The $RO(C_4)$ cohomology of the infinite real projective space

Following the Hu-Kriz method of computing the $C_2$ genuine dual Steenrod algebra $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$, we calculate the $C_4$ equivariant Bredon cohomology of the classifying space $\mathbf R P^{\infty \rho}=B_{C_4}\Sigma_{2}$ as an $RO(C_4)$ graded Green-functor. We prove that as a module over the homology of a point (which we also compute), this cohomology is not flat. As a result, it can't be used as a test module for obtaining generators in $(H\mathbf F_2)_{\bigstar}(H\mathbf F_2)$ as Hu-Kriz do in the $C_2$ case.

Historically, computations in stable equivariant homotopy theory have been much more difficult than their nonequivariant counterparts, even when the groups involved are as simple as possible (i.e.cyclic).In recent years, there has been a resurgence in such calculations for power 2-cyclic groups C 2 n , owing to the crucial involvement of C 8 -equivariant homology in the solution of the Kervaire invariant problem [HHR16].
The case of G = C 2 is the simplest and most studied one, partially due to its connections to motivic homotopy theory over R by means of realization functors [HO14].It all starts with the RO(C 2 ) homology of a point, which was initially described in [Lew88].The types of modules over it that can arise as the equivariant homology of spaces were described in [CMay18], and this description was subsequently used in the computation of the RO(C 2 ) homology of C 2 -surfaces in [Haz19].The C 2 -equivariant dual Steenrod algebra (in characteristic 2) was computed in [HK96] and gives rise to a C 2 -equivariant Adams spectral sequence that has been more recently leveraged in [IWX20].Another application of the Hu-Kriz computation is the definition of equivariant Dyer-Lashof operations by [Wil19] in the F 2 -homology of C 2 -spectra with symmetric multiplication.Many of these results rely on the homology of certain spaces being free as modules over the homology of a point, and there is a robust theory of such free spectra described in [Hil19].
The case of G = C 4 has been much less explored and is indeed considerably more complicated.This can already be seen in the homology of a point in integer coefficients (see [Zeng17] and [Geo19]) and the case of F 2 coefficients is not much better (compare subsections 3.1 and 3.2 for the C 2 and C 4 cases respectively).The greater complexity in the ground ring (or to be more precise, ground Green functor), means that modules over it can also be more complicated and indeed, certain freeness results that are easy to obtain in the C 2 case no longer hold when generalized to C 4 (compare subsection 4.1 with sections 6 and 7).
The computation of the dual Steenrod algebra relies on the construction of Milnor generators.Nonequivariantly, the Milnor generators ξ i of the mod 2 dual Steenrod algebra can be defined through the completed coaction of the dual Steenrod algebra on the cohomology of BΣ 2 = R P ∞ : H * (BC 2+ ; F 2 ) = F 2 [x] and the completed coaction In the C 2 -equivariant case, the space replacing BΣ 2 is the equivariant classifying space B C 2 Σ 2 .This is still R P ∞ but now equipped with a nontrivial C 2 action (described in subsection 4.1).Over the homology of a point, we no longer have a polynomial algebra on a single generator x, but rather a polynomial algebra on two generators c, b modulo the relation where a σ , u σ are the C 2 -Euler and orientation classes respectively (defined in section 2).As a module, this is still free over the homology of a point, and the completed coaction is The τ i , ξ i are the C 2 -equivariant analogues of the Milnor generators, and Hu-Kriz show that they span the genuine dual Steenrod algebra.
For C 4 , the cohomology of B C 4 Σ 2 is significantly more complicated (see section 7) and most importantly is not a free module over the homology of a point.In fact, it's not even flat (Proposition 5.3) bringing into question whether we even have a coaction by the dual Steenrod algebra in this case.
There is another related reason to consider the space B C 4 Σ 2 .In [Wil19], the author describes a framework for equivariant total power operations over an H F 2 module A equipped with a symmetric multiplication.The total power operation is induced from a map of spectra A → A tΣ [2]   where (−) tΣ [2] is a variant Tate construction defined in [Wil19].
In the nonequivariant case, A → A tΣ [2] induces a map A * → A * ((x)) and the Dyer-Lashof operations Q i can be obtained as the components of this map: In the C 2 equivariant case, we have a map A ⋆ → A ⋆ [c, b ± ]/(c 2 = a σ c + u σ b) and we get power operations For C 4 we would have to use the cohomology of B C 4 Σ 2 (localized at a certain class) but that is no longer free, meaning that the resulting power operations would have extra relations between them and further complicating the other arguments in [Wil19].
The computation of H ⋆ (B C 4 Σ 2+ ; F 2 ) also serves for a test case of RO(G) homology computations for equivariant classifying spaces where G is not of prime order.We refer the reader to [Shu14], [Cho18], [Wil19], [SW21] for such computations in the G = C p case.
As for the organization of this paper, section 2 describes the conventions and notations that we shall be using throughout this document, as well as the Tate diagram for a group G and a G-equivariant spectrum.
Subsections 3.1 and 3.3 describe the Tate diagram for C 2 and C 4 respectively using coefficients in the constant Mackey functor F 2 .
In section 4 we define equivariant classifying spaces B G H and briefly explain the elementary computation of the cohomology of B C 2 Σ 2 (this argument also appears in [Wil19]).
In section 5 we present the result of the computation of H ⋆ (B C 4 Σ 2+ ; F 2 ) and prove that it's not flat as a Mackey functor module over (H F 2 ) ⋆ .Sections 6 and 7 contain the proofs of the computation of the cohomology of B C 4 Σ 2 .
We have included two appendices in the end; Appendix A contains pictures of the spectral sequence converging to H ⋆ (B C 4 Σ 2+ ; F 2 ) while Appendix B contains a detailed description of H ⋆ (S 0 ; F 2 ), which is the ground Green functor over which all our Mackey functors are modules.
To aid in the creation of these appendices, we extensively used the computer program of [Geo19] available here.In fact, we have introduced new functionality in the software that computes the RO(G)-graded homology of spaces such as B C 4 Σ 2 given an explicit equivariant CW decomposition (such as we discuss in subsection 6.1).This assisted in the discovery of a nontrivial d 2 differential in the spectral sequence of B C 4 Σ 2 (see Remark 7.6), although the provided proof is independent of the computer computation.
Acknowledgment.We would like to thank Dylan Wilson for answering our questions regarding his paper [Wil19] as well as [HK96].We would also like to thank Peter May for his numerous editing suggestions, that vastly improved the readability of this paper.

Conventions and notations
We will use the letter k to denote the field F 2 , the constant Mackey functor k = F 2 and the corresponding Eilenberg-MacLane spectrum Hk.The meaning should always be clear from the context.
All our homology and cohomology will be in k coefficients.
The data of a C 4 Mackey functor M can be represented by a diagram displaying the values of M on orbits, its restriction and transfer maps and the actions of the Weyl groups.We shall refer to M(C 4 /C 4 ), M(C 4 /C 2 ), M(C 4 /e) as the top, middle and bottom levels of the Mackey functor M respectively.The Mackey functor diagram takes the form: If X is a G-spectrum then X ⋆ denotes the RO(G)-graded G-Mackey functor defined on orbits as The index ⋆ will always be an element of the real representation ring RO(G).RO(C 4 ) is spanned by the irreducible representations 1, σ, λ where σ is the 1dimensional sign representation and λ is the 2-dimensional representation given by rotation by π/2.
−V the Euler class induced by the inclusion of north and south poles S 0 ֒→ S V ; also denote by u V ∈ k C 4 |V|−V the orientation class generating the Mackey functor k |V|−V = k.
We will use the notation āV , ūV to denote the restrictions of a V , u V to middle level, and ūV to denote the restriction of u V to bottom level.
We also write a σ 2 ∈ k C 2 −σ 2 and u σ 2 ∈ k C 2 1−σ 2 for the C 2 Euler and orientation classes, where σ 2 is the sign representation of C 2 .
The Gold Relation ( [HHR16]) takes the following form in k coefficients: Let EG be a contractible free G-space and ẼG be the cofiber of the collapse map EG + → S 0 .We use the notation The Tate diagram ([GM95]) then takes the form:

≃
The square on the right is a homotopy pullback diagram and is called the Tate square.
Applying π G ⋆ on the Tate diagram gives 3.2.The RO(C 4 ) homology of a point.The RO(C 4 ) homology of a point (in k coefficients) is significantly more complicated than the RO(C 2 ) one (see [Geo19] for the integer coefficient case).Appendix B contains a very detailed description of it, and the goal in this subsection is to provide a more compact version.The top level is: where the indices i, j, m range in 0, 1, 2, ... and ǫ ranges in 0, 1.The use of .
= as opposed to = is meant to signify some subtlety present in (1) that needs to be clarified before the equality can be used.This subtlety has to do with how quotients are defined (cf [Geo19]) and how elements multiply (the multiplicative relations).We begin this process of interpreting (1) with the definition of θ: θ = Tr 4 2 ( ū−2 σ ).We further introduce the elements a λ With this notation, the second curly bracket in (1) contains elements of the form x n,1 a i λ , x n,1 a σ a i λ and the third contains The behavior of the x n,m depends crucially on whether m = 1 or not: x n,1 u σ = 0 but x n,m u σ = 0 for m > 1; the x n,1 are infinitely a σ divisible since: σ a λ while the x n,m , m > 1, can only be divided by a σ once.That's why we separate them into two distinct summands in (1).
The third curly bracket in (1) for ǫ = 0 consists of quotients of The quotients in the RHS of (1) are all chosen coherently (cf [Geo19]), that is we always have the cancellation property: To compute any product of two elements in the RHS of (1) we follow the following procedure: • If both elements involve θ then the product is automatically 0.
Here, √ āλ ūλ is the (unique) element whose square is āλ ūλ and v is defined by The interpretation of (2) is complete.In terms of the notation of the C 2 generators, Finally the bottom level is very simple: The map k h → k in the Tate diagram induces We define the equivariant classifying space For G = C 2 , the spaces B C 2 Σ 2 are used in the computation of the C 2 dual Steenrod algebra by ) and for the construction of the total C 2 -Dyer-Lashof operations in [Wil19].Both use the computation where c, b are classes in cohomological degrees σ 2 , 1 + σ 2 respectively.Let us note here that B C 2 Σ 2 is R P ∞ with a nontrivial C 2 action; the restrictions of c, b are the generators of degree 1, 2 of k * (R P ∞ ).
We shall now summarize this computation, since part of it will be needed for the analogous computation when G = C 4 which takes place in sections 5-7.
Let σ, τ be the sign representations of C 2 , Σ 2 respectively and ρ = 1 + σ.Then E C 2 Σ 2 = S(∞(ρ ⊗ τ)); the graph subgroups of C 2 × Σ 2 are C 2 , ∆ and their orbits correspond to the cells and whose quotients (after adjoining disjoint basepoints) are Quotiening Σ 2 gives a filtration for B C 2 Σ 2+ with Applying k ⋆ yields a spectral sequence of modules over the Green functor k ⋆ .The fact that the differentials are module maps gives E 1 = E 2 for degree reasons.Furthermore, the vanishing of the RO(C 2 ) homology of a point in a certain range gives E 2 = E ∞ .The E ∞ page is free as a module over the Green functor k ⋆ , hence there can't be any extension problems and we get the module structure: It's easier to prove (using the homotopy fixed point spectral sequence) that: where w has cohomological degree 1.The map k → k h from section 2 induces which is localization with u σ 2 being inverted.Thus we can see that c = e σ maps to u σ 2 w (or a σ 2 + u σ 2 w), b = e ρ maps to a σ 2 w + u σ 2 w 2 and conclude that: For degree reasons, we can see that (we can add a σ 2 to c to force ǫ(c) = 0).The primitive elements are spanned by c, b 2 i .

The cohomology of B C 4 Σ 2
In the next section we shall construct a cellular decomposition of B C 4 Σ 2 giving rise to a spectral sequence computing k ⋆ (B C 4 Σ 2+ ).Here's the result of the computation, describing k ⋆ (B C 4 Σ 2+ ) as a Green functor algebra over k ⋆ : Proposition 5.1.There exist elements e a , e u , e λ , e ρ in degrees σ + λ, σ The relation set S consists of two types of relations (we use indices i, j ≥ 0): • Module relations: As for the Mackey functor structure, the Weyl group C 4 /C 2 action on the generators is trivial and we have: • Mackey Functor relations: with trivial Weyl group C 4 action and Mackey functor relations obtained by applying Res 4 1 to the multiplicative relations of S: 1 (e u ) 4 Note: For every quotient y/x there is a defining relation x • (y/x) = y.We have omitted these implicit module relations from the description above.
The best description of the middle level is in terms of the generators c, b of The correspondence of generators is: We can also express the map to homotopy fixed points in terms of our generators: Proposition 5.2.There is a choice of the degree 1 element w in given on top level by multiplication with a 2 σ /a λ and determined on the lower levels by restricting (so it's multiplication with v ū2 σ on the middle level and 0 on the bottom level).If M is a flat R-module then we have an exact sequence The restriction functor Res 4 2 from R modules to Res 4 2 R modules is exact and symmetric monoidal, so we replace M, R, Ker( f ) by Res 4 2 M, Res 4 2 R, Res 4 2 Ker( f ) respectively and have an exact sequence of C 2 Mackey functors.Using the notation involving the C 2 generators c, b and writing a = a σ 2 , u = u σ 2 , we have M = ⊕ i≥0 R{b 2i , cb 2i+1 } ⊕ ⊕ i≥0 R{ab 2i+1 , ub 2i+1 , acb 2i , ucb 2i }/ ∼.The map f maps each summand to itself, so we may replace M by R{c, ab, ub, acb, ucb}/ ∼ and continue to have the same exact sequence as above.The top level then is: and v acts trivially on ab, ub, ac, uc i.e. on M(C 2 /C 2 ) so we get This contradicts that ab = e ′ λ+1 is not divisible by any element of the ideal I (e ′ λ+1 is only divisible by ū±i σ ∈ R which are not in I).

A cellular decomposition of
We denote the generators of C 4 and Σ 2 by g and h respectively; let also τ be the sign representation of Σ 2 and ρ = 1 + σ + λ the regular representation of C 4 .
We now describe a cellular decomposition of E C 4 Σ 2+ where the orbits are • Start with {(x 1 )} the union of two points (1), (−1) and the basepoint.This is The cofiber is the wedge of two circles, corresponding to x 2 being positive or negative, and the action is After applying the self equivalence given by f (x 1 , +) = (x 1 , +) and f (x 1 , −) = (−x 1 , −), the action becomes the cofiber is the wedge of four spheres corresponding to the sign of the nonzero coordinate among the last two coordinates.If we number the spheres from 1 to 4 and use (x, y) i coordinates to denote them i = 1, 2, 3, 4 then Applying the self equivalence the action becomes g(x, y) i = (−y, x) i+1 and h(x, y) i = (x, y) i+2 i.e. we have and the cofiber is the wedge of four S 3 's corresponding to the signs of x 3 , x 4 .Analogously to the item above, we get the space the cofiber is the wedge of two S 4 's corresponding to the sign of x 5 and we get . And so on...We get the decomposition of B C 4 Σ 2+ where the associated graded is: We note however that the type II decomposition of B C 4 Σ 2 obtained this way is rather inefficient and not minimal.For example, consider the subspace spanned by homogeneous coordinates {(x 1 : x 2 : x 3 : x 4 )}; this is obtained from the subspace {(x 1 : x 2 : 0 : x 4 ), (x 1 : x 2 : x 3 : 0)} by attaching a cell (C 4 /C 2 ) + ∧ S 1+λ .The sphere S 1+λ itself has a top dimensional cell C 4+ ∧ S 3 ; combining the two we get 2 cells in dimension 3. On the other hand, {(x 1 : x 2 : x 3 : x 4 )} can also be obtained from the subspace {(x 1 : x 2 : 0 : x 4 ), (x 1 : x 2 : x 3 : 0), (x 1 : 0 : x 3 : x 4 )} by attaching C 4+ ∧ S 3 i.e. 1 cell in dimension 3. Working out the lower dimensions, we can see that the type II decomposition of R P σ+λ = {(x 1 : x 2 : x 3 : x 4 )} obtained by expanding the type I decomposition has 12 total cells, while it is possible to obtain a more efficient type I decomposition with 9 total cells.

The spectral sequence for
The differential d r has (V, s) bidegree (1, r) so it goes 1 unit to the right and r units up in (V, s) coordinates.
Before we can write down the E 1 page, we will need some notation: For a G-Mackey functor M and subgroup H ⊆ G, M G/H denotes the G-Mackey functor defined on orbits as M G/H (G/K) = M(G/H × G/K); the restriction, transfer and Weyl group action in M G/H are induced from those in M. For G = C 4 and H = C 2 , the bottom level of M C 4 /C 2 is: where x, y are used to distinguish the two copies of M(C 4 /e), i.e. so that any element of M C 4 /C 2 (C 4 /e) can be uniquely written as mx + m ′ y for m, m ′ ∈ M(C 4 /e).The Weyl group W C 4 e = C 4 acts as We can then describe M C 4 /C 2 in terms of M and the computation of the restriction and transfer on x, which are shown in the following diagram: ) module via extension of scalars along the restriction map Res 4 2 : R(C 4 /C 4 ) → R(C 4 /C 2 ).
7.1.The E 1 page.The rows in the E 1 page are: We will write e jρ , e jρ+σ , e jρ+λ , e jρ+λ+1 for the unit elements corresponding to the E 1 terms above, living in degrees V = jρ, jρ + σ, jρ + λ, jρ + λ + 1 and filtrations s = 4j, 4j + 1, 4j + 2, 4j + 3 respectively.We also write ēV , ēV for their restrictions to the middle and bottom levels respectively.In this way: and the three levels of the Mackey functor E ⋆, * 1 , from top to bottom, are: At this point, the reader may want to look over pictures of the E 1 page that we have included in the Appendix A. 7.2.The d 1 differentials.In this subsection, we explain how the d 1 differentials on each level are computed.We shall need this crucial remark: The equivariant cohomology of this space is known from subsection 4.1 and we shall use this result to compute the middle level spectral sequence for B C 4 Σ 2 .Further restricting to the trivial group e ⊆ C 4 , we get the nonequivariant space R P ∞ and this will be used to compute the bottom level spectral sequence.
First of all, the bottom level spectral sequence is concentrated on the diagonal and the nontrivial d 1 's are k{x, gx} → k{x, gx}, x → x + gx (since k * (R P ∞ ) is k in every nonnegative degree).
The d 1 's on middle and top level are computed from the fact that they are k ⋆ module maps, hence determined on e jρ , e jρ+σ , ū−i for the top level (ǫ, ǫ ′ = 0, 1), and on ējρ , ējρ+σ , ējρ+λ x, ējρ+λ+1 x for the middle level.We remark that because k does not suffice to compute the top level d 1 on e jρ , e jρ+σ , e jρ+λ , e jρ+λ+1 .
The d 1 differentials from row 4j to row 4j + 1 are all determined by the differential d 1 : ke jρ → k 1−σ e jρ+σ .Note that k 1−σ is generated by 0|u −1 σ | ū−1 σ (this notation was defined in [Geo19] and expresses the generators of all three levels from top to bottom separated by vertical columns).The d 1 is trivial on bottom level, and using the fact that it commutes with restriction we can see that it's trivial in all levels.
Similarly, the d 1 differentials from row 4j + 1 to row 4j + 2 are all determined by d 1 : The differential is trivial on the bottom level, but on middle level the C 2 computation gives k σ C 2 (B C 4 Σ 2+ ) = 0 forcing the differential to be nontrivial (the only other way to kill E σ−λ+1,4j+2 1 (C 4 /C 2 ) = k 2 is for the d 1 differential from row 4j + 2 to 4j + 3 to be the identity k 2 → k 2 on middle level, which can't happen as we show in the next paragraph).Thus: The d 1 differentials from row 4j + 2 to row 4j + 3 are determined by On bottom level, these d 1 's all are x → x + gx and the commutation with restriction and transfer gives: āλ ūλ e jρ+λ (x + gx)) = 0 Finally, the d 1 differentials from row 4j + 3 to row 4j + 4 are determined by These are trivial on the bottom level and by the commutation with restriction and transfer we can see that they are trivial on all levels.
This settles the E 1 page computation.
∞ = 0 for some t > s, then there are multiple lifts of α.In that case, we pick the lift for which there are no exotic restrictions (if possible).For example, if Res 2 1 (α) = 0 in E ∞ and there is a unique lift β of α such that Res 2 1 (β) = 0, then we use β as our lift of α.With this in mind, and the computation of the E 2 terms, we have: • The elements ējρ survive the spectral sequence and lift uniquely to elements ējρ in k Remark 7.2.We should explain the notation used for the generators above.First, the elements ēj,u , ējρ+λ will turn out to be the restrictions of top level elements e j,u , e jρ+λ respectively, both in E ∞ and in k ⋆ C 4 (B C 4 Σ 2+ ), hence their notation.Second, the elements e j,au , e ′ jρ+λ+1 are never restrictions, neither in E ∞ nor in , so their notation is rather ad-hoc: the au in e j,au serves as a reminder of the √ āλ ūλ in e j,au = √ āλ ūλ ējρ+σ , while the prime ′ in e ′ jρ+λ+1 is used to distinguish them from the top level generators e jρ+λ+1 that the e ′ jρ+λ+1 transfer to.
Finally, the elements ẽj,a are restrictions of top level elements e j,a in E ∞ , but not in k ⋆ C 4 (B C 4 Σ 2+ ) due to nontrivial Mackey functor extensions (exotic restrictions).That's why we denote them by ẽj,a as opposed to ēj,a ; the ēj,a are reserved for Res 4 2 (e j,a ) = ẽj,a + ūσ e ′ jρ+λ+1 (see Lemma 7.12) .
e jρ 2 +σ 2 } (see subsection 4.1).We shall write our middle level C 4 generators in terms of the C 2 generators.Proposition 7.3.We have: Proof.The map f : homeomorphism and induces a map on filtrations: (the downwards arrows are f while the arrows in the opposite direction are f −1 ).
To keep the notation tidy, we verify the correspondence of generators for j = 0.
Proof.The elements a λ e jρ+σ can only support d 3 (a λ e jρ+σ ) = e (j+1)ρ and applying restriction shows that this cannot happen.
Both lifts have the same restriction, which by Lemma 7.12 is computed to be ẽj,a + ūσ e ′ jρ+λ+1 (the proof of the Lemma works regardless of the survival of a σ e jρ+σ ).Now one of those lifts, that we shall call β, satisfies: which contradicts the computation of the module structure of the middle level.
Proof.We work page by page.On E 2 we have: where ǫ i = 0, 1. Multiplying by a σ and using that a σ ū−i σ e jρ+λ = 0 and that a σ ū−i σ √ āλ ūλ e jρ+λ = 0 shows that ǫ 1 = ǫ 2 = ǫ 3 = 0. On E 3 we have: where again ǫ i = 0, 1.We see that ǫ 1 = ǫ 2 = 0 by multiplication with a σ , while ǫ 3 = 0 can be seen by multiplying with a 2 σ .The pattern of higher differentials is the same as in E 2 , E 3 and the same arguments show that there are no higher differentials.
In conclusion: where i, j ≥ 0 and ǫ, ǫ ′ = 0, 1.We have relations: If on the other hand E t,V ∞ = 0 for some t > s, then there are multiple choices of lifts of α.
When it comes to fractions y/x, we should make sure our choices of lifts are "coherent".Let us explain what that means with an example: The element u λ e σ has a unique lift x 0 while (u λ /u i σ )e σ has multiple distinct lifts if i ≥ 5.If we choose x i to lift (u λ /u i σ )e σ then it will always be true that u i σ x i = x 0 ; however, we shouldn't write x i = x 0 /u i σ unless we can also guarantee that: This expresses the coherence of fractions (also discussed in subsection 3.2 and Appendix B) which is the cancellation property: This holds on E ∞ and we also want it to hold on k ).One more property enjoyed by the (u λ /u i σ )e σ is that a 2 σ (u λ /u i σ )e σ = 0; it turns out that there are unique lifts x i of (u λ /u i σ )e σ such that a 2 σ x i = 0 and those lifts also satisfy the coherence property u σ x i = x i−1 : Proposition 7.9.For i, j ≥ 0, there are unique lifts e j,u /u i σ , e jρ+λ /u i σ of the elements (u λ /u i σ )e jρ+σ , ū−i σ e jρ+λ respectively that satisfy: These lifts are also coherent.
Uniqueness: If α, α ′ are two lifts of (u λ /u i σ )e jρ+σ then their difference is a finite sum p of elements β ′ e V where each Coherence: Unfix i and let x i be the lift of (u λ /u i σ )e jρ+σ with a 2 σ x i = 0. Then u σ x i is a lift of (u λ /u i−1 σ )e jρ+σ and a 2 σ (u σ x i ) = 0 hence by uniqueness: The case of ū−i σ e jρ+λ is near identical to what we did above for (u λ /u i σ )e jρ+σ .The changes are as follows.First, s 0 > 4j + 2 (instead of s 0 > 4j + 1).Next we can see that s 0 > 4j + 4 if i > 1, and multiplying by u σ also proves the i = 0, 1 cases (this replaces the argument that showed s 0 > 4j + 3).The rest of the arguments are identical.

Top level generators.
The elements e jρ have unique lifts to k ⋆ C 4 (B C 4 Σ 2+ ) that we continue to denote by e jρ .
On the other hand, for each j ≥ 0 there are two possible lifts of a λ e jρ+σ .There is no good way to make a unique choice at this point, so we shall write e j,a for either.
Before we can lift the rest of the E ∞ generators, we will need the following exotic restriction: Lemma 7.12.Both choices of e j,a have the same (exotic) restriction: Proof.The two choices of e j,a differ by u σ e jρ+λ+1 = Tr 4 2 ( ūσ e ′ jρ+λ+1 ) hence have the same restriction.From the E ∞ page: Res 4 2 (e j,a ) = ẽj,a + ǫ ūσ e Proof.Fix i, j ≥ 0 and fix ⋆ to be the degree of the element This element is by definition in filtration 4j + 1, however its projection to E 4j+1,⋆ ∞ is: āλ ūλ e jρ+λ so it suffices to check that . Multiplying by u i σ reduces us to the case i = 0 and then: using Lemma 7.12 and the middle level computation of subsection 7.4.Projecting this restriction to E 4j+2,⋆ ∞ returns ūσ √ āλ ūλ ējρ+λ = 0 as desired.
Coherence of e jρ+λ √ /u i σ follows from the coherence of u λ /u i σ and e j,u /u i σ .
Lemma 7.14.The elements Proof.The fact that these transfers are lifts follows from the E ∞ page; coherence follows from the Frobenius relations.We check the equality directly: We used the middle level relation āλ e j,au = √ āλ ūλ ẽj,a + āλ ūσ ējρ+λ and the fact that ū−i σ ējρ+λ is the restriction of e jρ+λ /u i σ which follows from the same fact on E ∞ .Lemmas 7.11, 7.13, 7.14 combined with Corollary 7.8 prove Proposition 7.10.
Proof.We can see directly that there are no Mackey functor extensions for e jρ , e j,u /u i σ and e jρ+λ /u i σ .The rest were established in the previous two subsections, apart from: To see this, recall that a 2 σ (e j,u /u i σ ) = 0 hence a σ (e j,u /u i σ ) is a transfer.Moreover, a σ (e j,u /u i σ ) = 0 which is seen on the E ∞ page, and the only way that a σ (e j,u /u i σ ) can be a nonzero transfer is for a σ (e j,u /u i σ ) = Tr 4 2 (e j,au ū−i σ ).
7.9.Top level module relations.With the exception of relations expressing coherence (u σ (e j,u /u i σ ) = e j,u /u i−1 σ and u σ (e jρ+λ /u i σ ) = e jρ+λ /u i−1 σ ), the rest of the module relations are: ) is generated by: e jρ , e j,a , e j,u u i σ , e jρ+λ u i σ under the relations: Proof.For m > 0, we have the possible extensions: where each * denotes a nonnegative index (with different instances of * being possibly different indices) and each ǫ * = 0, 1.Thus, multiplication by a λ is an isomorphism for both sides which reduces us to m = 1.For m = 1 and i > 0 there are no extensions i.e. ǫ * = 0 for all * .This establishes Similarly, if m > 0, we have the possible extensions: and multiplying with a m λ reduces us to: and substituting gives the desired relation.For i = m = 1 we get (a 2 σ /a λ )(e jρ+λ+1 /u σ ) = 0 which lifts the E ∞ relations v ūσ e jρ+λ = v ū2 σ e jρ+λ+1 = 0.
As special cases we get the relations: Proof.First of all, e jρ = (e ρ ) j since there are no extensions in degree jρ (to see that (e ρ ) j = 0 apply restriction).Let A be the algebra span of e a , e u /u i σ , e λ /u i σ , e ρ .To see that e j,a ∈ A observe: e jρ e a = ǫa σ a λ e jρ + e j,a + ǫ ′ u σ e jρ+λ+1 we get that e j,a ∈ A regardless of the status of ǫ, ǫ ′ .Now suppose by induction that all elements in filtration ≤ 4j are in A. We have that: where • • • are in filtration < 4j + 1 hence in A. Since e * ρ , e * ,a ∈ A for any * ≥ 0, we get e j,u /u i σ ∈ A. This establishes that everything in filtration ≤ 4j + 1 is in A. Finally, where • • • are in filtration < 4j + 2, so by the same argument e jρ+λ /u i σ ∈ A as well.This completes the induction step.
Inverting u σ , u λ gives k hC 4 ⋆ [e ρ , e a , e u , e λ ] modulo relations, which is isomorphic to is given by: Proof.Using the C 2 result (see subsection 4.1), we have the correspondence on the middle level generators: 2 (e a ) → ūσ ( ūλ w 3 + āλ w) • ēρ → ūσ ( āλ w 2 + ūλ w 4 ) from which we can deduce that the correspondence on top level is: where the ǫ i range in 0, 1.
where ǫ i = 0, 1 and • • • is the sum of elements mapping to 0 in homotopy fixed points, but all having denominator a 2 σ .Mapping to homotopy fixed points shows ǫ 0 = ǫ 1 = 0 and ǫ 2 = 1, while multiplying by a 2 σ trivializes the LHS (by a 2 σ (e u /u i σ ) = 0) and thus shows that • • • = 0.The same argument applied to: There are two ways to show that ǫ 0 = 0: The first is to multiply with a σ u i+j−2 σ and compute a σ e λ (e u /u 2 σ ) using a σ e λ = Tr 4 2 (e ′ λ+1 ) together with the Frobenius relation and our knowledge of the multiplicative structure of the middle level from subsection 7.4.The alternative is to observe that in the spectral sequence, if a, b live in filtrations ≥ n then so does ab.Before the modifications to the generators done in the proof of Proposition 7.18, e u /u i σ , e a were in filtration ≥ 1 and e λ /u i σ were in filtration ≥ 2. Thus, with the original generators, the extension for e λ e u /u 2 σ does not involve the filtration 0 term a 2 λ θ/a σ .This is true even after performing the modifications prescribed in the proof of Proposition 7.18, since said modifications never involve terms with θ.Thus ǫ 0 = 0.
Similarly we have: for i ≥ 3, and mapping to homotopy fixed points and multiplying by a 2 σ shows e a e u u i Multiplying by a σ and using that a σ (e u /u i σ ) = Tr 4 2 (e au ū−i σ ) shows that ǫ 1 = 0. To show ǫ 0 = 0 we use the filtration argument above.
These arguments also work with: to complete the proof.
We also have the nontrivial Bockstein: Appendix A. Pictures of the spectral sequence In this appendix, we have included pictures of the E 1 page of the spectral sequence from section 7.In each page, the three levels of the spectral sequence are drawn in three separate figures from top to bottom, using (V, s) coordinates.For notational simplicity and due to limited space, we suppress the e V 's and x, gx's from the generators.The e V 's can be recovered by looking at the filtration s (e.g. in filtration s = 4j we get e jρ ) and to denote the presence of 2-dimensional vector spaces k{x, gx} we write k 2 next to each generator.
For example, in the very first picture there is an element x 0,1 /u 2 σ in coordinates (5, 5).This represents the fact that the top level of E 5,5 1 is generated by (x 0,1 /u 2 σ )e ρ+σ .In the picture directly below and in the same coordinates we have v ū−2 σ meaning that the middle level of E 5,5 1 is generated by (v ū−2 σ ) ēρ+σ .We have In the same picture, if we look at coordinates (2, 2) we see v, vk 2 , ū−1 λ k 2 in the top, middle and bottom levels respectively.This represents that the three levels of E 2,2 1 are generated by ve λ (x + gx) for the top, v ēλ x, v ēλ gx for the middle, and ū−1 In this appendix, we write down the detailed computation of k ⋆ for ⋆ ∈ RO(C 4 ).We use the following notation for Mackey functors (compare with [Geo19]).
Henceforth n, m ≥ 0. We employ the notation a|b|c to denote the generators of all three levels of a Mackey functor, from top to bottom, used in [Geo19].In this subsection, we investigate the subtleties regarding quotients y/x, similar to what we did in [Geo19] for the integer coefficient case.The crux of the matter is as follows: If we have ax = y in k C 4 ⋆ then we can immediately conclude that a = y/x as long as a is the unique element in its RO(C 4 ) degree satisfying ax = y.Unfortunately, as we can see from the detailed description of k C 4 ⋆ , there are degrees ⋆ for which k C 4 ⋆ is a two dimensional vector space, generated by elements a, b both satisfying ax = bx = y; in this case a, b are both candidates for y/x and we need to distinguish them somehow.This is done by looking at the products of a, b with other Euler/orientation classes.
For a concrete example, take k C 4 −2+4σ−λ which is k 2 with generators a, b such that u σ a = u σ b = u λ u 3 σ so both a, b are candidates for u λ /u 4 σ (for degree reasons, there is a unique choice for u λ /u 3 σ ).To distinguish a, b, we use multiplication by a 2 σ : for one generator, say a, we have a 2 σ a = 0 while for the other generator we get a 2 σ b = θa λ .So now a 2 σ (a + b) = a 2 σ b = θa λ and both a + b, b are candidates for (θa λ )/a 2 σ .However, θ/a 2 σ is defined uniquely and we insist Here, * ≥ 0 is a generic index i.e. the 12 total instances of * can all be different; the important thing is that the * 's are chosen so that these three elements are in the same RO(C 4 ) degree.
We can also distinguish between Tate diagram for C 2 and C 4 3.1.The Tate diagram for C 2 .For X = k and G = C 2 the corners of the Tate square are:

3. 3 .
The Tate diagram for C 4 .Using the notation of the previous subsection, the corners of the Tate square are: k

λ
ēλ gx for the bottom level.We have Tr 4 2 (v ēλ x) = ve λ (x + gx) These pictures are all obtained automatically by the computer program of[Geo19] available here.

σ
by u σ and a σ multiplication, although it's easier to use that only the first of the three elements is a transfer.λ multiplication (which for large enough i annihilates only the second term) and a σ multiplication (which annihilates only the first term).We similarly distinguish + u σ a λ e ρ + a σ a λ e aThe middle level of k ⋆ (B C 4 Σ 2+ ) is generated by the restrictions of e a , e u , e λ , e ρ , which we denote by ēa , ēu , ēλ , ēρ respectively, and two quotients as follows: The cellular decomposition of B C 4 Σ 2 we just established, consists of one cell in every dimension, whereby "cell" we mean a space of the form (C 4 /H) + ∧ S V where H is a subgroup of C 4 and V is a real non-virtual C 4 -representation; let us call this a "type I" decomposition.It is also possible to obtain a decomposition using only "trivial spheres", namely with cells of the form (C 4 /H) + ∧ S n ; we shall refer to this as a "type II" decomposition.A type I decomposition can be used to produce a type II decomposition by using the type II decompositions of each type I cell (C 4 /H) + ∧ S V .This is useful for computer-based calculations, since type II decompositions lead to chain complexes as opposed to spectral sequences (k ⋆ (B C 4 Σ 2+ ) in a finite range (this can be helpful with our spectral sequence calculations: see Remark 7.6).
[Geo19]ltration gives a spectral sequence of k ⋆ modules converging to k ⋆ (B C 4 Σ 2+ ) that we shall analyze in the next section.6.1.A decomposition using trivial spheres.*((C 4 /H) + ∧ S V ) is concentrated in a single degree if and only if V is trivial).Equipped with a type I decomposition, the computer program of[Geo19]can calculate the additive structure of k 7.3.Bottom level computation.We can immediately conclude that the bottom level spectral sequence collapses in E 2 , giving a single k in every RO(C 4 ) degree.Thus there are no extension problems and the C 4 (Weyl group) action is trivial.7.4.Middle level computation.By remark 7.1 we can immediately conclude that the middle level spectral sequence collapses on E 2 = E ∞ .If we have a middle level element α ).•The elements ūi For each j ≥ 0, the element āλ ējρ+σ has two distinct lifts.On E ∞ we have that Res 2 1 ( āλ ējρ+σ ) = 0 and on k ⋆ C 2 (B C 4 Σ 2+ ) only one of the two lifts has trivial restriction.We denote that lift by ẽj,a .C 4 Σ 2+ ), that we denote by e j,au .•The elements ējρ+λ x, ējρ+λ gx don't survive while the elements ējρ+λ (x + gx) do.They lift uniquely to elements in k C 4 Σ 2+ ) that we denote by ējρ+λ .We have the relation v ējρ+λ = 0 • The elements ējρ+λ+1 (x + gx) don't survive while the elements ējρ+λ+1 x = ējρ+λ+1 gx do.They lift uniquely to elements in k ⋆ C 2 (B C 4 Σ 2+ ) that we denote by e -The elements ūλ ējρ+σ lift uniquely to elements ēj,u ink ⋆ C 2 (B C 4 Σ 2+ ).-⋆ C 2 (B ⋆ C 2 (B ′ jρ+λ+1 . Lemma 7.12, we shall see that ẽa + ūσ e ′ λ+1 is the restriction of a top level generator e a , which we denote by ēa .We can replace the generator ẽa by the element ēa and get the relation: (B C 4 Σ 2+ ) found in Proposition 5.1.For our convenience, we shall continue to use the generators ẽj,a , e j,au , e ′ jρ+λ+1 in the following subsections instead of their replacements.In this subsection, we compute the top level of the E ∞ page.From subsection 7.2, we know that (the top level of) the E 2 page is generated by e jρ , αe jρ+σ , ū−i 7.5.Top level differentials.
e jρ+λ+1 and coherence follows from the Frobenius relations.Next, we see directly from the E ∞ page that e jρ+λ is not in the image of the transfer Tr 4 2 .Since Ker(a σ ) = Im(Tr 4 2 ) in k ⋆ C 4 (B C 4 Σ 2+ ), we must have a module extension of the form: σ (B C 4 Σ 2+ ) is generated by e a , e u /u i σ , e λ /u i σ , e ρ .
w]/a 2 σ , |w| = 1 There are two possible choices for w, differing by a σ u −1 σ , but both work equally well for the following arguments.After potentially replacing the generators e a , e u /u i σ , e λ /u i σ with algebra generators in the same degrees of k ⋆ C 4 (B C 4 Σ 2+ ) and satisfying the same already established relations, the localization map k Proposition 7.19.We have the multiplicative relations in k ⋆ C 4 (B C 4 Σ 2+ ): ) 2 = u σ e λ e ρ + a σ e u u σ e ρ + u σ a λ e ρ + a σ a λ e a and * + n is odd and * ≥ 0Q if n − 2m < * < n − 2 and * + n is even and * < 0 Q ♯ if n − 2m < * < n − 2 and * + n is odd and * < 0 L ⊕ k if * = n − 2m and n − 2m ≥ 0 and m ≥ L if * = n − 2mand n − 2m < 0 and m ≥ k if 0 ≤ * < n − 2m