The localized slice spectral sequence, norms of Real bordism, and the Segal conjecture

In this paper, we introduce the localized slice spectral sequence, a variant of the equivariant slice spectral sequence that computes geometric fixed points equipped with residue group actions. We prove convergence and recovery theorems for the localized slice spectral sequence and use it to analyze the norms of the Real bordism spectrum. As a consequence, we relate the Real bordism spectrum and its norms to a form of the $C_2$-Segal conjecture. We compute the localized slice spectral sequence of the $C_4$-norm of $BP_\mathbb{R}$ in a range and show that the Hill--Hopkins--Ravenel slice differentials is in one-to-one correspondence with a family of Tate differentials for $N_1^2 H{\mathbb{F}}_2$.


Introduction
The complex conjugation action on the complex bordism spectrum M U defines a C 2 -spectrum M U R , the Real bordism spectrum of Landweber, Fujii, and Araki [Lan68,Fuj76,Ara79]. Its norms M U ((C 2 n )) := N C 2 n C2 M U R = N 2 n 2 M U R have played a central role in the solution of the Kervaire invariant one problem [HHR16]. After localizing at 2, the norm M U ((C 2 n )) splits as a wedge of suspensions of BP ((C 2 n )) := N 2 n 2 BP R , where BP R is the Real Brown-Peterson spectrum.
The spectra M U ((C 2 n )) and BP ((C 2 n )) connect many fundamental objects and computations in non-equivariant stable homotopy theory to equivariant stable homotopy theory. The fixed points of these norms are ring spectra, and their Hurewicz images detect families of elements in the stable homotopy groups of spheres [HHR16,Hil15,LSWX19]. The Lubin-Tate spectra at prime 2 with finite group actions can also be built from these norms and their quotients [HS20,BHSZ21]. They produce higher height analogues of topological K-theory and play a fundamental role in chromatic homotopy theory.

MEIER, SHI, AND ZENG
To compute the equivariant homotopy groups of M U ((C 2 n )) and BP ((C 2 n )) , Hill, Hopkins, and Ravenel introduced the equivariant slice spectral sequence [HHR16]. However, due to the complexity of the equivariant computations, besides M U R and BP R , we still know relatively little about the behavior of their norms. For example, we are still far from a complete understanding of the equivariant homotopy groups of BP ((C4)) .
Our project arose from the desire to systematically understand the equivariant homotopy groups of M U ((C 2 n )) and BP ((C 2 n )) . The goal of this paper is two-fold: first, we establish our main computational tool, the localized slice spectral sequence. This is a variant of the slice spectral sequence that is easier for computations while at the same time recovers the original slice differentials. Second, as an application of the localized slice spectral sequence, we focus on the C 4 -norm BP ((C4)) . We compute its localized slice spectral sequence in a range and build a new connection to the Segal conjecture at C 2 . As a consequence, we establish correspondences between families of slice differentials for BP ((C4)) and families of differentials in the Tate spectral sequence for N 2 1 HF 2 .

Fixed points and geometric fixed points.
It is well-known in equivariant stable homotopy theory that a map between G-spectra is a weak equivalence if and only if for all subgroups H ⊂ G, it induces (non-equivariant) weak equivalences on all H-fixed points or H-geometric fixed points. Despite this fact, fixed points and geometric fixed points behave very differently.
The fixed points of a G-spectrum X can be difficult to understand. For a suspension spectrum, its fixed points can be described by using the tom Dieck splitting [LMSM86,Section V.11], but such a splitting does not exist in general. Nevertheless, by the Wirthmüller isomorphism, there are natural maps between fixed points of different subgroups of G. The induced maps on their homotopy groups can be assembled into an algebraic object π * X, called a Mackey functor. Organizing information in terms of Mackey functors is one of the most powerful ideas in equivariant stable homotopy theory, and this has produced new insights in both theory and computation (e.g. [GM20b,HHR16]).
As an important example, the C 2 -fixed points of the Real bordism spectrum M U R is computable but complicated [HK01,GM17]. For groups beyond C 2 , we still don't know very much about the fixed points of the norms M U ((C 2 n )) aside from the computations in [HHR16,HHR17,Hil15,HSWX18]. Nevertheless, these fixed points contain very rich information about the stable homotopy groups of spheres (such as the Kervaire invariant elements) and chromatic homotopy theory [HHR16,LSWX19,HS20,BHSZ21].
On the other hand, the geometric fixed points are easier to understand. The geometric fixed points functor Φ H : Sp G → Sp is compatible with the suspension spectrum functor, commutes with all homotopy colimits, and is symmetric monoidal.
For the Real bordism spectrum M U R , a straightforward geometric argument, based on the fact that the fixed points of the C 2 -Galois action on C is R, shows that the C 2 -geometric fixed points of M U R and BP R are M O (the unoriented bordism spectrum) and HF 2 , respectively. The geometric fixed points functor also behaves well with respect to the norm functor [HHR16,Proposition 2.57]. This renders the geometric fixed points of the norms M U ((C 2 n )) easy to understand.
Although the homotopy groups of the geometric fixed points for various subgroups do not form a Mackey functor, there are reconstruction theorems which recovers a G-spectrum from structures on its geometric fixed points [AK15,Gla15,AMGR17].
At this point, it is natural to ask the following questions: (1) How do the fixed points and the geometric fixed points of an equivariant spectrum interact with each other? (2) Computationally, how to recover the fixed points of equivariant spectra, such as norms of M U R , through their geometric fixed points, which are significantly easier to compute? In order to attack these questions, the first observation is that it is necessary to consider the H-geometric fixed points not only as a non-equivariant spectrum, but as a W G (H)-equivariant spectrum, where W G (H) is the Weyl group. In our examples of interest, H will be a normal subgroup of G, so that W G (H) ∼ = G/H. When the G-spectrum is of the form N G H X, we prove the following theorem.
Theorem 1.1. Let H ⊂ G be a normal subgroup and X be an H-spectrum. Then we have an equivalence of G/H-spectra . If X is an H-commutative ring spectrum, then this equivalence is an equivalence of G/H-commutative ring spectra.
This theorem is by no means difficult to prove, and in fact it only marks the starting point of our analysis. To understand how the H-fixed points and H-geometric fixed points interact with each other, we introduce our main computational tool: the localized slice spectral sequence.
Let X be a G-spectrum and H ⊂ G a normal subgroup. As a G/H-spectrum, Φ H X can be constructed as (

ẼF[H]∧X) H , whereẼF[H] is the universal space of the family F[H]
consisting of all subgroups that do not contain H. In many cases, including G cyclic, smashing withẼF[H] is equivalent to inverting an Euler class a V ∈ π G −V S 0 for V a certain G-representation. In particular, the residue fixed points (Φ H X) G/H are equivalent to the fixed points (a −1 V X) G . To define the localized slice spectral sequence, let P • X be the regular slice tower of X [HHR16] [Ull13]. The a V -localized slice spectral sequence of X is, by definition, the spectral sequence corresponding to the localized tower {a −1 V P • X}. It has E 2 -page E s,t 2 = π t−s a −1 V P t t X. Theorem 1.2. Let X be a C 2 n -spectrum and V be an actual C 2 n -representation. Then the a V -localized slice spectral sequence converges strongly to the homotopy groups π t−s a −1 V X. The localized slice spectral sequence serves as a bridge between the fixed points X G and the residue fixed points (Φ H X) G/H . More precisely, even though the localized slice spectral sequence only computes the geometric fixed points, its E 2 -page is closely related to the original slice spectral sequence, which computes the fixed points. From now on, we will denote the regular slice spectral sequence and the a V -localized slice spectral sequence of X by SliceSS(X) and a −1 V SliceSS(X), respectively. The following theorem directly follows from computations of the homotopy groups of HZ [HHR17, Section 3]. Theorem 1.3. Let X be a (−1)-connected C 2 n -spectrum whose slices are wedges of the form C 2 n + ∧ C 2 k Σ iρ k HZ, and λ be the 2-dimensional real C 2 n -representation that is rotation by π 2 n−1 . Then the localizing map SliceSS(X) −→ a −1 λ SliceSS(X) induces an isomorphism on the E 2 -page for classes whose filtration is greater than 0. On the 0-line, this map is surjective, with kernel consisting of elements in the image of the transfer T r C 2 n e .
By the slice theorem [HHR16, Theorem 6.1], the C 2 n -norms of M U R and BP R both satisfy the conditions of Theorem 1.3.
An upshot of Theorem 1.3 is that despite the fact that the fixed points are harder to compute than the geometric fixed points, if we already know the differentials in the localized slice spectral sequence, then we can use the isomorphism on the E 2 -page given by Theorem 1.3 to recover differentials in the original slice spectral sequence. This allows us to approach the computation of the fixed points X G from the residue fixed points (Φ H X) G/H . A subtlety that arises from the localized slice spectral sequence is its compatibility with multiplicative structures. More precisely, let R be a connective G-commutative ring spectrum. Ullman [Ull13] has shown that the slice tower of R is multiplicative. Therefore, the corresponding slice spectral sequence has all the desired multiplicative properties such as the Leibniz rule, the Frobenius relation [HHR17, Definition 2.3], and most importantly, the norm [HHR17, Corollary 4.8]. On the other hand, the localization a −1 V R can never be a G-commutative ring spectrum because its underlying spectrum is contractible.
To establish multiplicative properties for the localizations, we apply the theory of N ∞ -operads from [BH15]. More precisely, in Section 2.5, we establish a criterion generalizing the results of [HH14] and [Böh19]. As a consequence, we obtain the following theorem, which shows that a V -localization preserves algebra structures over a certain N ∞ -operad O that depends on the class a V .
Theorem 1.4. Let V be a G-representation. Assume that Ind H K Res G K V is a summand of a multiple of Res G H V for every K ⊂ H ⊂ G such that H/K is an admissible H-set. Then localization at a V preserves O-algebras.
Therefore, the homotopy of the a V -localization of an equivariant commutative ring spectrum such as M U ((C 2 n )) forms an incomplete Tambara functor [BH18], and the norm maps essential to our computation are still available. In Section 3.4, we draw consequences of the behavior of norms in the localized slice spectral sequence.
Aside from the localized slice spectral sequence a −1 λ SliceSS(X), the G/H-slice spectral sequence of Φ H X also computes the residue fixed points (Φ H X) G/H . Even though both spectral sequences compute the same homotopy groups, their behaviors can be very different. Surprisingly, we have the following theorem, which shows that after a modification of filtrations, there is map between the two spectral sequences.
Theorem 1.5. Let X be a C 2 n -spectrum, then there is a canonical map of spectral sequences a −1 λ SliceSS C 2 n (X) → P * C 2 n /C2 DSliceSS C 2 n /C2 (Φ C2 X) that converges to an isomorphism in homotopy groups. Here D is the doubling operation defined in Section 3.5, which slows down a tower by a factor of 2, and P * C 2 n /C2 is the pullback functor from [Hil12], which is recalled in Section 2.2.
In the second half of the paper, as an application of all the tools that we have developed, we will use the localized slice spectral sequence to analyze the norms of M U R . The Segal conjecture is a deep result in equivariant homotopy theory. In its original formulation, it was proven by Lin [Lin80] for the group C 2 and by Carlsson [Car84] for all finite groups, building on the works of May-McClure [MM82] and Adams-Gunawardena-Miller [AGM85]. When the group is C 2 , the most general formulation can be found in Lunøe-Nielsen-Rognes [LNR12] and Nikolaus-Scholze [NS18]: for every bounded below spectrum X, the Tate diagonal map X → (N 2 1 X) tC2 is a 2-adic equivalence. We are interested in the case when X = HF 2 , the mod 2 Eilenberg-Mac Lane spectrum. This case is intriguing for at least two reasons: first, Nikolaus-Scholze [NS18] show that the general formulation follows formally from this case. Second, even though the Segal conjecture implies the equivalence HF 2 (N 2 1 HF 2 ) tC2 , this is still a mystery from a computational perspective. More precisely, the Tate spectral sequence computing (N 2 1 HF 2 ) tC2 has E 2 -pageĤ * (C 2 ; A * ), the Tate cohomology of the dual Steenrod algebra A * with the conjugate C 2 -action. This cohomology is highly nontrivial and we currently don't even have a closed formula [Bru]. However, because of the equivalence HF 2 (N 2 1 HF 2 ) tC2 given by the Segal conjecture, every element besides 1 ∈ F 2 ∼ =Ĥ 0 (C 2 ; (A * ) 0 ) must either support or receive a differential.
Understanding equivariant equivalences from a computational perspective can be extremely useful. For example, in the case of BP R and its norms, it is relatively straightforward to establish the equivalence Φ C 2 n BP ((C 2 n )) Φ C2 BP R HF 2 . By working backwards, Hill-Hopkins-Ravenel used this equivalence to prove a family of differentials in the slice spectral sequence of BP ((C 2 n )) [HHR16, Theorem 9.9], from which their Periodicity Theorem and eventually the nonexistence of the Kervaire invariant elements followed.
By Theorem 1.1, we have a C 2 -equivalence For the left hand side, we can use the localized slice spectral sequence to compute the C 2 -fixed points of Φ C2 BP ((C4)) . We demonstrate this computation in a range (Theorem 4.4). Note that we can actually compute much further than the range we have shown, but the point is to give the readers a taste of the computations involved and to draw comparisons to the slice spectral sequence computations in [HHR16,HHR17,HSWX18]. After demonstrating these computations, we use Theorem 1.5 to establish a map between the slice spectral sequence of BP ((C4)) and the Tate spectral sequence of N 2 1 HF 2 . We prove that this map establishes a correspondence between families of differentials in the two spectral sequences.
Theorem 1.6. After the E 2 -page, the Hill-Hopkins-Ravenel slice differentials [HHR16, Theorem 9.9] are in one-to-one correspondence to a family of differentials on the first diagonal of slope (−1) in the Tate spectral sequence of N 2 1 HF 2 . This completely determines all the differentials in the Tate spectral sequence that originate from the first diagonal of slope (−1).
In the future, we wish to reverse the flow of information: to prove slice differentials from spectral sequences associated to N 2 1 HF 2 . Computations along this line appear in [BHL + 21] and will be refined in an upcoming article by the same authors. There are various methods to study the norms of HF 2 and their modules, such as the modified Adams spectral sequence [Rav84,BBLNR14] and the descent spectral sequence [HW21]. These methods allow one to understand modules over norms of HF 2 and BP R from different perspectives.
It is worth noting that in another direction, one can prove the C 2 -Segal conjecture by showing that N 4 2 M U R is cofree and using Theorem 1.1. This approach is taken by Carrick in [Car22]. Theorem 1.6 has an unexpected consequence. Let R be an arbitrary non-equivariant (−1)connected homotopy ring spectrum with π 0 R ∼ = Z (or a localization thereof not containing 1 2 ). We can use the (stable) EHP spectral sequence and the Tate spectral sequence of N 2 1 HF 2 to bound the length of differentials on powers of the Tate generator in the Tate spectral sequence of N 2 1 R. Theorem 1.7. Let v ∈Ĥ 2 (C 2 ; π 0 N 2 1 R) be the generator of the Tate cohomology, and l k be the length of differential that v 2 k supports in the Tate spectral sequence of N 2 1 R. Then where ρ(n) is the Radon-Hurwitz number.
1.4. Outline of paper. In Section 2, we recall a few basics of equivariant homotopy theory. In particular, we discuss the interplay between the norm functor, the geometric fixed points functor, and the pull back functor. We prove Theorem 1.1. We also investigate the multiplicative structure of localizations and give a criterion for a localization at an element to preserve multiplicative structures, thus proving Theorem 1.4.
In Section 3, we recall the spectra M U ((G)) and BP ((G)) and their slice spectral sequences. We then introduce the main computational tool for this paper, the localized slice spectral sequence. We prove Theorem 1.2, the strong convergence of the localized slice spectral sequence (Theorem 3.3). We also discuss exotic extensions and norms.
Sections 4 and 5 are dedicated to the computation of the localized slice spectral sequence of a −1 λ BP ((C4)) . In Section 4, we give an outline of the computation and list our main results (Theorem 4.1 and Theorem 4.4). The detailed computations are in Section 5. While computing differentials, we make full use of the Mackey functor structure of the spectral sequence. Certain differentials are proven using exotic extensions and norms by methods established in Section 3.
In Section 6, we turn our attention to the Tate spectral sequence of N 2 1 HF 2 . We use the computation of the localized slice spectral sequence of BP ((C4)) to prove families of differentials and compute the Tate spectral sequence in a range. In particular, Theorem 1.6 is proven as Theorem 6.6, which describes the first infinite family of differentials in the Tate spectral sequence.
Acknowledgments. The authors would like to thank Bob Bruner for sharing his computation on the Tate generators of the dual Steenrod algebra, and J.D. Quigley for sharing his computation of the Adams spectral sequence of N 2 1 HF 2 . The authors would furthermore like to thank Agnès Beaudry, Christian Carrick, Gijs Heuts, Mike Hill, Tyler Lawson, Guchuan Li, Viet-Cuong Pham, Doug Ravenel, John Rognes and Jonathan Rubin for helpful conversations. Finally, we would like to thank the anonymous referee for the many detailed suggestions. The second author was supported by National Science Foundation grant DMS-2104844.

Conventions.
(1) Given a finite group G, all representations will be finite-dimensional and orthogonal. Per default actions will be from the left. (2) We denote by ρ G the real regular representation of a finite group G and we abbreviate ρ C2 to ρ 2 . (3) We will often use the abbrevation BP ((C4)) for N 4 2 BP R and more generally BP ((G)) for N G C2 BP R .
(4) All spectral sequences use the Adams grading.
(5) We use the regular slice filtration and its corresponding tower and spectral sequence defined in [Ull13] throughout the paper, often omitting "regular".
2. Equivariant stable homotopy theory 2.1. A few basics. We work in the category of genuine G-spectra for a finite group G, and our particular model will be the category of orthogonal G-spectra Sp G . For us these will be simply Gobjects in orthogonal spectra as in [Sch14], which will often be just called G-spectra. This category is equivalent to the categories of orthogonal G-spectra considered in [MM02] and [HHR16]. In particular, we are able to evaluate a G-spectrum at an arbitrary G-representation to obtain a G-space. We refer to the three cited sources for general background on G-equivariant stable homotopy theory, of which we will recall some for the convenience of the reader. For each G-representation V , we denote by S V its one-point compactification. Denoting further by ρ G the regular representation, we obtain for each subgroup H ⊂ G and each G-spectrum its homotopy groups π H n (X) = colim k [S kρ G +n , X(kρ G )] H . These assemble into a Mackey functor π n (X). A map of G-spectra is an equivalence if it induces an isomorphism on all π n . Inverting the equivalences of G-spectra in the 1-categorical sense yields the genuine equivariant stable homotopy category Ho(Sp G ) and inverting them in the ∞categorical sense the ∞-category of G-spectra Sp ∞ G . These constructions are well-behaved as there is a stable model structure on Sp G with the weak equivalences we just described [MM02,Theorem III.4.2]. The fibrant objects are precisely the Ω-G-spectra. In the main body of the paper we will implicitly work in Ho(Sp G ) or Sp ∞ G ; in particular, commutative squares are meant to be only commutative up to (possibly specified) homotopy.
By [MM02,Proposition V.3.4], the categorical fixed point construction Sp G → Sp is a right Quillen functor. We call the right derived functor (−) G : Sp ∞ G → Sp ∞ the (genuine) fixed points. We can define fixed point functors for subgroups H ⊂ G by applying first the restriction functor Sp G → Sp H and then the H-fixed point functor. One easily shows that π n X H ∼ = π H n X. Thus, a map is an equivalence if it is an equivalence on all fixed points.
Note that if H ⊂ G is normal, the categorical fixed points carry a residual G/H-action. The resulting functor Sp G → Sp G/H is a right Quillen functor as well [MM02,p. 81] and thus H-fixed points actually define a functor Sp ∞ G → Sp ∞ G/H . The left adjoint of this is the inflation functor p * associated to the projection p : G → G/H.
As π H n translates filtered homotopy colimits into colimits, we see that fixed points Sp ∞ G → Sp ∞ preserve filtered homotopy colimits. As they preserve homotopy limits as well (as they are induced by a Quillen right adjoint) and are a functor between stable ∞-categories, they preserve all finite homotopy colimits [Lur17, Proposition 1.
Examples of such families include the case F = {e} of just the trivial group, where we denote EF by EG, and the case F = P of all proper subgroups. To each family, we can associate furthermore the cofiberẼF of EF + → S 0 , which is again characterized by its fixed points For each family F and every G-spectrum we have an associated isotropy separation diagram, whose rows are parts of cofiber sequences: Upon taking fixed points, we can identify some of the entries with well-known constructions. If EF = EG, then (X EF+ ) G is the spectrum of homotopy fixed points X hG and (X ∧ EF + ) G is (by the Adams isomorphism) the spectrum of homotopy orbits X hG . Moreover, one calls in this case (X EF+ ∧ẼF) G the Tate construction and denotes it by X tG . If F = P, then (X ∧ EP) G is called the geometric fixed points and denoted by Φ G X.
Let H ⊂ G be normal. As mentioned above, H-fixed points define a functor Sp ∞ G → Sp ∞ G/H . We want to define a similar version for geometric fixed points. Let F[H] be the family of all subgroups of G not containing H. We consider the functor This agrees with our previous definition when H = G since F[G] = P. Another important special case is G = C 2 n and H = C 2 ; thenẼF[H] = EG.
As the geometric fixed points functor Φ H : Sp ∞ G → Sp ∞ G/H is the composition of smashing with a space and taking fixed points, it preserves all homotopy colimits as well.
This property implies that Φ H must possess a right adjoint, which was constructed in [Hil12, Definition 4.1] as the pullback functor where p * is the functor induced by the projection p : Here we view V also as an element of RO(G) by pullback along G → G/H.
Proof. Essentially by definition, π K V (P * G/H X) ∼ = [Σ V G/K + , P * G/H X] G . By the containment H ⊂ K, all points in G/K are H-fixed and moreover V H = V . Hence we get Φ H Σ V G/K + Σ V (G/H)/(K/H) + . By the adjointness of Φ H and P * G/H we thus obtain the result. 2.3. Universal properties of G-spectra. In [GM20a, Corollary C.7], Gepner and the firstnamed author established a universal property for symmetric monoidal colimit-preserving functors out of Sp ∞ G . We will need a variant of this for functors just preserving filtered colimits. Localizing the 1-category of pointed finite G-CW-complexes at G-homotopy equivalences yields an ∞-category S fin,G * . This ∞-category is essentially small. For every essentially small ∞category C, we can freely adjoin filtered colimits to obtain an ∞-category Ind(C) [Lur09, Section 5.3]. The inclusion S fin,G * → S G * into the ∞-category of pointed G-spaces induces a functor Ind(S fin,G * ) → S G * . Since S fin,G * consists of compact objects inside S G * and generates S G * under filtered colimits, the functor is an equivalence.
Let us explain to obtain G-spectra and finite G-spectra as stabilization of S G * and S fin,G * respectively. Let U be a complete G-universe and denote by Sub U the poset of finite-dimensional subrepresentations. Following [GM20a, Appendix C], we can consider functors T and T fin from Sub U to Cat ω ∞ (resp. Cat ∞ ), sending each V ∈ Sub U to S G * (resp. S fin,G * ) and each inclusion V ⊂ W to smashing with S W −V . Here, Cat ω ∞ is the ∞-category of compactly generated ∞-categories with compact object preserving left adjoints as morphisms, and W − V is the orthogonal complement of V in W . As explained in [GM20a, Appendix C], colim Sub U T carries a canonical symmetric monoidal structure, which is as a symmetric monoidal ∞-category canonically equivalent to Sp G ∞ . Denote colim Sub U T fin by Sp G,fin ∞ . General properties of colimits in Cat ω ∞ ( [Lur09, Proposition 5.5.7.10]) imply that the functor Sp G,fin ∞ → Sp G ∞ extends to an equivalence Ind(Sp G,fin ∞ ) Sp G ∞ . This yields directly: Lemma 2.2. Let D be an ∞-category with filtered colimits. The space of functors Sp G ∞ → D preserving filtered colimits is equivalent to that of functors Sp fin,G ∞ → D.
Remark 2.3. With our convention that G is always finite, we could simplify the colimit colim Sub U T to the colimit of the directed system and similarly for S fin,G * . For possible future applications, we chose however to present the proofs in this section in a way that applies to all compact Lie groups.
We want to discuss a universal property of Sp fin,G ∞ using symmetric monoidal structures. For this, we need the following result of Robalo. Recall here that an object X in a symmetric monoidal ∞-category is symmetric if the cyclic permutation of X ⊗ X ⊗ X is homotopic to the identity Proposition 2.4. Let C be a small symmetric monoidal ∞-category and X ∈ C symmetric. Then C[X −1 ] := colim C X⊗ −−→ C X⊗ −−→ · · · has a symmetric monoidal structure such that C → C[X −1 ] refines to a symmetric monoidal functor, which is initial among all those that send X to an invertible object.

Norms and pullbacks.
In this section, we will identify certain localizations of norm functors with pullbacks of norms from quotient groups. In the case of BP ((G)) this is a central ingredient of this paper.
First, we will recall the norm construction. For a group G, let BG denote the category with one object and having G as morphisms. Given an arbitrary symmetric monoidal category (C, ⊗, 1), there is for a subgroup H ⊂ G a norm functor from H-objects to G-objects, where the G-action is induced by the right G-action on G. In the case of spaces or sets, one can identify X × H G with Map H (G, X) and for based spaces or sets, one can likewise identify X ∧ H G with Map * H (G, X). In the case of orthogonal spectra, one can by [HHR16,Proposition B.105] left derive the functor (−) ∧ H G to obtain a functor N G H . (Often, N G H is also used for the corresponding underived functor, but the derived functor will be more important for us.) The functor N G H commutes with filtered (homotopy) colimits by [HHR16, Propositions A.53, B.89]. Note moreover that N G H Σ ∞ X Σ ∞ Map * H (G, X) (if X is cofibrant or at least well-pointed) as Σ ∞ is symmetric monoidal.
Lemma 2.8. Let G be a finite group, K, H ⊂ G be two subgroups and X be a (based) topological H-space. Let H\G/K = {Hg 1 K, . . . , Hg l K}. Then there are natural (based) homeomorphisms where the K-action on the mapping spaces is induced by the right K-action on G. In particular, if H = K is normal, we obtain a natural G/H-equivariant homeomorphism Proof. The first two statements follow from the H-K-equivariant decomposition of G into l i=1 Hg i K. For the last one observe that if H = K is normal, H\G/K = G/H and G/H permutes the factors of the decomposition in (1).
To put the following theorem and its corollary into context, recall from [HHR16,Proposition B.213 and H is normal. Before we do so in Corollary 2.11, we provide a version that gives an equivalence on the level of G-spectra, i.e. before taking fixed points.
Theorem 2.9. Let H ⊂ G be a normal subgroup and X be an H-spectrum. Then we have an equivalence of G-spectraẼ Proof. We have for all H-spectra X. Indeed: If X is a suspension spectrum, this reduces to the space-level statement Map * H (G, X) H Map * (G/H, X H ), which is part of Lemma 2.8. Both sides of (2) are symmetric monoidal and commute with filtered homotopy colimits. Thus Corollary 2.6 implies the claim.
Applying P * G/H to (2), it suffices to check that Corollary 2.10. Let K ⊂ H ⊂ G be subgroups and assume that H ⊂ G is normal. Let moreover X be a K-spectrum. Then there is an equivalence of G-spectrã Proof. This follows from Theorem 2.9 by applying it to N H K X. Here, we use Taking H-fixed points we obtain a strengthened form of Theorem 1.1: Corollary 2.11. Let K ⊂ H ⊂ G be subgroups and assume that H ⊂ G is normal. Let moreover X be a K-spectrum. Then there is an equivalence of G/H-spectra Remark 2.12. An alternative proof of this result is possible using [Yua23, Theorem 2.7].
As we will recall below, there is a C 2 -spectrum BP R with geometric fixed points HF 2 . For G = C 4 and H = C 2 , we can expressẼF[H] as S ∞λ , where λ is the 2-dimension representation of C 4 corresponding to rotation by an angle of π 2 . Denoting the norm N C4 C2 BP R by BP ((C4)) , we obtain our main example for Theorem 2.9.
Corollary 2.13. There is an equivalence We end this section with a different kind of compatibility of norms and pullbacks.
Proposition 2.14. Let K ⊂ H ⊂ G be subgroups such that K is normal in G. Then there is a natural equivalence N G H P * Proof. Since both N G H P * H/K and P * G/K N G/K H/K commute with filtered colimits and are symmetric monoidal, it suffices (as in the proof of Theorem 2.9) to provide a natural equivalence of their restriction to suspension spectra. We compute is admissible, and the groups π H R assemble into an RO(G)-graded incomplete Tambara functor.
As already observed in [McC96], localizations only need to preserve naive E ∞ -structures, but not G-E ∞ -structures. Later, [HH14] gave a criterion when localizations indeed preserve G-E ∞structures and this was extended in [Böh19] to N ∞ -algebras, albeit only for localizations of elements in degree 0. In this section, we will extend this work to elements in non-trivial degree and follow the proof strategy of [Böh19, Proposition 2.30].
Let us first recall what localizing at some x ∈ π G V S means. We say that a G-spectrum E is x-local if x acts invertibly on E or, equivalently, on π G E. Given a G-spectrum E, we construct its x-localization as In particular, we can reformulate Corollary 2.13 as We will use the following specialization of a criterion of [GW18, Corollary 7.10]: Proposition 2.17. Localization at x preserves O-algebras if and only if To reformulate this criterion, we need the following lemma.
we obtain precisely Here we have used that the norm of a representation sphere is computed by induction. As both N H K and Res G K preserve filtered homotopy colimits, the result follows.
becomes a unit after inverting Res G H (x) and just must divide a power of it.
we assume that f ∧ x −1 S is an equivalence. As N H K and Res G H are symmetric monoidal, we see that We specialize now to the case that x is the Euler class a V : S 0 → S V . In this case we Thus to see which multiplicative structure localization at a V preserves, we only have to understand divisibility relations between Euler classes. In particular, we obtain the following corollary: Remark 2.21. While this corollary is everything we need, one can be more precise. For a H-representation V , let F fix V be the family of subgroups K ⊂ H such that V K = 0. Thus, (This is a weaker condition than W being contained in a multiple of V : for example, take G = C 8 and W and V be the two-dimensional real representation corresponding to rotation by 1 8 · 2π and 3 8 · 2π, respectively, which both have trivial fixed point family.) Specializing Proposition 2.19 thus yields: For a G-representation V , localization at a V preserves O-algebras if and only if F fix Example 2.22. Let G = C 2 n and λ = λ n be the two-dimensional representation of C 2 n given by rotation by an angle of 2π 2 n . We observe that Res Thus localizing at a λ preserves O-algebras if the following holds: H/K is H-admissible if and only if K = e. In particular, we see that for any commutative C 2 n -spectrum R, the localization a −1 λ R admits norms from π C 2 k * to π C 2 n * for 0 < k < n, but will not admit norms from π e * unless the target is zero. The example we care most about is a −1 λ M U ((C 2 n )) . These considerations have consequences for the multiplicative behaviour of the pullback functor is a graph subgroup of (C 2 n /C 2 ) × Σ n . This means that H/K is H-admissible if and only if K = e. Note further that Using the paragraph above we see that P * C 2 n /C2 R retains the structure of a p * O-algebra.
Likewise we can apply our considerations to the geometric fixed point functor. With p * O as above, we see that for a G-commutative ring spectrum R, the localization a −1 λ R retains an action of p * O and thus Φ C2 R (a −1 λ R) C2 has the structure of a O-algebra. Thus Φ C2 R is equivalent to a G/C 2 -commutative ring spectrum.
3. The slice spectral sequence and the localized slice spectral sequence 3.1. The slice spectral sequence of M U ((C 2 n )) and BP ((C 2 n )) . Our main computational tool in this paper is a modification of the equivariant slice spectral sequence of Hill-Hopkins-Ravenel. In this subsection, we list some important facts about the slice filtration for norms of M U R and BP R , which we will need for the rest of the paper. For a detailed construction of the slice spectral sequence and its properties, see [HHR16, Section 4] and [HHR17].
Let G = C 2 n be the cyclic group of order 2 n , with generator γ. The spectrum M U ((G)) is defined as Hill, Hopkins, and Ravenel [HHR16, Section 5] constructed elements . .], Here, G · x denotes the set {x, γx, γ 2 x, . . . , γ 2 n−1 x}, and the Weyl action is given by is an associative algebra map from the free associative algebra Applying the norm and using the norm-restriction adjunction, this gives a G-equivariant associative algebra map . Smashing these maps together produces an associative algebra map Note that by construction, A is a wedge of representation spheres, indexed by monomials in the r i s. By the Slice Theorem [HHR16, Theorem 6.1], the slice filtration of M U ((G)) is the filtration associated with the powers of the augmentation ideal of A. The slice associated graded for M U ((G)) is the graded spectrum where the degree of a summand corresponding to a monomial in the r i generators and their conjugates is the underlying degree.
As a consequence of the slice theorem, the slice spectral sequence for the RO(G)-graded homotopy groups of M U ((G)) has E 2 -term the RO(G)-graded homology of S 0 [G · r 1 , G · r 2 , . . .] with coefficients in the constant Mackey functor Z. To compute this, note that S 0 [G · r 1 , G · r 2 , . . .] can be decomposed into a wedge sum of slice cells of the form where p ranges over a set of representatives for the orbits of monomials in the γ j r i generators, and H p ⊂ G is the stabilizer of p (mod 2). Therefore, the E 2 -page of the integer graded slice spectral sequence can be computed completely by writing down explicit equivariant chain complexes for the representation spheres S |p| |Hp | ρ Hp . The exact same story holds for norms of BP R as well. By [HK01, Theorems 2.25, 2.33], the classical Quillen idempotent M U −→ M U lifts to a multiplicative idempotent M U R → M U R with image BP R , resulting in particular in a multiplicative C 2 -equivariant map The exact same technique as the one used in [HHR16, Section 5] shows that there are generators . Throughout the paper, the generators t i are chosen to be the coefficients of the canonical isomorphism from the formal group law of the first BP R component to the formal group law of the second BP R -component. In the case when G = C 4 , it is the canonical isomorphism from the formal group law F L to F R , where F L is induced by the map and F R is induced by the map Remark 3.1. Our specific choice of the formal group law and the generators t i is because we would like to control their geometric fixed points (See Proposition 6.2). Nevertheless, we would like to remark that the proofs and formulas in both [HHR16] and [BHSZ21] work for any choice of formal group law and the correspondingt i generators we get for π C2 * ρ2 BP ((G)) , as long as the conditions in [HHR16, Proposition 5.45] are satisfied.
Just like M U ((G)) , we can build an equivariant refinement from which the Slice Theorem implies that the slice associated graded for BP ((G)) is the graded Remark 3.2. The regular slice towers of M U ((G)) and BP ((G)) are isomorphic to their slice towers in [HHR16], since all the HHR-slices of them are regular slices. For a proof of the slice theorem in terms of the regular slice filtration, see [HHR21,Chapter 12.4] Since the slice filtration is an equivariant filtration, the slice spectral sequence is a spectral sequence of RO(G)-graded Mackey functors. Moreover, the slice spectral sequences for M U ((G)) and BP ((G)) are multiplicative spectral sequences and the natural maps between them are multiplicative as well (see [HHR16,Section 4.7]), and the slice spectral sequence for BP ((G)) is a spectral sequence of modules over the spectral sequence of M U ((G)) in Mackey functors.
3.2. The localized spectral sequence. In this subsection, we introduce a variant of the slice spectral sequence which we call the localized slice spectral sequence. This will be our main computational tool to compute a −1 λ BP ((C4)) in the later sections. Let λ 2 n−i denote the 2-dimensional real C 2 n -representation corresponding to rotation by π 2 n−i and σ denote the real sign representation of C 2 n . Given a C 2 n -spectrum X, we have an equivalencẽ The following theorem shows that one can compute the homotopy groups ofẼF[C 2 i ] ∧ X = a −1 λ 2 n−i X by smashing the slice tower of X withẼF[C 2 i ]. The resulting localized slice spectral sequence will converge to the homotopy groups of a −1 λ 2 n−i X.
Theorem 3.3. Let X be a C 2 n -spectrum, and let {P • } denote the (regular) slice tower for X. Consider the tower The spectral sequence associated to {Q • } converges strongly to the homotopy groups ofẼF[C 2 i ] ∧ X.
We will first show that the spectral sequence converges to the limit, lim ← − (S ∞λ ∧ P • X). Since smash products commute with colimits, we have the equivalence and so the colimit of the tower is contractible. The slices P n n X satisfy P n n X ≥ n for all n. Furthermore, since S ∞λ ≥ 0, we also have S ∞λ ∧ P n n X ≥ n by [HHR16,Proposition 4.26]. 1 Applying Proposition 4.40 in [HHR16] to S ∞λ ∧ P n n X shows that the homotopy groups π k (S ∞λ ∧ P n n X) = 0 if n ≥ 0 and k < n |G| , n < 0 and k < n.
This gives a vanishing line on the E 2 -page of the spectral sequence. It follows that the spectral sequence converges strongly to the homotopy groups of the limit, To finish our proof, it suffices to show that the map is an equivalence. Consider the cofiber sequence The proof of this result and of the part of [HHR16, Proposition 4.40] we need are still valid for the regular slice filtration instead of the slice filtration as used in [HHR16].
The cofiber sequence above induces the following long exact sequence in homotopy groups: It follows from this long exact sequence and the discussion above that n + 1 ≥ 0 and k < n+1 |G| , n + 1 < 0 and k < n + 1.
This means that for any k, the kth homotopy groups of S ∞λ ∧ X and S ∞λ ∧ P n X will be isomorphic when n is large enough. In particular, the map S ∞λ ∧ P n+1 X → S ∞λ ∧ P n X will induce an isomorphism on π k . It is then immediate that the system π k (S ∞λ ∧ P • X) satisfies the Mittag-Leffler condition and therefore for n large. Another way to observe this is by using the localized slice spectral sequence. As we have shown, the spectral sequence associated to the tower By [HHR16,Proposition 4.40], the homotopy groups π n−s (S ∞λ ∧ P n n X) do not contribute to π k lim ← − (S ∞λ ∧ P • X) when n ≥ 0 and k < n |G| , or when n < 0 and k < n (see Figure 1). Therefore, n ≥ 0 and k < n |G| , n < 0 and k < n.
For any k, consider the diagram We have proven that when n is large enough (n > k), the vertical arrow and the diagonal arrow are isomorphisms. Therefore, the horizontal arrow induces an isomorphism for all k. It follows that S ∞λ ∧ X lim ← − (S ∞λ ∧ P • X), as desired.
From the discussion in [Ull13, Section I.4] it follows that the localized slice spectral sequences of M U ((G)) and BP ((G)) (and more generally of G-ring spectra) are multiplicative spectral sequences.
3.3. Exotic transfers. If the transfer of a given class in the slice spectral sequence is zero, it might still support a non-trivial exotic transfer in a higher filtration. Understanding these is both crucial for understanding the Mackey functor structure of the spectral sequence and helpful to deduce differentials and extensions inside the spectral sequence. While the concept of exotic transfers is pretty transparent for permanent cycles, it is slightly more subtle for exotic transfers just happening on finite pages. Following the lead of [BBHS20] (in the case of the Picard spectral sequence), we will give a precise definition of this phenomenon and show how it behaves with respect to differentials. It turns out that it is no more difficult to treat a more general setting, which specializes to several different known spectral sequences and allows also for more general operations than just transfers.
In this subsection, we will first state a general definition of exotic w-operations and prove some general results. Then, we will specialize to the case of cyclic 2-groups and prove a variant of [HHR17,Theorem 4.4] that also work for exotic transfers and restrictions on finite pages.
We consider a tower of G-spectra. Recall that to this we can associate a spectral sequence as follows: are defined as the restrictions of the boundary maps (coming from the cofiber sequence X ). See e.g. [Lur17, Section 1.2.2] for some details in the setting of an ascending filtration. Our setting specializes in particular to the following spectral sequences: (1) Given a spectrum Z with a G-action, set X i = (τ ≤i Z) EG+ . We recover the homotopy fixed point spectral sequence.
(2) Given a spectrum Z with a G-action, set X i = (τ ≤i Z ∧ẼG) EG+ . We recover the Tate spectral sequence.
(3) Given a G-spectrum Z, set X i = P i Z, the slice tower. We obtain the slice spectral sequence. (4) Given a C 2 n -spectrum Z and 1 ≤ j ≤ n, set X i =ẼF[C 2 j ] ∧ P i Z. We obtain the localized slice spectral sequence. This will be the main example of relevance for us.
We fix an arbitrary map Σ ∞ G/K → Σ ∞ G/H and denote the resulting operation π H n → π K n by w. The most important case for us will be H ⊂ K and w = Tr K H . But equally well w could be a restriction map, multiplication by a fixed element such as 2, or any combination of these.
For notational simplicity, we will restrict for our treatment of exotic w-operations to integer degrees. By suspending by a representation sphere, one can easily translate our definitions and results to the RO(G)-grading.
Definition 3.4. Let x ∈ E s,t r (G/H), and let 0 ≤ p ≤ r − 2 and 0 ≤ q ≤ p. We may lift the corre- If p > 0, we speak of an exotic w-operation, which, depending on w, might be an exotic transfer, exotic restriction etc. 2 If the page jump is zero, we omit the mention of it.
This definition can be illustrated with the following diagram: Remark 3.5.
(1) The w-operations of filtration jump 0 are just the algebraic w-operations on the E r -page as inherited from the E 2 -page. This is why we call the w-operations of higher filtration jump exotic.
(2) The most classical case of exotic w-operations is the limiting case when r = ∞. If x ∈ E s,t ∞ (G/H) and X denotes lim t X t , we can actually lift x to x ∈ π H t−s X (which is further than to π H t−s X t+p as required by the previous definition). If w( x) = 0, it must be detected in some E s+p,t+p ∞ (G/K) and the resulting element is an example of a woperation of filtration jump p on x. In the case when x is just on a finite page, we can suitably truncate the original spectral sequence to force x to be a permanent cycle that survives to the E ∞ -page. We will do this in the proof of Lemma 3.7.
(3) Even in the classical situation of the last item, exotic w-operations are in general not unique; in other words, w( x) will depend on the choice of lift x. With notation as in the last bullet point, suppose for example that there exists z ∈ E s+i,t+i ∞ (G/H) for 0 < i < p such that z supports a non-exotic w-operation. If we lift z to z ∈ π H t−s X, then w( x + z) will be detected by w(z) ∈ E s+i,t+i ∞ (G/K), while x + z lifts x. In the extreme case, x might even be zero. In Lemma 3.7, we will prove a criterion that ensures the uniqueness of exotic w-operations. This criterion is often fulfilled in practice.
(4) A w-operation z = w( x) of filtration jump p and page jump 0 defines a w-operation of filtration jump p and page jump q if d r (z) = · · · = d r+q−1 (z) = 0 by just mapping z ∈ π K t−s X t+p t+p−(r−2) down to π K t−s X t+p t+p−q−(r−2) . All w-operations of page jump q are of this form.
The following lemma holds by definition.
Lemma 3.6. Let x ∈ E s,t r (G/H) be a d r -cycle and denote by x its image in E s,t r+1 (G/H). Let z ∈ E s+p,t+p r+q be a w-operation on x of filtration jump p and page jump q ≥ 1. Then z is a w-operation on x of filtration jump p and page jump q − 1.
The following is the uniqueness result for exotic w-operations that we will use.
Lemma 3.7. Let x ∈ E s,t r (G/H) and 0 < p ≤ r − 2. Suppose every class in E s+k,t+k 2 (G/H) for 0 < k < p is either hit by a differential of length at most r + k − 1 or supports a differential of length at most p − k + 1. Denoting by I the image of all (r + p)-cycles in w(E s+p,t+p 2 (G/H)) in E s+p,t+p r+p (G/K), then there is at most one class in E s+p,t+p r+p (G/K)/I that is a w-operation of x of filtration jump p and page jump p.
Proof. Consider the towers X • and X • with . Via the maps of spectral sequences, differentials in the original spectral sequence enforce corresponding differentials in the E-spectral sequence in the range t − (r − 2) ≤ t ≤ t + p. In particular, E s,t r injects into E s,t r . Note moreover that the E-spectral sequence converges to π * X t+p t−(r−2) . Our assumptions imply that E s+k,t+k ∞ (G/H) = 0 for 0 < k < p and moreover E s,t r = E s,t ∞ . Thus, we can lift the image of x in E s,t r uniquely to π H t−s X t+p t−(r−2) modulo E s+p,t+p ∞ . The latter term is a quotient of E s+p,t+p 2 = E s+p,t+p 2 . In summary, we have shown that we can lift x uniquely to x ∈ π H t−s X t+p t−(r−2) modulo the image from π H t−s X t+p t+p . Thus, w( x) ∈ π K t−s X t+p t−(r−2) is indeed well-defined modulo the image of w(π H t−s X t+p t+p ) = w(E s+p,t+p 2 ).
Remark 3.8. One can probably formulate a sharper criterion for the uniqueness of exotic woperations, without requiring that all classes between x and its target vanish. The essential point is to require that there are no interleaving w-operations such as classes in E s+k,t+k r with 0 < k < p that admit nonzero w-operations of filtration jump smaller than p − k. Moreover, one would have to enlarge I to include exotic w-operations as well. We refrain from making this precise.
Proposition 3.9. Let x ∈ E s,t r (G/H) and z a class with d r (z) = x. Suppose d r+q (w(z)) is zero for q < p. Then d r+p (w(z)) is a w-operation of x of filtration jump p and page jump p.
Proof. We choose a lift of z ∈ π H t−s+1 X t−r+1 t−2r+3 to z ∈ π H t−s+1 X t−1 t−r+1 . As δ( z) in the diagram below is a lift of x, contemplating the fate of w( z) passing along the two different travel paths from the upper left corner to the lower right corner proves the proposition.
While our definition and results so far are very general (and our proofs would also apply to other settings than equivariant homotopy theory), we will now formulate a result that is specific to cyclic 2-groups. For the following proposition, both the statement and the proof are variants of [HHR17,Theorem 4.4], but also work for exotic transfers and restrictions on finite pages and circumvent a mistake in [HHR17, Lemma 4.5]. 3 Proposition 3.10. Let G be a cyclic 2-group, H ⊂ G an index 2 subgroup, and V ∈ RO(G).
of filtration jump (at most) r − 1.
Proof. For the first part, by shifting the tower and applying suspension if necessary, we can fix the bidegree of y to be (r − 1, r − 1). The term E r−1,r−1 . Smashing the long exact sequence associated with the cofiber sequence G/H + → S 0 aσ −→ S σ with X r−1 0 and taking homotopy groups, we get the long exact sequence From this long exact sequence, we see that a σ y = 0 implies y = Tr( w) with w ∈ π H 0 X r−1 0 . By definition, this defines an element w ∈ E 0,0 r+1 (G/H) such that y is an exotic transfer of w of filtration jump r − 1.
For the second part, we can fix the bidegree of z to be (r − 1, r − 1) by shifting the tower and applying suspension if necessary to view z as an element in π H 0 X r−1 0 . Using the long exact sequence induced by G/H + → S 0 aσ −→ S σ again, we see that z is the restriction of some v ∈ π G 1−σ X r−1 0 . By definition, this defines an element v ∈ E 0,1−σ r+1 (G/G) such that z is an exotic restriction of v of filtration jump r − 1.
3 With notation as in the cited lemma, a counterexample is the following: Fix an object A. Take A i,j to be Σ −1 A, zero or A, depending on whether i + j is smaller, equal or larger than 2. The a i,j and b i,j are id if possible, with the exception of a 2,1 being an arbitrary self-equivalence of A, which is not equivalent to ± id. Take further W = A and f 3 = id. Then f 1 exists (and can be taken to be id), but f 1 and f 2 cannot simultaneously exist. Strictly speaking, the cited lemma is ambiguous on whether it claims that f 1 and f 2 exist simultaneously if f 1 exists, but this seems to be the way that it is later used in [HHR17,Theorem 4.4]. 4 The "at most" is actually unnecessary here, as the proof shows that y is an exotic transfer of filtration jump r − 1. We write it for emphasis though since y might be very well also an exotic transfer of smaller filtration jump. This is related to the non-uniqueness described in Item 3 of Remark 3.5. Thus the statement is best used in conjunction with a uniqueness result like Lemma 3.7.

MEIER, SHI, AND ZENG
Let us give an example of a possible workflow working with exotic transfers, which we will apply in Proposition 5.19.
Workflow 3.11. Let G be a cyclic 2-group and H ⊂ G of index 2. Let y ∈ E s+r−1,t+r−1 r (G/G) and r > r. We assume the following: (1) a σ y is nonzero and is hit by a d r -differential; (2) y persists to a nonzero class in the E r +r−1 -page, which we denote by the same name; (3) every class in E s+k,t+k 2 (G/H) for 0 < k < r − 1 is either hit by a differential of length at most r + k − 1 or supports a differential of length at most r − k; (4) y ∈ E s+r−1,t+r−1 2r−1 is not the image of a (2r − 1)-cycle in E 2 which is the transfer of a class in E s+r−1,t+r−1 2 . By (1), a σ y vanishes on E r+1 . Thus, by Proposition 3.10, there exists x ∈ E s,t r+1 (G/H) such that y ∈ E s+r−1,t+r−1 r+1 (G/G) is an exotic transfer of x of filtration jump r − 1. Applying Lemma 3.7 in conjunction with (3) and (4), we see that x cannot be zero (as zero is the unique exotic transfer of zero under our assumptions); in case that there is only one non-zero element in the relevant bidegree, this already uniquely determines x. Suppose now further that: (5) x = d r (a); (6) d r +q (Tr G H a) = 0 for 0 ≤ q < r − 1. Then Proposition 3.9 implies that d r +r−1 (Tr G H (a)) is an exotic transfer of x in the same degree as y ∈ E s,t r +r−1 and thus must be y by Lemma 3.7 again. 3.4. The behaviour of norms. This section is about the behaviour of norms in the (regular) slice spectral sequence and its localized variant. We will formulate a generalization of [Ull13, Chapter I.5] and then discuss how it applies to Ullman's original setting (the regular slice spectral sequence), to the localized slice spectral sequence and the homotopy fixed point spectral sequence.
We will first work in an abstract setting: Let (X i ) be a tower of G-spectra and E * , * * be the associated spectral sequence as in the preceding subsection. Set X ∞ = lim i X i and X n = X ∞ n . Let H ⊂ G be a subgroup of index h. We assume that we have maps N G H X n → X hn and N G H X n n → X hn hn that are (up to homotopy) compatible with the maps X n → X n−1 and X n → X n n . (Here we leave the restriction maps implicit.) We call this a norm structure. It induces norm Proposition 3.12. Let x ∈ E 2 (G/H) be an element representing zero in E r+2 (G/H). Then N G H (x) represents zero in E rh+2 (G/G). Proof. The proof is the same as that of [Ull13, Proposition I.5.17].
Example 3.13. Our first example of this setting is the regular slice tower of [Ull13], which coincides with the slice tower of [HHR16] for norms of M U R and BP R -thus there should be no danger of confusion if we use the same notation P i X for the regular slice tower.
Ullman constructs in [Ull13, Corollaries I.5.10 and I.5.11] for every H-spectrum X natural compatible maps N G H P n X → P nh N G H X and N G H P n n X → P nh nh N G H X. Moreover the square commutes, as N G H P n X is ≥ hn by [Ull13, Corollary I.5.8] and both maps into N G H P n X → P hn−h N G H X are compatible with the respective maps to N G H X. Let R be a G-spectrum with a map N G H Res G H R → R. The composite N G H P n Res G H R → P nh N G H Res G H R → P nh R and its analogue for P n n define a norm structure on the regular slice tower of R. This applies in particular if R is a G-commutative ring spectrum.
Example 3.14. Let R be a G-commutative ring spectrum with G = C 2 n . We will define a norm structure on the tower X i = a −1 λ P i X defining the localized regular slice spectral sequence. Using the observations above for the regular slice spectral sequence, it suffices to produce natural maps N G H Res G H a −1 λ P n R → a −1 λ N G H Res G H P hn R and similarly for P n n . As N G H and Res G H are monoidal, by Lemma 2.18 it thus suffices to provide a natural map We remark that we have not used the full strength of our considerations in Section 2.5 here, but we expect that these will be necessary for deeper considerations about norms.
Example 3.15. Lastly we define a norm structure on the homotopy fixed point spectral sequence. Observe first that there is for H-spectra X a natural map where the latter map is an equivalence as Res G e N G H X → Res G e N G H X EH+ is an equivalence. Recall that the tower defining the homotopy fixed point spectral sequence for a spectrum R is defined by X n = (τ ≤n R) EG+ . We observe that we have natural equivalences X n (P n R) EG+ and X n n (P n n R) EG+ for (P n R) n the regular slice tower. Combining these equivalences with the natural map from the last paragraph, the norm structure from Example 3.13 induces a norm structure on the homotopy fixed point spectral sequence.
We will use the following proposition without further comment.
Proposition 3.16. Both in the regular slice spectral sequence and in the localized regular slice spectral sequence of a G-commutative ring spectrum, the norms are multiplicative: N G H (xy) = N G H (x)N G H (y). Proof. This follows from the commutativity of for G-spectra X and Y . This in turn follows as there is up to homotopy just one map Given two towers (X n ) and (Y n ) with norm structures, a morphism of towers (X n ) → (Y n ) is compatible with the norm structures if the diagrams commute for all n and similarly for X n n and Y n n . Such a morphism induces in particular a morphism of spectral sequence that is compatible with the norms on the E 2 -terms.
Example 3.17. Given any spectrum X, there is a natural map from the regular slice tower to the tower defining the homotopy fixed point spectral sequence, namely P n X → (P n X) EG+ . In case that X is a G-commutative ring spectrum (or more generally a spectrum admitting a map N G H Res G H X → X), this map of towers is (essentially by construction) compatible with the norm structures introduced in Example 3.13 and Example 3.15.
3.5. Comparison of spectral sequences. When computing localizations of a norm, we can apply different spectral sequences. For instance, in the isomorphism of Theorem 2.9, the left hand sideẼF[H] ∧ N G H X can be computed by the localized slice spectral sequence we just built, while the right hand side can be computed by the pullback of the (G/H)equivariant slice spectral sequence of N G/H e Φ H X. In this section, we give a comparison map between these spectral sequences, which we will use in understanding the homotopy fixed points and the Tate spectral sequence of N G/H e Φ H X. Such comparison can only be made by regrading the slice tower. In the cases of relevance for us this takes the shape of the following doubling process: Let P • be a tower, we define DP • , the doubled tower of P • , as DP 2n+ := P n for = 0, 1. We also use D as a prefix of a spectral sequence obtained from a tower as the spectral sequence of the doubled tower.
In the following theorem we will use both the slice tower P • and the pullback functor P *

G/C2
from Section 2.2; the double usage of P will hopefully not cause any confusion to the reader.
Theorem 3.18. Let G = C 2 n , X ∈ Sp G and Y = Φ C2 X ∈ Sp G/C2 . Let P • X and P • Y be their slice towers in the corresponding categories. Then there is a commutative diagram of towers such that the mapẼG∧P • X → P * G/C2 DP • Y converges to the G-equivalenceẼG∧X → P * G/C2 Y . In particular, the induced map on the C 2 -level spectral sequences of P • X → P * G/C2 DP • Y converges to the geometric fixed points map Proof. To construct the map P • X → P * G/C2 DP • Y , consider the composition X → a −1 λ X P * G/C2 Y → P * G/C2 DP 2i Y. We only need to show P * G/C2 DP 2i Y ≤ Slice 2i for each i (the analogous statement for DP 2i+1 follows from this). This can be checked by testing against slice cells of dimension more than 2i. By induction we can assume that our claim is true after restriction to any proper subgroup of G, so we can ignore induced slice cells. Thus it suffices to check that [S kρ G , P * G/C2 DP 2i Y ] G = 0 for k|G| > 2i.
The following equivalence of G-spectra is essential to our proof: EG ∧ S kρ G P * G/C2 S kρ G/C 2 . It comes from the fact that both sides are equivalent to the representation sphere S ∞λ+ρ G/C 2 . The left hand side of the equivalence is a localization of a slice cell of dimenson k|G| while the right hand side is a pullback of a slice cell of dimension k|G| 2 . This difference is the reason of doubling the tower of Y .
Using this equivalence, we have a series of equivalences of mapping sets: The change-of-group isomorphism comes from the fact that P * G/C2 is fully faithful on homotopy categories, and the last isomorphism is because S kρ G/C 2 is a slice cell of dimension > i in G/C 2spectra.
By construction, the map P • X → P * G/C2 DP • Y converges to the map X → a −1 λ X P * G/C2 Y . Since everything in the tower P * G/C2 DP • Y is already a λ -local, the tower map factors through the a λ -localizationẼG ∧ P • X.
Proposition 3.19. Let G = C 2 n and X ∈ Sp G a G-commutative ring spectrum. Then the tower P * G/C2 DP • Φ C2 X has a norm structure in the sense of Section 3.4 and the maps P • X → P * G/C2 DP • Φ C2 X and a −1 λ P • X → P * G/C2 DP • Φ C2 X from Theorem 3.18 are compatible with norms from subgroups containing C 2 .
Proof. Let H ⊂ G be a subgroup of index h such that C 2 ⊂ H. Then we obtain maps H/C2 P 2n 2n Φ C2 X → P * G/C2 P 2hn 2hn Φ C2 X, which are compatible in the necessary sense. Here we use the norm structure on the regular slice tower from Example 3.13, the G/C 2 -commutative ring structure on Φ C2 X from Example 2.22 and the commutation of norms and pullbacks from Proposition 2.14.
To show that P • X → P * G/C2 DP • Φ C2 X is compatible with norm structures, note first that the diagram The outer rectangle is obtained from the previous diagram by applying Φ C2 (and using the maps P n X → X and P hn X → X) and thus commutes. Given the connectivity estimate [Ull13, Corollary I.5.8] and the universal property of P 2hn , we see that Φ C2 N G H P n X → Φ C2 X factors through P 2hn Φ C2 in an essentially unique way, so the left square also has to commute. By the adjointness of Φ C2 and P * G/C2 this implies the commutativity of The proof of the commutativity for the corresponding square for P n n is completely analogous. The a λ -inverted case follows again because the target is a λ -local.

The localized slice spectral sequences of BP ((G)) : summary of results
We now turn to analyze the localized slice spectral sequence of BP ((G)) for G = C 2 n . From now on, everything will be implicitly 2-localized. In this section, we list our main results and give an outline of the computation. Detailed computations of the results stated in this section are in Section 5.
As we discussed in Section 3, the Slice Theorem [HHR16, Theorem 6.1] implies that the slice associated graded of BP ((C 2 n )) is where t i ∈ π C2 (2 i −1)ρ2 BP ((C 2 n )) (see also [HHR16, Section 2.4] for details). For the rest of the paper, we use λ for the 2-dimensional real representation of C 2 n which is rotation by π 2 n−1 , and σ for the 1-dimensional sign representation of G. We use σ 2 for the sign representation of the unique subgroup C 2 in G. Let i < j ≤ n, we will use Res 2 j 2 i , Tr 2 j 2 i and N 2 j 2 i for restrictions, transfers and norms between C 2 i and C 2 j as subgroups of G. If their subscript and superscript are omitted, they mean the restriction, transfer and norm between C 2 and C 4 .
(1) Let G = C 2 n and H = C 2 be the subgroup of order 2 inside G. There is a RO(G/H)graded spectral sequence of Mackey functors a −1 λ SliceSS(BP ((G)) ) that converges to the RO(G/H)-graded homotopy Mackey functor of N G/H e HF 2 . The E 2 -page of this spectral sequence is a −1 λ HZ [G · t 1 , G · t 2 , · · · ].
(2) The integral E 2 -page of a −1 λ SliceSS(BP ((G)) ) is bounded by the vanishing lines s = (2 n − 1)(t − s) and s = −(t − s) in Adams grading. In other words, at stem t − s, the classes with filtrations greater than (2 n − 1)(t − s) or less than −(t − s) are all zero.
(3) On the integral E 2 -page, the a λ -localizing map SliceSS(BP ((G)) ) → a −1 λ SliceSS(BP ((G)) ) induces an isomorphism of classes in positive filtrations. The kernel of this map consists of transfer classes in SliceSS(BP ((G)) ) from the trivial subgroup in filtration 0. These classes are all permanent cycles.
The top vanishing line s = (2 n − 1)(t − s) follows from the fact that π i (S kρ G +lλ ∧ HZ) = 0 for k, l ≥ 0 and i < k (See [HHR16, Theorem 4.42]). For the second vanishing line y = −x, note that in stem t − s, classes in filtration less than −(t − s) are contributed by slices of negative dimension, but BP ((G)) has no negative slices. This proves (2).
To prove (3), by unpacking the description of the E 2 -page, we need to show that for k, l ≥ 0, the a λ -multiplication map a λ : π G i (S kρ G +lλ ∧ HZ) −→ π G i (S kρ G +(l+1)λ ∧ HZ) is an isomorphism for k ≤ i < k|G| + 2l and is surjective with kernel consisting of transfer classes from trivial subgroup for i = k|G| + 2l. Using the cellular structures and their corresponding chain complexes described in [HHR17, Section 3], we see that when k ≤ i ≤ k|G| + 2l, a λ induces isomorphism on the cellular chain complexes, therefore it induces isomorphism on homology for k ≤ i < k|G| + 2l and surjection on homology for i = k|G| + 2l with the kernel exactly the image of T r 2 n 1 . Since the underlying tower of the slice tower is the Postnikov tower, all the class in the trivial subgroup and their transfers are permanent cycles.
Remark 4.2. In fact, (2) and (3) of Theorem 4.1 hold in a greater generality. For instance, they are true for any (−1)-connected G-spectrum. We will investigate properties of the localized slice spectral sequences in a future paper.
By [LNR12] and [BBLNR14], all C 2 n norms of HF 2 are cofree, therefore we will not distinguish between their fixed points and homotopy fixed points.
For the rest of the paper, we focus on the case G = C 4 .
Theorem 4.4. The first 8 stems of π C4 * (a −1 λ BP ((C4)) ) ∼ = π C2 * N 2 1 HF 2 are shown in the following chart: On the E ∞ -page of the localized spectral sequence, the black subgroups are those generated by non-exotic transfers from A * = π * (HF 2 ∧ HF 2 ), and the red subgroups consist of everything else. For the Mackey functor structure, see Figure 6.
In [Rog], Rognes shows that the unit map S 0 → (N 2 1 HF 2 ) hC2 induces a splitting injection on mod 2 homology as an A * -comodule thus a splitting injection on the E 2 -page of the Adams spectral sequence. Therefore, the ring spectrum (N 2 1 HF 2 ) hC2 (a −1 λ BP ((C4)) ) C4 detects all Hopf invariant one elements. They all restrict to 0, since the underlying Adams spectral sequence of HF 2 ∧ HF 2 is concentrated in filtration 0. Therefore, they are detected by red subgroups in the corresponding degree.
The proof of Theorem 4.4 is by computing a −1 λ SliceSS(BP ((C4)) ) and is given in the next section. The most relevant differentials in the spectral sequence are listed in the following table: Proposition 5.8 In this section, we compute a −1 λ SliceSS(BP ((C4)) ) and prove Theorem 4.4. Our approach is similar to that of [HHR17] and [HSWX18]. When going through the computations in this section, the following guiding principles are useful to keep in mind. We hope these points would serve as a road map that will be helpful to the readers who are new to these types of computations.
(1) The E 2 -page of the spectral sequence can be obtained by computing the RO(C 4 )-graded homotopy groups of a −1 λ HZ.
(2) The C 2 -level spectral sequence, a −1 σ2 SliceSS(BP R ∧ BP R ), is easy to compute, as it is completely determined by the Hill-Hopkins-Ravenel slice differentials.
(3) In the positive cone part of a −1 λ SliceSS(BP ((C4)) ) (which includes the entire integer-graded spectral sequence), the only algebra generators that are not permanent cycles are essentially classes of forms u V and u V a σ . Therefore, we only need to focus on finding differentials on these classes, and then use the Leibniz rule. This is why even though the integer-graded spectral sequence is the computation of interest, we often move to analyze certain classes in RO(C 4 )-degrees.
(4) Many of the differentials are proven by using the C 2 -level spectral sequence, and using the restrictions and transfers on the E 2 -page. More precisely, if one knows that d r (Res C4 C2 x) = y, then x must support a differential of length at most r. Similarly, if d r (x) = y, and Tr C4 C2 (y) is not zero on the E 2 -page, then it must be killed by a differential of length at most r. (5) The remaining differentials and extension are proven by using the Hill-Hopkins-Ravenel norm and the theory of exotic restrictions and transfers. We would like to also remark that the differentials proven in this section determine all the differentials in the integer-graded spectral sequence in our range of interest. There are other differentials in the RO(C 4 )-graded page (both in the positive cone and outside the positive cone) that don't influence the integer-graded page of the spectral sequence.

5.1.
Computing the E 2 -page. We will first give a complete algebraic description of the E 2page of a −1 λ SliceSS(BP ((C4)) ) in terms of generators and relations. To do so, by Theorem 4.1, we need to describe the C 2 -homotopy groups π (a −1 σ2 HZ) and the C 4 -homotopy groups π (a −1 λ HZ). Proposition 5.1. We have π C2 (a −1 σ2 HZ) = F 2 [u 2σ2 , a ±1 σ2 ]. The Mackey functor structure is determined by the contractibility of the underlying spectrum.

MEIER, SHI, AND ZENG
This proposition is proved by a standard Tate cohomology computation, see [Gre18, Section 2.C] for details.
Let S be the subring of Proposition 5.2. We have The Green functor structure is determined by the following facts: (1) The C 2 -restriction of a −1 λ HZ is the spectrum a −1 σ2 HZ in Proposition 5.1. (2) The C 2 -restrictions of the classes u λ and u 2σ are u 2σ2 and 1, respectively.
(3) Given V ∈ RO(C 4 ), there is an exact sequence (see [HHR17,Lemma 4 In other words, the kernel of a σ -multiplication is the image of the transfer from C 2 to C 4 , and the image of a σ -multiplication is the kernel of the restriction from C 4 to C 2 .
The proof of Proposition 5.2 and a more explicit presentation of the Mackey functor are given in [Zen,Proposition 6.7]. Fortunately, in most of the paper we only need the "positive cone" of the coefficient Green functor, that is, the part = a+bσ +cλ for b ≤ 0. The Green functor structure of this part is computed in [HHR17, Section 3]. However, the other part also plays an important role on the computation, see for example the proofs of Proposition 5.14 and Proposition 5.21.

MEIER, SHI, AND ZENG
Although we mostly care the most about the C 4 -equivariant homotopy groups of a −1 λ BP ((C4)) , there are two advantages for computing a −1 λ SliceSS(BP ((C4)) ) as a spectral sequence of Mackey functors: (1) The Mackey functor structure can transport certain differentials on the C 2 -level to differentials on the C 4 -level.
(2) The Mackey functor structure and d r -differentials can result in exotic extensions of filtration r − 1 (see Section 3.3). We will see (1) in the computations of d 3 , d 7 , and d 15 -differentials below.
(2) will be used to prove certain extensions forming the (Z/4)s in Theorem 4.4, see Proposition 5.14 and 5.21. : π H V X → π G W X, as a part of the homotopy Mackey functor structure. In our computation we will omit writing W when it is clear from the context what W is.
(1) The underlying . More precisely, the E 2 -page of the underlying non-equivariant spectral sequence is trivial, and the E 2 -page of the C 2 -spectral sequence is F 2 [u 2σ2 , a ±1 σ2 ][t 1 , γt 1 , t 2 , γt 2 , · · · ]. The elements u 2σ2 , t i and γt i have filtration 0, while a σ2 has filtration 1. 5 (2) All the differentials in a −1 σ2 SliceSS(BP R ∧ BP R ) are determined by a σ2 , t i and γt i being permanent cycles, the differentials and the Leibniz formula (for notational convenience, we let t 0 = γt 0 = 1). The E 2 k+1 -page has the form In particular, in the integral grading, all the stem-n non-trivial permanent cycles are located in filtration n. 5 We recall the convention here that the filtration of an element in π H V P n n X in the slice spectral sequence for some X is in filtration n − dim R V . In particular the classes a V will be always in filtration dim R V .
For (2), we use the Hill-Hopkins-Ravenel slice differential theorem [HHR16, Theorem 9.9] and the formula in [BHSZ21, Theorem 1.1] that expresses thev i -generators in terms of the t igenerators (our v i and t i are t C2 i and t C4 i respectively in [BHSZ21]). The Hill-Hopkins-Ravenel slice differential theorem states that in the slice spectral sequence of BP R , there are differentials The formula in [BHSZ21, Theorem 3.1] shows that under the left unit map The left unit map induces a map SliceSS(BP R ∧ BP R ) of spectral sequences. We will use naturality and induction to obtain the differentials and the description of the E 2 k+1 -page.
To start the induction process, note that the description of the E 2 -page is already given in (1). Now assume that we have obtained a description of the E 2 k -page. For degree reasons, the next potential differential is of length exactly 2 k+1 − 1. The differential formula for a −1 σ2 SliceSS(BP R ) above shows that for any polynomial P ∈ F 2 [t 1 , γt 1 , · · · ]/(v 1 , v 2 , · · · , v k−1 ) and l an odd number, we have the differential . The source and the target of this differential are always non-zero on the E 2 k -page because the sequence (v 1 , v 2 , · · · ) is a regular sequence in the polynomial ring F 2 [t 1 , γt 1 , · · · ]. Taking the quotient of the kernel and cokernel of this differential, we see that the E 2 k+1 -page has the above description.
(3) is a direct consequence of (2) by letting k → ∞. See Figure 4 for the integral E 2 and E ∞ -pages of this spectral sequence.
Remark 5.5. In Proposition 6.2 we show that the C 2 -geometric fixed points of the t i and γt i generators are the ξ i and ζ i generators in the mod 2 dual Steenrod algebra A * . Therefore, the formula 5.3. The C 4 -spectral sequence: d 3 , d 5 and d 7 -differentials. The rest of this section is dedicated to computing the first 8 stems of the C 4 -Mackey functor homotopy groups of a −1 λ BP ((C4)) . The result is stated in Theorem 4.4. By Section 3.4, we are free to use the norm structure from C 2 to C 4 in the localized slice spectral sequence.
As a consequence of the slice theorem [HHR16, Theorem 6.1], the 0-th slice of M U ((G)) is HZ and π 0 M U ((G)) ∼ = Z. Therefore, every Mackey functor in the (localized) slice spectral sequence Proposition 5.6. Let K ⊂ H ⊂ G, and x be an element in the G/H-level of a Mackey functor either in the (localized) slice spectral sequence or the homotopy of a M U ((G)) -module, then Tr H K (Res H K (x)) = [H : K]x. Before getting to the page-by-page computation, we note that all the differentials on the classes u 2 k 2σ for k ≥ 0 are already known by the work of Hill-Hopkins-Ravenel. Their theorem is originally formulated for the slice spectral sequence for M U ((C4)) and the exact same statement and proof carries over to SliceSS(BP ((C4)) ) and a −1 λ SliceSS(BP ((C4)) ). Theorem 5.7 ( [HHR16, Theorem 9.9]). For k ≥ 0 and i < 2 k+3 − 3, d i (u 2 k 2σ ) = 0 and Now we start the page-by-page computation. First, note that for degree reasons all the differential lengths will be odd. Proof. By Theorem 5.4, the restriction Res 4 2 (u λ ) = u 2σ2 supports the differential d 3 (u 2σ2 ) = (t 1 + γt 1 )a 3 σ2 in the C 2 -spectral sequence. By naturality and degree reasons, the class u λ must also support a d 3 -differential in the C 4 -spectral sequence whose target restricts to the class (t 1 + γt 1 )a 3 σ2 . The only class that restricts to (t 1 + γt 1 )a 3 σ2 with RO(C 4 )-degree 1 − λ is Tr 4 2 (t 1 a 3 σ2 ).
Proof. This is a direct consequence of Proposition 5.8, the Frobenius relation [HHR17, Definition 2.3] and the Leibniz rule.
As displayed in Figure 5, this corollary gives all other d 3 -differentials. We now explain them in detail.
In terms of Mackey functors, the d 3 -differentials give the following exact sequences: Here are examples of d 3 -differentials corresponding to each exact sequence above: d 3 (u λ ) = Tr 4 2 (t 1 a 3 σ2 ) d 3 (Tr 4 2 (t 1 a σ2 )u λ ) = Tr 4 2 (t 1 (t 1 + γt 1 )a 4 σ2 ) d 3 (u 2σ2 a σ2 ) = (t 1 + γt 1 )a 4 σ2 . Note that the last differential is a C 2 -differential, but it has an effect on the C 4 -level Mackey functor structure. By results in Section 3.3, the d 3 -differentials also give certain exotic restrictions of filtration jump at most 2 (that is, the image of the restriction is of filtration at most 2 higher than the source). For example, consider the element N (t 1 )u λ a σ at (3, 1). This class is a d 3 -cycle. By Proposition 5.8, the class N (t 1 )u λ supports the d 3 -differential d 3 (N (t 1 )u λ ) = Tr 4 2 (t 2 1 γt 1 a 3 σ2 ).
By Proposition 3.10, the class t 2 1 γt 1 a 3 σ2 receives an exotic restriction of filtration jump at most 2 in integral degree, and the only possible source is N (t 1 )u λ a σ . The same argument applies to all 2-torsions classes with (t − s, s)-bidegrees (3 + 4i + 4j, 1 + 4i − 4j) for i, j ≥ 0. The exotic restrictions are represented by the vertical green dashed lines in Figure 5. if one evaluates the exact sequence of Mackey functors at C 4 /C 4 . Notice that in the category of Mackey functors, there are essentially two nontrivial extensions between • and , but only the one above fits into Proposition 5.6.
For readers who are familiar with Lubin-Tate E-theories and topological modular forms, the family of 2-extensions above is a generalization of the type of 2-extension between the class ν at (3, 1) and the class 2ν at (3, 3) in the homotopy fixed points spectral sequences of E hC4 2 and T M F 0 (5) (see [BBHS20] and [BO16]).
In summary, the d 3 -differentials can be described as follows: (1) On C 2 -level, it is the first differential in Theorem 5.4.
(2) The Green functor structure of the spectral sequence gives d 3 -differentials on the C 4 -level, by Proposition 5.8 and Corollary 5.9. After these d 3 -differentials, there is no room for further d 3 -differentials.
(3) Every d 3 -differential of the form • →• gives an extension of filtration 2 by the above remark. Now we will prove the d 5 -differentials. There are two different types of d 5 -differentials. The first type is given by Theorem 5.7: σ . Since N (t 1 ) and a λ are both permanent cycles, on the integral page for our range, it gives the following d 5 -differential at (4, 4): d 5 (N (t 1 ) 2 u 2σ a 2 λ ) = N (t 1 ) 2 a 3 λ a 3 σ , and it repeats by multiplying by N (t 1 )a λ a σ . In Figure 5, these are the d 5 -differentials with sources on or above the line of slope 1.
The second type of d 5 -differentials is given by the following proposition.
Remark 5.12. Although u 2 λ and u 2 λ a σ support differentials of the same length, this is not true in general. For example, we will see soon that u 4 λ supports a d 7 -differential, while u 4 λ a σ supports a d 13 -differential.
Corollary 5.13. d 5 (u 3 λ a σ ) = 2N (t 1 )u λ u 2σ a 3 λ . Proof. First, we will show that u λ a σ is a nontrivial permanent cycle. Since the target of the d 3 -differential on u λ is a transfer class, it is killed by a σ , and therefore u λ a σ is a d 3 -cycle. The only potential non-trivial differential that u λ a σ can support is the d 5 -differential d 5 (u λ a σ ) = N (t 1 )a 2 λ a 2 σ . If this differential happens, then multiplying a σ on both sides and using the gold relation will produce the differential d 5 (2u 2σ a λ ) = N (t 1 )a 2 λ a 3 σ . This is a contradiction to Theorem 5.7.
In Figure 5, this d 5 -differential implies the d 5 -differential on the class N (t 1 )u 3 λ a −2 λ a σ at (7, −3). Notice that the class N (t 1 )u λ u 2σ a 2 λ supports a d 3 -differential and the class 2N (t 1 )u λ u 2σ a 2 λ is killed by a d 5 -differential. In the integral grading, this happens to the Z/4 in (6, 2).
There are extensions of filtration jump 4 induced by the d 5 -differentials.
For the first claim, note that d 5 (N (t 1 )u 2σ a λ ) = N (t 1 ) 2 a 2 λ a 3 σ , and N (t 1 ) 2 a 2 λ a 2 σ is a nontrivial d 5 -cycle. Therefore, N (t 1 ) 2 a 2 λ a 2 σ is the target of an exotic transfer of filtration jump 4 in E 6 , and the only possible source is t 2 1 a 2 σ2 .

NORMS OF REAL BORDISM AND THE SEGAL CONJECTURE 41
For the second claim, first note that by Proposition 5.2 (also see Figure 3) and the gold relation, We have the d 5 -differential To prove this differential, consider the class u 2 λ u2σ a −2 λ . This class supports a d 5 -differential because after multiplying it by u 2 2σ a 2 λ (which is a d 5 -cycle), the class u 2 λ u 2σ supports the d 5 -differential d 5 (u 2 λ u 2σ ) = N (t 1 )u λ u 2σ a 2 λ a σ by Proposition 5.11. Therefore Multiplying both sides by a σ , we have u λ u 2σ a 2 σ = 2N (t 1 )a λ = Tr 4 2 (Res 4 2 (N (t 1 )a λ )) = Tr 4 2 (t 1 γt 1 a 2 σ2 ) = Tr 4 2 (t 2 1 a 2 σ2 ) The last equation holds because by Theorem 5.4, t 1 = γt 1 after the d 3 -differentials in the C 2spectral sequence. Therefore, t 2 1 a 2 σ2 must receive an exotic restriction of filtration jump 4 in the integral degree, and the only source of the restriction is 2u λ a −1 λ .
In Figure 6, the exotic restrictions and transfers are the green and blue dashed lines, respectively.
Remark 5.15. Similar to Remark 5.10, the exotic restrictions and transfers also give extensions of abelian groups on the C 4 -level. The situation is more subtle here because each individual exotic extension doesn't involve non-trivial extensions of abelian groups at any level. When we combine the two extensions together, however, we obtain an abelian group extension of filtration 8 from (2, −2) to (2, 6): 0 → Z/2 → Z/4 → Z/2 → 0, and 2(2u λ a −1 λ ) = N (t 1 ) 2 a 2 λ a 2 σ in homotopy. This extension is similar to the extension in the 22-stem of E hC4 2 and T M F 0 (5). (See [BBHS20, Figure 10] and [BO16, Section 2]).
We will now prove the d 7 -differentials. While we state them in some RO(C 4 )-graded page first, we recommend that the reader multiplies with appropriate powers of a λ whenever possible to visualize the arguments in Figure 5. Proof. We will prove the first differential. The second differential is proven by the exact same method. On the C 2 -level, we have the d 7 -differential by Theorem 5.4. Taking transfer on the target and using naturality, the class Tr 4,3−2λ 2 (a 7 σ2 (t 2 + t 3 1 + γt 2 )) = Tr 4,3−2λ 2 (a 7 σ2 t 3 1 ) must be killed by a differential of length at most 7. For degree reasons, it must be the d 7differential with source 2u 2 λ . Remark 5.17. These differentials can also be proved by combining Proposition 3.9 and Remark 5.10. One sets the exotic w-operation to be multiplication by 2, and Remark 5.10 shows that the exotic restriction gives such an exotic multiplication.
In Figure 5, The d 7 -differentials in Proposition 5.16 and the underlying C 2 -level d 7 -differentials in Theorem 5.4 are supported by the classes at (4 + i, −4 + i) for i ≥ 0.
Proposition 5.18. d 7 (u 4 λ ) = u 2 λ Tr 4 2 (t 3 1 a 7 σ2 ). Proof. We will prove in Proposition 5.22 that there is a nontrivial d 13 -differential on the class u 4 λ a σ (we can already prove it at this point, but for organization reasons we prove it later). This implies that the class u 4 λ must support a differential of length at most 13. For degree reasons, the claimed d 7 -differential is the only possibility.

5.4.
The C 4 -spectral sequence: higher differentials and extensions. We will now prove the higher differentials in our range (see Figure 6). The next possible differential is a d 13differential from Theorem 5.7: d 13 (u 2 2σ ) = N (t 2 )a 3 λ a 7 σ . However, we won't see this differential in Figure 6. This is because its first appearance in the integer graded spectral sequence is on the class (10, 14), which is outside of our range. Note also that even though some classes at (8, 8) contain u 2 2σ , they don't support d 13 -differentials. We will give a detailed discussion of the classes at (8, 8) in Section 5.5.
The class N (t 1 ) 3 u λ u 2 2σ a 6 λ a σ in filtration 13 is killed by a d 5 -differential from Proposition 5.11: . It follows that the class N (t 2 )u λ u 2 2σ a 6 λ a σ is the only possible target. Remark 5.20. The class u 4 λ u 2σ is a permanent cycle in the homotopy fixed points spectral sequence of E hC4 2 (see [BBHS20,Proposition 5.23]) because N (t 2 ) is zero there.
Although this d 13 doesn't imply any differentials in our range, it is used in proving extensions.
(2) There is an exotic restriction in stem 6 of filtration 12, Res 4 2 (2u 3 λ a −3 λ ) = t 2 γt 2 a 6 σ2 . Proof. The proof is similar to that of Proposition 5.14. The exotic transfer comes from applying Proposition 3.10 to the d 13 -differential d 13 N (t 2 )u 2 2σ a 3 λ = N (t 2 ) 2 a 6 λ a 7 σ in Theorem 5.7. For the exotic restriction, first note that 2u 3 λ a −3 λ = u 4 λ u2σ a −4 λ a σ a σ by the gold relation. We will prove that the class u 4 λ u2σ a −4 λ a σ supports a d 13 -differential. To do so, we multiply this class by u 2 2σ a 4 λ . After multiplying the differential in Proposition 5.19 by a σ , we have d 13 (u 4 λ u 2σ a σ ) = 2N (t 2 )u 3 2σ a 7 λ . As by the gold relation u 2 λ kills d 13 (u 2 2σ a 4 λ ), we can use the Leibniz rule to obtain the d 13differential On the E 2 -page, 2N (t 2 )u 2σ a 3 λ = T r 4 2 (t 2 γt 2 a 6 σ2 ). By Proposition 3.10, t 2 γt 2 a 6 σ2 must receive an exotic restrion of filtration jump 12, and the only possible source is 2u 3 λ a −3 λ (see Figure 6). In Figure 6, they are the exotic restriction from the class (6, −6) to (6, 6) and the exotic transfer from (6, 6) to (6, 18). Since these extensions involve elements containing t 2 , we expect similar extensions in the homotopy fixed points spectral sequence of E hC4 4 by [BHSZ21, Theorem 1.1].
Proposition 5.22. d 13 (u 4 λ a σ ) = N (t 2 + t 2 1 γt 1 + γt 2 )u 2 2σ a 7 λ . Proof. Consider the C 2 -differential . Applying Proposition 3.12 to its target, we see that its norm N (t 2 + t 2 1 γt 1 + γ(t 2 ))a 7 λ must be killed by a differential of length 13 or shorter. Since the restriction of this element is killed by d 7 , it must be killed by a differential of length between 7 and 13. Since u 2 2σ supports a d 13 , if d r (x) = N (t 2 +t 2 1 γt 1 +γ(t 2 ))a 7 λ happens for r < 13, one can multiply both sides by u 2 2σ . However, for degree reasons N (t 2 + t 2 1 γt 1 + γt 2 )u 2 2σ a 7 λ cannot be hit by a differential shorter than a d 13 . Thus this element and hence also N (t 2 + t 2 1 γt 1 + γ(t 2 ))a 7 λ must be hit by a d 13 and the only possible source is u 4 λ a σ . On the integer graded page, this contributes to the d 13 -differential supported by the class N (t 1 )u 4 λ a −3 λ a σ at (9, −5). The last differential in our range is a d 15 -differential.
Proposition 5.23. We have the d 15 -differential In the the C 2 -spectral sequence, we have the d 15 -differential σ2 . Applying the transfer shows that the class Tr 4 2 (t C2 3 a 15 σ2 ) must be killed by a differential of length at most 15. By naturality and degree reasons, the only possible source is the class 2u 4 λ = Tr 4 2 (u 4 2σ2 ). In Figure 6, this contributes to the d 15 -differential supported by the class 2u 4 λ a −4 λ at (8, −8) (the d 15 -differential supported by the class at (9, −7) is a C 2 -level differential).
These are all the differentials and extensions in the first 8 stems. Now we will discuss in detail the generators and relations in degree (8, 8) after each differential in order to illustrate the technical aspect of tracking differentials in the localized slice spectral sequences. 5.5. The classes at (8, 8). Since our discussion here focuses on a single degree, we will omit the powers of a V and u V classes on each monomial, except in formulas of differentials. That is, we omit u 2 2σ a 4 λ on C 4 -classes and a 8 σ2 on C 2 -classes. (4) Tr 4 2 (t 2 γt 2 t 2 1 ); (5) Tr 4 2 (t 3 t 1 ), Tr 4 2 (t 3 γt 1 ). At the C 2 -level, the d 3 -differentials identifies t 1 with γt 1 . At the C 4 -level, the effect of the d 3 -differentials are as follows: (1) All the classes in (1) are identified with 2N (t 1 ) 4 ; (2) all the classes in (2) are identified to be the same; (3) all the classes in (3) are identified to be the same; (4) the class Tr 4 2 (t 2 γt 2 t 2 1 ) is identified with 2N (t 2 )N (t 1 ); (5) all the classes in (5) are identified to be the same.
In other words, it identifies the classes Tr 4 2 (t 2 2 t 2 1 ) and Tr 4 2 (t 2 t 5 1 ). The d 7 -differential on the class t In the quotient we need to choose our generators carefully: The • is generated by N (t 1 ) 4 + Tr 4 2 (t 2 2 t 2 1 ), because the image of • identifies the restriction of N (t 1 ) 4 with the restriction of Tr 4 2 (t 2 2 t 2 1 ). Therefore their sum is the unique element in C 4 -level that has trivial restriction. The is generated by N (t 2 )N (t 1 ), as it still has nontrivial restriction. The two• are generated by Tr 4 2 (t 2 2 t 2 1 ) and Tr 4 2 (t 3 t 1 ). The next differential is a d 13 -differential supported by the class N (t 1 )u 4 λ a −3 λ a σ at (9, −5). By Proposition 5.22, the target of this differential is the class N (t 1 )N (t 2 + t 3 1 + γt 2 )u 2 2σ a 4 λ . The restriction of this class is t 1 γt 1 (t 2 + t 3 1 + γt 2 )(γt 2 + γt 3 1 − t 2 ), which, after the d 3 -differentials, is

NORMS OF REAL BORDISM AND THE SEGAL CONJECTURE 47
As we have discussed above, this class is killed by the d 7 -differentials supported by the class t 5 1 . It follows that the target of the d 13 -differential is the generator of •, the unique nontrivial element that restricts to 0.

5.6.
A family of permanent cycles. We will now present families of nontrivial permanent cycles in a −1 λ SliceSS(BP ((C4)) ). These families will be used in the proof of Theorem 6.6.
Proof. We have the following commutative diagram of pointed C 4 -spaces where θ is the C 4 -equivariant 2-folded branched cover. Since θa λ = a 2 σ is invertible, a λ is invertible.
By Proposition 5.2 and the gold relation, the class u 2 k−1 2σ a −(2 k+1 −1)+i σ is in the image of π C4 a −1 λ HZ → π C4 a −1 σ HZ only when a σ has a non-negative power, i.e. i ≥ 2 k+1 − 1. Therefore by naturality, if the class N (t k )a i σ , 0 ≤ i < 2 k+1 − 1 is killed in a −1 λ SliceSS(BP ((C4)) ), the differential killing it must be of length longer than 2 k+2 − 3. However, by Proposition 5.2 and Theorem 4.1, the potential source of such a differential must be trivial in the E 2 -page. Therefore the classes N (t k )a i σ for k > 0 and 0 ≤ i ≤ 2 k+1 − 1 are nontrivial permanent cycles.
6. The Tate spectral sequence of N 2 1 HF 2 The goal of this section is to advance our knowledge of the Tate spectral sequence of N 2 1 HF 2 . We compute it in a range and also give all differentials originating from the first diagonal of slope −1. Our main method is comparison with the localized slice spectral sequence, a method we describe first.
The above argument, along with Lemma 5.24, gives the following comparison square, which is central to our computation in this section. (3) Proposition 6.1. In the comparison square, both horizontal maps converge to isomorphisms in homotopy groups.
The bottom map is the a σ -localization of the top map, thus also converges to an isomorphism.
The bottom map in the comparison square is particularly interesting: We completely understand a −1 σ SliceSS(BP ((C4)) ), which is determined by the fact that it computes π * HF 2 . All differentials are derived from the slice differential theorem [HHR16, Theorem 9.9, Remark 9.11]. On the other hand, the Tate spectral sequence of N 2 1 HF 2 is very mysterious: its E 2 -page is determined by the Tate cohomologyĤ * (C 2 ; A * ), for which we do not know a closed formula yet. Nevertheless, the Segal conjecture shows that the Tate spectral sequence converges to π * HF 2 , meaning almost everything kills each other by differentials. Using Theorem 3.18, we can apply our understanding of the slice spectral sequence to understand partially how differentials work in the Tate spectral sequence. Figure 7 consists of the integral E 2 -pages of the four spectral sequences in the comparison square. Red elements in the homotopy fixed points and the Tate spectral sequences are those in the image of the horizontal maps. We prove these claims in Corollary 6.5.
Using the comparison square, we establish an infinite family of differential in the Tate spectral sequence. We also compute all differentials in the Tate spectral sequence in the same range we computed a −1 λ SliceSS(BP ((C4)) ) in Section 5. Specifically, we show all differentials hitting elements from stem 0 to 8 which map non-trivially into the homotopy orbit spectral sequence. In the (doubled) Tate spectral sequence of Figure 7, they are elements below slope 1 from stem 0 to 8.
Because of the comparison square, we make our statements and arguments entirely in the spectral sequences P * 4/2 DHFPSS and P * 4/2 DTateSS. The translation back to the C 2 -homotopy fixed points and the Tate spectral sequence is straightforward. As a reference, TateSS(N 2 1 HF 2 ) with known differentials is shown as Figure 9.
To start the computation, we want to understand how the maps of the comparison square behave on the E 2 -page. By Theorem 3.18, they are determined by the C 2 -geometric fixed points of elements in π C2 BP ((C4)) .
Proof. We will show Φ C2 (t i ) = ξ i ; the formula Φ C2 (γt i ) = ζ i follows from the fact that the residue C 4 /C 2 -action on Φ C2 BP ((C4)) becomes the conjugate action on the dual Steenrod algebra.
Let e : S 0 → BP R be the unit map and F 1 and F 2 be the formal groups laws on π C2 * ρ2 BP R ∧BP R induced by the map respectively, and letx 1 ,x 2 be the corresponding power series generators. As in Section 3, the elements t i are defined asx Taking Φ C2 mapsx 1 andx 2 to the two M O-orientation on HF 2 ∧ HF 2 . The following lemma completes the proof. respectively. Then we have Proof. Identify (HF 2 ∧ HF 2 ) * (RP ∞ ) with A * [[x 1 ]], and write x 2 = ∞ j=0 a j x j+1 1 . We will show that a 2 i −1 = ξ i and all other a j 's are 0. First, since the power series ∞ j=0 a j x j+1 1 is an automorphism of the additive formal group law in an F 2 -algebra, we must have a 0 = 1, and a j = 0 for j = 2 i − 1. Let I be an admissible sequence and define to be the composition One can verify directly that θ I has the following properties: • θ I (x 2 ) = x 2 n 2 if I = (2 n−1 , 2 n−2 , ..., 2, 1), and θ I (x 2 ) = 0 otherwise. • θ I (x 1 ) = x 1 if I = (0) and θ I (x 1 ) = 0 otherwise. Comparing the coefficient of x 2 n 1 in both expressions, we see that 1 = θ I (a 2 n −1 ) = a 2 n −1 , Sq I . Now, if I is any other admissible sequence with |I| = 2 n − 1, then θ I (x 2 ) = 0 and thus 0 = θ I (a 2 n −1 ) = a 2 n −1 , Sq I . This is exactly the definition of ξ n , see [MT68, Chapter 6, Proposition 1].
We pause here to clarify notations in P * 4/2 DHFPSS(N 2 1 HF 2 ) and P * 4/2 DTateSS(N 2 1 HF 2 ). The C 2 -level of this spectral sequence is the spectral sequence of the doubled Postnikov tower of HF 2 ∧ HF 2 , treated as a C 2 -equivariant spectral sequence whose underlying level is trivial (and thus a σ2 acts invertibly). Therefore, given an element x ∈ A * , there are elements in different RO(G)-grading differing by powers of a σ2 that deserve the name x. We name the corresponding element in the integral grading by x, and name all others by a i σ2 x for some i ∈ Z. Notice that in this way, ξ n has stem and filtration 2 n − 1 since we are working in the doubled spectral sequence. Under this notation, the map a −1 λ SliceSS(BP ((C4)) ) → P * 4/2 DHFPSS(N 2 1 HF 2 ) sends t i to a −(2 i −1) σ2 ξ i (on the C 2 -level), as follows from Theorem 3.18 and Proposition 6.2: since the target spectral sequence collapses on C 2 -level, the image of t i is determined by its RO(C 2 )degree and its image under Φ C2 . In the C 4 -level, we need to be extra careful. By taking N = N 4 2 on t i → a −(2 i −1) σ ξ i , we see that where N (ξ i ) is in RO(C 4 )-degree (2 i − 1)(1 + σ) and filtration 2(2 i − 1). The complication comes from the fact that there are other generators of Tate cohomology than N (ξ i ). For example, the element ξ 1 is invariant under the conjugate action and thus gives a generator ofĤ 0 (C 2 ; (A * ) 1 ). For such generators in degree i of A * , we will use the notation b i , and define that they are in the integral grading. For example, the generator ofĤ 0 (C 2 ; (A * ) 1 ) is named b 1 , and has bidegree (1, 1) in the double of the homotopy fixed points and the Tate spectral sequence. Since the square of b 1 restricts to ξ 2 1 = ξ 1 ζ 1 , we have (for degree reasons) a multiplicative relation b 2 1 = N (ξ 1 )u σ , where u σ is a generator of the Tate cohomology of trivial modulê H (C 2 ; F 2 ) ∼ = F 2 [a ± σ , u ± σ ]. The generator u σ has degree 1 − σ and a σ has degree −σ. The classical integral graded Tate cohomologyĤ with degree 1 generator x is related to the RO(C 2 )-graded cohomology via x = u σ a −1 σ . Since the sign representation on C 4 /C 2 pulls back to the sign representation on C 4 , we use the same notations u σ and a σ in the pullback of the homotopy fixed points and the Tate spectral sequence.
The proof is purely combinatorical and is irrelevant to other parts of the paper. It uses computations and ideas from [CW00].
Proof. We argue by monomial degrees in A * = F 2 [ξ 1 , ξ 2 , · · · ] and the Milnor conjugate formula The conjugate formula tells us that the transfer of a monomial (i.e. the sum of the monomial and its conjugate) in the ξ i can only increase its monomial degree. It also tells us that the monomial with minimal monomial degree in (ξ i ζ i ) k is ξ 2k i . Therefore, (ξ i ζ i ) k being in the image of transfer can only happen when ξ 2k i appears in the transfer of a monomial, which has smaller monomial degree and the same topological degree.
To streamline the computation, we define that a monomial P has bidegree (a, b) if P has monomial degree b and topological degree a − b. In this way, ξ i has bidegree (2 i , 1). To find monomials which have smaller monomial degree and the same topological degree, we can look at the binary expansion of a. Let t be a positive integer, and Bin(t) be the number of 1s in the binary expansion of t. If a monomial has bidegree (a, b), then both a must be even and Bin(a) ≤ b.
We will only check the highest power of ξ i ζ i listed in the statement of the proposition, since if (ξ i ζ i ) k is nontrivial in Tate cohomology, then (ξ i ζ i ) j for j ≤ k are all nontrivial.
The class of ξ 1 ζ 1 is obviously nontrivial in Tate cohomology, so we start our argument with (ξ 2 ζ 2 ) 2 . Writing it as a polynomial in the ξ i , the leading term is ξ 4 2 , which has bidegree (16, 4). We only need to check if there is a nontrivial monomial in bidegree (14, 2). Since Bin(14) = 3 > 2, there is no monomial in this degree. Therefore (ξ 2 ζ 2 ) 2 is nontrivial in the Tate cohomology.
The proof can certainly be generalized. For example, (ξ i ζ i ) 8 are nontrivial in the Tate cohomology for i > 11. However, what we proved is sufficient for our computation.
The exact same argument gives the following differentials.