ(p) of λ, and we get a G-equivalence

hocolim_{j} S^{jλ^{p}} ⟶^{≃} S^{∞λ^{p}} = ẼḠ ,

where the right hand side is implicitly viewed as a G-space by pullback along G → Ḡ. The G-map S^{jλ^{p}} → S^{(j+1)λ^{p}} in the colimit system is given by smashing S^{jλ^{p}} with the one-point compactification z: S^{0} → S^{λ^{p}} of the inclusion {0} ⊂ λ^{p}.

Lemma 3.2 Let X be a G-spectrum. There is a natural homotopy cofiber sequence

\xymatrix{holim_{j} (Σ^{−jλ^{p}}X)^{G} \ar[r] & (X^{C})Ḡ \ar[r]^{Γ_{n−1}} & (X^{C})^{hḠ} ,}

where the right hand map is Γ_{n−1} for the Ḡ-spectrum X^{C}.

Proof By mapping the Ḡ-homotopy cofiber sequence EḠ_{+} → S^{0} → ẼḠ into X^{C}, we get the homotopy (co-)fiber sequence

F (ẼḠ, X^{C})Ḡ → (X^{C})Ḡ ⟶^{Γ_{n−1}} F (EḠ_{+}, X^{C})Ḡ .

At the left hand side we have a natural chain of equivalences

F (ẼḠ, X^{C})Ḡ ≃ F (hocolim_{j} S^{jλ̄}, X^{C})Ḡ ≃ holim_{j} F (S^{jλ̄}, X^{C})Ḡ

≃ holim_{j}(Σ^{−jλ̄}(X^{C}))Ḡ ≃ holim_{j}((Σ^{−jλ^{p}}X)^{C})Ḡ ≃ holim_{j} (Σ^{−jλ^{p}}X)^{G} .

This gives the asserted homotopy cofiber sequence.

Proposition 3.3 Let X be a G-spectrum. There is a vertical map of homotopy cofiber sequences

\xymatrix{holim_{j} Φ^{C}(Σ^{−jλ^{p}}X)Ḡ \ar[r] \ar[d] & Φ^{C}(X)Ḡ \ar[r]^{Γ_{n−1}} \ar[d]^{Γ̂_{n}} & Φ^{C}(X)^{hḠ} \ar[d]^{(Γ̂_{1})^{hḠ}} \\ holim_{j} (Σ^{−jλ^{p}}X)^{tG} \ar[r] & X^{tG} \ar[r]^{Γ_{n−1}} & (X^{tC})^{hḠ}.}

The right hand horizontal maps are Γ_{n−1} for the Ḡ-spectra Φ^{C}(X) ≃ [ẼG ∧ X]^{C} and X^{tC} = [ẼG ∧ F (EG_{+}, X)]^{C}, respectively.

Proof We replace X in the lemma above by the G-spectra ẼG ∧ X and ẼG ∧ F (EG_{+}, X). This gives the two claimed homotopy cofiber sequences, in view of the Ḡ-equivalences

[Σ^{−jλ^{p}}(ẼG ∧ X)]^{C} ≃ [ẼG ∧ Σ^{−jλ^{p}}X]^{C} ≃ Φ^{C}(Σ^{−jλ^{p}}X)

and

[Σ^{−jλ^{p}}(ẼG ∧ F (EG_{+}, X))]^{C} ≃ [ẼG ∧ F (EG_{+}, Σ^{−jλ^{p}}X)]^{C} = (Σ^{−jλ^{p}}X)^{tC} ,

respectively. These follow from the G-dualizability of S^{jλ^{p}}.

Lemma 3.4 If Γ̂_{1} : Φ^{C}(X) → X^{tC} is (W, k)-coconnected, then (Γ̂_{1})^{hḠ} is (W, k)-coconnected.

Proof This is a special case of a more general result. The homotopy fixed point spectral sequence

E_{s,t}^{2} = H^{−s}(G; π_{t}(Y)) ⟹ π_{s+t}(Y^{hG})

shows that Y^{hG} is k-coconnected whenever Y is a k-coconnected G-spectrum. Commutation of function spectra, homotopy fibers and homotopy fixed points shows that f^{hG}: Y_{1}^{hG} → Y_{2}^{hG} is (W, k)-coconnected whenever f: Y_{1} → Y_{2} is a (W, k)-coconnected G-map. The lemma follows by applying this to the case of the Ḡ-map Γ̂_{1}.

Definition 3.5 The Greenlees filtration [9, p. 437] of ẼG = S^{∞λ} is an integerindexed G-cellular filtration of spectra, whose 2i-th term is S^{iλ} for each i. The (2i + 1)-th term is obtained from S^{iλ} by attaching a single G-free (2i + 1)-cell, and S^{(i+1)λ} is in turn obtained from it by attaching a single G-free (2i+2)-cell. The composite G-map S^{iλ} → S^{(i+1)λ} is given by smashing S^{iλ} with the onepoint compactification τ: S^{0} → S^{λ} of the inclusion {0} ⊂ λ. The Greenlees filtration induces an increasing filtration of X^{tG} = [ẼG ∧ F (EG_{+}, X)]^{G}, and a tower of homotopy cofibers with (2i + 1)-th term

X^{tG}⟨i⟩ = [ẼG/S^{iλ} ∧ F (EG_{+}, X)]^{G} ,  (3.1)

which we call the Tate tower. The associated spectral sequence is the homological G-equivariant Tate spectral sequence

Ê_{s,t}^{2} = Ĥ^{−s}(G; H_{t}(X))

converging to the continuous homology groups

H_{∗}^{c}(X^{tG}) = lim_{i} H_{∗}(X^{tG}⟨i⟩)

of X^{tG}, when X is a bounded below spectrum with H_{∗}(X) of finite type. See [17, Def. 2.3, Prop. 4.15]. Note that i tends to −∞ in this limit. We shall also refer to the continuous cohomology groups

H_{c}^{∗}(X^{tG}) = colim_{i} H^{∗}(X^{tG}⟨i⟩) ,

and note that H_{∗}^{c}(X^{tG}) ≅ H_{c}^{∗}(X^{tG})^{∗} (the Hom dual) when H_{∗}(X) is bounded below and of finite type, because then each H_{∗}(X^{tG}⟨i⟩) is also of finite type.

Definition 3.6 Let the G-map ξ: S^{λ} → S^{λ^{p}} of representation spheres be the suspension of the standard degree p covering map π: S(λ) → S(λ^{p}) of unit circles, as in the following vertical map of horizontal G-homotopy cofiber sequences:

[figure: S(λ)_{+}; π_{+}; S^{0}; τ; S^{λ}; ξ; S(λ^{p})_{+}; S^{0}; z; S^{λ^{p}} .]

We note that ξ is not induced by a linear map. It has degree p on the top cell, so ξ_{∗}: H_{∗}(S^{λ}) → H_{∗}(S^{λ^{p}}) is the zero homomorphism, since we work with reduced homology and mod p coefficients.

Proposition 3.7 Let X be a G-spectrum with H_{∗}(X) bounded below. Then

lim_{j} H_{∗}^{c}((Σ^{−jλ^{p}}X)^{tG}) = lim_{i,j} H_{∗}((Σ^{−jλ^{p}}X)^{tG}⟨i⟩) = 0

and

colim_{j} H_{c}^{∗}((Σ^{−jλ^{p}}X)^{tG}) = colim_{i,j} H^{∗}((Σ^{−jλ^{p}}X)^{tG}⟨i⟩) = 0 .

Proof In the notation of (3.1) we have a natural equivalence

ν: (Σ^{−jλ^{p}}X)^{tG}⟨i⟩ ⟶^{≃} (Σ^{j(λ−λ^{p})}X)^{tG}⟨i−j⟩

for each i and j. It is obtained from the G-equivalence

\frac{S^{∞λ} ∧ S^{jλ}}{S^{iλ}} ∧ F (EG_{+}, Σ^{−jλ^{p}}X) ⟶^{≃} \frac{S^{∞λ}}{S^{(i−j)λ}} ∧ F (EG_{+}, S^{jλ} ∧ Σ^{−jλ^{p}}X)

(see [14, III.1]) by passage to G-fixed points. Under these equivalences, the z-tower map

z: (Σ^{−(j+1)λ^{p}}X)^{tG}⟨i⟩ → (Σ^{−jλ^{p}}X)^{tG}⟨i⟩

induced by smashing with z: S^{0} → S^{λ^{p}} is compatible with the composite of the Tate tower map

τ: (Σ^{(j+1)(λ−λ^{p})}X)^{tG}⟨i−j−1⟩ → (Σ^{(j+1)(λ−λ^{p})}X)^{tG}⟨i−j⟩

induced by smashing with τ: S^{0} → S^{λ}, and the ξ-tower map

ξ: (Σ^{(j+1)(λ−λ^{p})}X)^{tG}⟨i−j⟩ → (Σ^{j(λ−λ^{p})}X)^{tG}⟨i−j⟩

induced by smashing with ξ: S^{λ} → S^{λ^{p}}, in the sense that the following diagram commutes up to homotopy:

\xymatrix{[ẼG/S^{iλ} ∧ F (EG_{+}, Σ^{−(j+1)λ^{p}}X)]^{G} \ar[dr]^{ν}_{≃} \ar[ddd]_{z} \\  & [ẼG/S^{(i−j−1)λ} ∧ F (EG_{+}, Σ^{(j+1)(λ−λ^{p})}X)]^{G} \ar[d]^{τ} \\  & [ẼG/S^{(i−j)λ} ∧ F (EG_{+}, Σ^{(j+1)(λ−λ^{p})}X)]^{G} \ar[dd]^{ξ} \\ [ẼG/S^{iλ} ∧ F (EG_{+}, Σ^{−jλ^{p}}X)]^{G} \ar[dr]_{ν}^{≃} \\  & [ẼG/S^{(i−j)λ} ∧ F (EG_{+}, Σ^{j(λ−λ^{p})}X)]^{G} .}

To see this, note that the diagram

[figure: S^{λ} ∧ F (EG_{+}, S^{0} ∧ X); ≃; ν; z; S^{0} ∧ F (EG_{+}, S^{λ} ∧ X); τ; S^{λ} ∧ F (EG_{+}, S^{λ} ∧ X); ξ; S^{λ} ∧ F (EG_{+}, S^{λ^{p}} ∧ X); S^{λ} ∧ F (EG_{+}, S^{λ^{p}} ∧ X)]

commutes up to G-homotopy (because the twist map S^{λ} ∧ S^{λ} ≅ S^{λ} ∧ S^{λ} is G-homotopic to the identity), replace X with Σ^{jλ−(j+1)λ^{p}}X, smash with ẼG/S^{(i−j−1)λ} and pass to G-fixed points.

Passing to homology, we get that z_{∗} is strictly compatible with the composite ξ_{∗}τ_{∗} under the isomorphisms ν_{∗}. Next pass to continuous homology by forming limits over i along the homomorphisms τ_{∗}. Then the homomorphism

z_{∗}: H_{∗}^{c}((Σ^{−(j+1)λ^{p}}X)^{tG}) → H_{∗}^{c}((Σ^{−jλ^{p}}X)^{tG})

is identified with the homomorphism

ξ_{∗}: H_{∗}^{c}((Σ^{(j+1)(λ−λ^{p})}X)^{tG}) → H_{∗}^{c}((Σ^{j(λ−λ^{p})}X)^{tG}) ,  (3.2)

so it suffices to show that the limit over j of the latter homomorphisms is zero. Let

Ê_{s,t}^{2}(j) = Ĥ^{−s}(G; H_{t}(Σ^{j(λ−λ^{p})}X)) ⟹ H_{s+t}^{c}((Σ^{j(λ−λ^{p})}X)^{tG})

be the homological Tate spectral sequence for the j-th term in the ξ-tower.

By naturality of the Tate spectral sequence, the homomorphism ξ_{∗} above is compatible with the spectral sequence map Ê_{∗∗}^{2}(j + 1) → Ê_{∗∗}^{2}(j) that is induced on Tate cohomology by the G-module homomorphism

ξ_{∗}: H_{∗}(Σ^{(j+1)(λ−λ^{p})}X) → H_{∗}(Σ^{j(λ−λ^{p})}X) .

This homomorphism is zero, since ξ_{∗}: H_{∗}(S^{λ}) → H_{∗}(S^{λ^{p}}) is zero. Hence the map of spectral sequences is also zero. It follows that the homomorphism ξ_{∗} in (3.2) strictly reduces the Tate filtration (= s) of each nonzero continuous homology class. Equivalently, ξ_{∗} strictly increases the vertical degree (= t) of the spectral sequence representative of each nonzero class.

By assumption, there is an integer ℓ such that H_{t}(X) = 0 for all t < ℓ. Then Ê_{s,t}^{2}(j) = Ê_{s,t}^{∞}(j) = 0 for t < ℓ and any j. If x = (x_{j})_{j} is an arbitrary element of lim_{j} H_{∗}^{c}((Σ^{j(λ−λ^{p})}X)^{tG}), then x_{j} = ξ_{∗}^{m}(x_{j+m}) for each m ≥ 0. If x_{j} is represented in vertical degree t, then x_{j+m} must be represented in vertical degree ≤ (t−m). Choosing m so large that t−m < ℓ, it follows that x_{j+m} = 0, which implies x_{j} = 0. Repeating the argument for each j we see that x = 0, so lim_{j} H_{∗}^{c}((Σ^{j(λ−λ^{p})}X)^{tG}) must be the trivial group.

Let M = colim_{j} H_{c}^{∗}((Σ^{−jλ^{p}}X)^{tG}). Then the Hom dual M^{∗} is the limit group we just showed is zero, and M injects into its double Hom dual M^{∗∗}, so M = 0 as well.

Proposition 3.8 Let X be a G-spectrum such that π_{∗}(X) is bounded below and H_{∗}(X) is of finite type. Then the p-adic completion Y_{p}of

Y = holim_{j} (Σ^{−jλ^{p}}X)^{tG}

is contractible. Hence the map Γ_{n−1} : X^{tG} → (X^{tC})^{hḠ} is a p-adic equivalence.

Proof The spectrum Y is the homotopy limit over i and j of the spectra

(Σ^{−jλ^{p}}X)^{tG}⟨i⟩ = [ẼG/S^{iλ} ∧ F (EG_{+}, Σ^{−jλ^{p}}X)]^{G} ,

which can be rewritten as

(ẼG/S^{iλ} ∧ Σ^{−jλ^{p}}X)_{hG}

by the Adams equivalence [14, II.8.4], since ẼG/S^{iλ} is a free G-CW spectrum. Each of these is bounded below with mod p homology of finite type. Hence there is an inverse limit Adams spectral sequence

E_{2}^{∗∗} = Ext_{𝒜}^{∗∗}(M, 𝔽_{p}) ⟹ π_{∗}(Y_{p})

converging to the p-adic homotopy of that homotopy limit (see [8, Prop. 7.1] and [17, Prop. 2.2]), where

M = colim_{j} H_{c}^{∗}((Σ^{−jλ^{p}}X)^{tG}) .

The latter 𝒜 -module was shown to be zero in Proposition 3.7, hence the E_{2}-term is zero and Y_{p} is contractible. The second conclusion follows from Proposition 3.3.

Proof of Theorem 2.4 Consider the diagram in Proposition 3.3. By assumption, the maps

Γ_{n−1} : Φ^{C}(X)Ḡ → Φ^{C}(X)^{hḠ}  and  Γ_{1} : X^{C} → X^{hC}

are (W, k)-coconnected. Hence Γ̂_{1} : Φ^{C}(X) → X^{tC} is (W, k)-coconnected, so by Lemma 3.4 also

(Γ̂_{1})^{hḠ}: Φ^{C}(X)^{hḠ} → (X^{tC})^{hḠ}

is (W, k)-coconnected. By Proposition 3.8, the map Γ_{n−1} : X^{tG} → (X^{tC})^{hḠ} is a p-adic equivalence, hence (W, −∞)-coconnected, by our standing assumption that W is in the localizing ideal of spectra generated by S^{−1}/p^{∞}. It follows easily that Γ̂_{n}: Φ^{C}(X)Ḡ → X^{tG} is (W, k)-coconnected, which is equivalent to Γ_{n}: X^{G} → X^{hG} being (W, k)-coconnected.

Proof of Theorem 2.5 This follows by induction on n, using Theorem 2.4 and the observation that

Φ^{C_{p}}(Φ^{C_{p^{e}}} (X)) ≅ Φ^{C_{p^{e+1}}} (X)

for all 0 ≤ e < n.

Proof of Theorem 2.7 This follows from Theorem 2.5 in the case X = B^{∧p^{n}}, W = S^{−1}/p^{∞} and k = −∞, once we show that for each 0 ≤ e < n there is a C_{p^{n−e}}-equivalence

Y = Φ^{C_{p^{e}}} (B^{∧p^{n}}) ≃ B^{∧p^{n−e}} ,

the right hand side is bounded below with mod p homology of finite type, and Γ_{1} : Y^{C_{p}} → Y^{hC_{p}} is a p-adic equivalence. The first claim follows from the proof in simplicial degree 0 of [12, Prop. 2.5]. Writing Y ≃ Z^{∧p}, where Z = B^{∧p^{n−e−1}} is bounded below with H_{∗}(Z) of finite type, the other claims also follow, since Γ_{1} : (Z^{∧p})^{C_{p}} → (Z^{∧p})^{hC_{p}} is a p-adic equivalence by [17, Thm. 5.13], generalizing [6, §II.5].

Proof of Theorem 2.8 There is a C_{p^{n−1}}-equivalence

r: Φ^{C_{p}}T HH(B) ⟶^{≃} T HH(B)

(the cyclotomic structure map of T HH(B), see [12, §2.5]), whose e-fold iterate is a C_{p^{n−e}}-equivalence Φ^{C_{p^{e}}} (T HH(B)) ≃ T HH(B). It is clear from the simplicial definition that T HH(B) is connective and has mod p homology of finite type, hence the theorem follows from Theorem 2.5.

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