The $ RO (C 4) $ Cohomology of the Infinite Real Projective Space

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1. Introduction 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 C2n , owing to the crucial involvement of C8-equivariant homology in the solution of the Kervaire invariant problem [HHR16]. The case of G = C2 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(C2) 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(C2) homology of C2-surfaces in [Haz19]. The C2-equivariant dual Steenrod algebra (in characteristic 2) was computed in [HK96] and gives rise to a C2-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 F [Wil19] in the 2-homology of C2-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 = C4 has been much less explored and is indeed considerably more complicated. This can already be seen in the homology of a point in integer F coefficients (see [Zeng17] and [Geo19]) and the case of 2 coefficients is not much better (compare subsections 3.1 and 3.2 for the C2 and C4 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 C2 case no longer hold when generalized to C4 (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 ξi of the mod 2 dual Steenrod algebra can be defined through the completed coaction of the dual R ∞ F F Steenrod algebra on the cohomology of BΣ2 = P : H∗(BC2+; 2) = 2[x] and F F F the completed coaction 2[x] (H 2) (H 2)[[x]] is: → ∗ 2i x ∑ x ξi 7→ i ⊗

In the C2-equivariant case, the space replacing BΣ2 is the equivariant classifying Σ R ∞ space BC2 2. This is still P but now equipped with a nontrivial C2 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

2 c = aσc + uσb 2 where aσ, uσ are the C2-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

2i c c 1 + ∑ b τi 7→ ⊗ i ⊗ 2i b ∑ b ξi 7→ i ⊗

The τi, ξi are the C2-equivariant analogues of the Milnor generators, and Hu-Kriz show that they span the genuine dual Steenrod algebra. Σ For C4, the cohomology of BC4 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 BC Σ2. In[Wil19], the 4 F author describes a framework for equivariant total power operations over an H 2 module A equipped with a symmetric multiplication. The total power operation is induced from a map of spectra Σ A At [2] Σ → where ( )t [2] is a variant Tate construction defined in [Wil19]. − Σ In the nonequivariant case, A At [2] induces a map A A ((x)) and the → ∗ → ∗ Dyer-Lashof operations Qi can be obtained as the components of this map: Q(x) = ∑ Qi(x)xi i 2 In the C2 equivariant case, we have a map A⋆ A⋆[c, b±]/(c = aσc + uσb) and we get power operations → Q(x) = ∑ Qiρ(x)bi + ∑ Qiρ+σ(x)cbi i i F 2 Σ When A = H 2, A⋆[c, b±]/(c = aσc + uσb) is the cohomology of BC2 2 localized at the class b. Σ For C4 we would have to use the cohomology of BC4 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]. ⋆ Σ F The computation of H (BC4 2+; 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 = Cp 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 C2 and C4 respectively F using coefficients in the constant Mackey functor 2. In section 4 we define equivariant classifying spaces BGH and briefly explain Σ the elementary computation of the cohomology of BC2 2 (this argument also appears in [Wil19]). 3 ⋆ Σ F In section 5 we present the result of the computation of H (BC4 2+; 2) and F prove that it’s not flat as a Mackey functor module over (H 2)⋆. Sections 6 and Σ 7 contain the proofs of the computation of the cohomology of BC4 2. We have included two appendices in the end; Appendix A contains pictures of ⋆ Σ F the spectral sequence converging to H (BC4 2+; 2) while Appendix B contains ⋆ 0 F a detailed description of H (S ; 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 Σ BC4 2 given an explicit equivariant CW decomposition (such as we discuss in subsection 6.1). This assisted in the discovery of a nontrivial d2 differential in Σ the spectral sequence of BC4 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 ques- tions 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 read- ability of this paper.

2. Conventions and notations F We will use the letter k to denote the ﬁeld 2, the constant Mackey functor F k = 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 C4 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(C4/C4), M(C4/C2), M(C4/e) as the top, middle and bottom levels of the Mackey functor M respectively. The Mackey functor diagram takes the form:

M(C4/C4)

4 4 Res2 Tr2 M = M(C4/C2) C4/C2

2 2 Res1 Tr1

M(C4/e) C4

If X is a G-spectrum then X⋆ denotes the RO(G)-graded G-Mackey functor deﬁned on orbits as H H ⋆ ⋆ H X⋆(G/H) = X = π (S− X)=[S , X] ⋆ ∧ The index ⋆ will always be an element of the real representation ring RO(G).

RO(C4) is spanned by the irreducible representations 1, σ, λ where σ is the 1- dimensional sign representation and λ is the 2-dimensional representation given by rotation by π/2. 4 C4 For V = σ or V = λ, denote by aV k V the Euler class induced by the ∈ − V C4 ֒ 0 inclusion of north and south poles S S ; also denote by uV k V V the → ∈ | |− orientation class generating the Mackey functor k V V = k. | |− We will use the notation a¯V, u¯V to denote the restrictions of aV, uV to middle level, and u¯V to denote the restriction of uV to bottom level. We also write a kC2 and u kC2 for the C Euler and orientation σ2 σ2 σ2 1 σ 2 ∈ − ∈ − 2 classes, where σ2 is the sign representation of C2. The Gold Relation ([HHR16]) takes the following form in k coefﬁcients:

2 aσuλ = 0 Let EG be a contractible free G-space and EG˜ be the coﬁber of the collapse map 0 h EG+ S . We use the notation X = EG+ X, X˜ = EG˜ X, X = F(EG+, X) → h ∧ ∧ and Xt = Xh. The Tatef diagram ([GM95]) then takes the form:

Xh X X˜

≃ h t Xh X X The square on the right is a homotopy pullback diagram and is called the Tate square. G Applying π⋆ on the Tate diagram gives

G G XhG⋆ X⋆ X˜ ⋆

≃ hG tG XhG⋆ X⋆ X⋆

3. The Tate diagram for C2 and C4

3.1. The Tate diagram for C2. For X = k and G = C2 the corners of the Tate square are:

C2 θσ2 k⋆ = k[aσ2 , uσ2 ] k i,j 0 ⊕ { i j } ≥ aσ2 uσ2 ˜C2 = [ ] k⋆ k aσ±2 , uσ2

hC2 k⋆ = k[aσ2 , uσ±2 ]

tC2 k⋆ = k[aσ±2 , uσ±2 ] where θ = Tr2(Res2(u ) 2). The map k k in the Tate diagram induces σ2 1 1 σ2 − h → 1 tC2 hC2 C2 k ⋆ = Σ− k /k k hC2 ⋆ ⋆ → ⋆ i j θσ2 a u− σ−2 σ2 j 1 7→ i − aσ2 uσ2 5 3.2. The RO(C4) homology of a point. The RO(C4) homology of a point (in k coefficients) is significantly more complicated than the RO(C2) 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: 2 θ a1+ǫ θ a1+ǫ C4 . uλ aσ θ aλ σ aλ σ k⋆ = k[aσ, uσ, aλ, uλ, , ] k[a±] k k[uσ±] (1) ui ai ⊕ λ { i j } ⊕ { i j } ⊕ { j 1+m } σ λ aσuσ uσaλ aλuλ 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 4 2 definition of θ: θ = Tr2(u¯σ− ). We further introduce the elements 4 n m x0,1 x = Tr (u¯− u¯− ) = , m 1 n,m 2 σ λ n m 1 ≥ uσuλ − where 2 2 θ aσ x0,1 = aσ = θ aλ aλ With this notation, the second curly bracket in (1) contains elements of the form xn,1 xn,1 i , i aλ aσaλ and the third contains x x n,m n,m > i , i , m 1 aλ aσaλ The behavior of the xn,m depends crucially on whether m = 1 or not: xn,1uσ = 0 but x u = 0 for m > 1; the x are infinitely a divisible since: n,m σ 6 n,1 σ xn,1 θ 2 = n aσ uσaλ while the xn,m, m > 1, can only be divided by aσ 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 θ aσ aλ x0,2uσ s := uσ = uλ aσ which is the mod 2 reduction of the element s from [Geo19]. Note that suλ = saλ = 0. The quotients in the RHS of (1) are all chosen coherently (cf [Geo19]), that is we always have the cancellation property: y y z = · xz x We also have that x z xz = y · w yw as long as xz = 0 (this condition is necessary: (θ/a )a = 0 is not (θa )/a as 6 λ σ 6 σ λ θaσ = 0). 6 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. • If neither element involves θ then perform all possible cancellations and use • the relation u a2 λ σ = 0 ui · j σ aλ If only one element involves θ perform all possible cancellations and use • x z xz = y · w yw as long as xz appears in (1). If the resulting element appears in (1) then that’s the product; if not then the product is 0. These are all the remarks needed to properly interpret the formula in (1) for the C4 top level k⋆ . The middle level is:

C2 . v k⋆ = k[a¯λ, u¯λ, √a¯λu¯λ, u¯σ±] k[u¯σ±] (2) ⊕ { i j ǫ } a¯λu¯λ√a¯λu¯λ

Here, √a¯λu¯λ is the (unique) element whose square is a¯λu¯λ and v is deﬁned by 2 1 v = Tr1(u¯λ− ). Furthermore, 4 aσuλ Tr2(√a¯λu¯λ) = uσ 4 Tr2(v) = x0,1 4 v s¯ := Res2(s) = √a¯λu¯λ 2 4 aσ 2 Res2 = vu¯σ aλ The interpretation of (2) is complete. In terms of the notation of the C2 generators, 2 2 √ a¯λ = aσ2 , u¯λ = uσ2 , a¯λu¯λ = aσ2 uσ2 , v = θσ2 Finally the bottom level is very simple: e k⋆ = k[u¯λ±, u¯σ±] For more details, consult the Appendix B.

3.3. The Tate diagram for C4. Using the notation of the previous subsection, the corners of the Tate square are: 2 θ a1+ǫ θ a1+ǫ C4 . uλ aσ θ aλ σ aλ σ k⋆ = k[aσ, uσ, aλ, uλ, , ] k[a±] k k[uσ±] ui ai ⊕ λ { i j } ⊕ { i j } ⊕ { j 1+m } σ λ aσuσ uσaλ aλuλ u θ ˜C4 1 C4 . λ k⋆ = a− k⋆ = k[aσ, uσ, a±, uλ, ] k[a±] λ λ i λ i j uσ ⊕ { aσuσ } hC4 2 k⋆ = k[aσ, uσ±, aλ, uλ±]/aσ

tC4 2 k⋆ = k[aσ, uσ±, aλ±, uλ±]/aσ 7 The map k k in the Tate diagram induces h → 1 tC4 hC4 C4 k ⋆ = Σ− k /k k hC4 ⋆ ⋆ → ⋆ θ aσ i j m aλ uσ− a− u− λ λ 7→ i 1 j 1 m uσ− aλ− uλ 4. Equivariant classifying spaces Γ EGK is a G K space that’s K-free and (EGK) is contractible for any subgroup Γ G K with× Γ ( 1 K) = 1 (a graph subgroup). The spaces E Σ are ⊆ × ∩ { } × { } G n those appearing in a G-E∞-operad. We deﬁne the equivariant classifying space

BGK = EGK/K Σ 4.1. The case of C2. For G = C2, the spaces BC2 2 are used in the computation of the C2 dual Steenrod algebra by Hu-Kriz ([HK96]) and for the construction of the total C2-Dyer-Lashof operations in [Wil19]. Both use the computation ⋆ ⋆ 2 k (B Σ ) = k [c, b]/(c = aσ c + uσ b) C2 C2 2+ C2 2 2 where c, b are classes in cohomological degrees σ2,1 + σ2 respectively. Let us note Σ R ∞ here that BC2 2 is P with a nontrivial C2 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 = C4 which takes place in sections 5-7. Let σ, τ be the sign representations of C2, Σ2 respectively and ρ = 1 + σ. Then Σ ∞ Σ ∆ EC2 2 = S( (ρ τ)); the graph subgroups of C2 2 are C2, and their orbits correspond to the⊗ cells × C Σ 2 × 2 = S(1 τ) C2 ⊗ C Σ 2 × 2 = S(σ τ) ∆ ⊗ Σ [Wil19] deﬁnes a ﬁltration on EC2 2 given by S(1 τ) S(ρ τ) S((ρ + 1) τ) S(2ρ τ) ⊗ ⊆ ⊗ ⊆ ⊗ ⊆ ⊗ ⊆··· and whose quotients (after adjoining disjoint basepoints) are

C2 Σ2 (j+1)ρ 1 1 gr2j+1 = × S ⊗ − ∆ + ∧ jρ 1 gr = Σ S ⊗ 2j 2+ ∧ Σ Σ Quotiening 2 gives a ﬁltration for BC2 2+ with (j+1)ρ 1 1 gr2j+1 = S ⊗ − jρ 1 gr2j = S ⊗ Applying k⋆ yields a spectral sequence ⋆ ⋆ E1 = k ejρ, ejρ+σ = k (B Σ ) { } ⇒ C2 2+ of modules over the Green functor k⋆. The fact that the differentials are module maps gives E1 = E2 for degree reasons. Furthermore, the vanishing of the RO(C2) 8 homology of a point in a certain range gives E2 = 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: ⋆ ⋆ k (B Σ ) = k ejρ, ejρ+σ C2 2+ { } It’s easier to prove (using the homotopy ﬁxed point spectral sequence) that: h⋆ Σ h⋆ k (BC2 2+) = k [w] where w has cohomological degree 1. The map k kh from section 2 induces → k⋆(B Σ ) kh⋆(B Σ ) C2 2+ → C2 2+ σ which is localization with uσ2 being inverted. Thus we can see that c = e maps ρ 2 to uσ2 w (or aσ2 + uσ2 w), b = e maps to aσ2 w + uσ2 w and conclude that:

⋆ ⋆ 2 k (B Σ ) = k [c, b]/(c = aσ c + uσ b) C2 C2 2+ C2 2 2 ⋆ ⋆ B Σ is a C -H-space so k (B Σ ) is a Hopf algebra (since it is ﬂat over k ). C2 2 2 C2 C2 2+ C2 For degree reasons, we can see that ∆(c) = c 1 + 1 c ⊗ ⊗ ∆(b) = b 1 + 1 b ⊗ ⊗ ǫ(c) = ǫ(b) = 0

(we can add aσ2 to c to force ǫ(c) = 0). The primitive elements are spanned by i c, b2 .

he cohomology of Σ 5. T BC4 2 Σ In the next section we shall construct a cellular decomposition of BC4 2 giv- ⋆ Σ ing rise to a spectral sequence computing k (BC4 2+). Here’s the result of the ⋆ Σ ⋆ computation, describing k (BC4 2+) as a Green functor algebra over k : Proposition 5.1. There exist elements ea, eu, eλ, eρ in degrees σ + λ, σ + λ 2, λ, ρ of ⋆ − k (B Σ ) respectively, such that C4 C4 2+ u λ ⋆ e e k ea, , , eρ C4 i i ⋆ h uσ uσ ii 0 k (BC Σ2+) = ≥ C4 4 S The relation set S consists of two types of relations (we use indices i, j 0): ≥ Module relations: • a2 eu σ = 0 j ui aλ σ θ 2 λ a aσ s a e λ ea + eu = σ i 2 j 1 i 1 j 2 j ui uσ− aλ− uσ− aλ− aλ σ 9 Multiplicative relations: • u u e e uλ λ i j = i+j 2 e uσ uσ uσ − eλ eu u eu = λ a + i j i+j e aλ i+j uσ uσ uσ uσ u a e uλ ρ uλ a e i = i 1 e + aσ i e uσ uσ− uσ eλ eλ u u eλ = λ ρ + λ a + i j i+j+1 e aσ i+j+2 e aλ i+j uσ uσ uσ uσ uσ eλ eu u a = ρ + λ ρ e i i+1 e aσ i+1 e uσ uσ uσ u a 2 λ ρ e ρ ρ a (e ) = uσe e + aσ e + uσaλe + aσaλe uσ ⋆ Σ a u λ ρ The middle level of k (BC4 2+) is generated by the restrictions of e , e , e , e , which we denote by e¯a, e¯u, e¯λ, e¯ρ respectively, and two quotients as follows: u 1 u λ ⋆ √a¯ u¯ e¯ a¯ u¯ e¯ + √a¯ u¯ e¯ k e¯a, e¯u, e¯λ, e¯ρ, λ λ , λ σ− λ λ ⋆ C2 u¯ u¯ k (B Σ ) = h λ λ i C2 C4 2+ 4 Res2(S) 4 4 Here, Res2(S) denotes the relation set obtained by applying the ring homomorphism Res2 on each relation of S. That is, we have the module relations for any i 0: ≥ v v u = λ = i e¯ i e¯ 0 a¯λ a¯λ and the multiplicative relations: u 2 2 λ (e¯ ) = u¯σu¯λe¯ λ u a u e¯ e¯ = u¯λe¯ + a¯λe¯ a u ρ e¯ e¯ = u¯λu¯σe¯ λ 2 1 ρ λ (e¯ ) = u¯λu¯σ− e¯ + a¯λe¯ a λ 1 u ρ e¯ e¯ = u¯σ− e¯ e¯ a 2 λ ρ ρ (e¯ ) = u¯σe¯ e¯ + u¯σa¯λe¯

As for the Mackey functor structure, the Weyl group C4/C2 action on the generators is trivial and we have: Mackey Functor relations: • √a¯ u¯ e¯u eu 4 i λ λ = Tr2 u¯σ− aσ i+1 u¯λ uσ a¯ u¯ 1e¯u + √a¯ u¯ e¯λ eλ 4 i λ σ− λ λ = Tr2 u¯σ− aσ i+1 u¯λ uσ Finally, the bottom level is ⋆ Σ ⋆ 4 u ke (BC4 2+) = ke [Res1(e )] 10 with trivial Weyl group C4 action and Mackey functor relations obtained by applying 4 Res1 to the multiplicative relations of S: 4 λ 2 1 4 u 2 Res1 e = u¯σ− u¯λ− Res1(e ) 4 a 2 2 4 u 3 Res1 e = u¯σ− u¯λ− Res1(e ) 4 ρ 3 3 4 u 4 Res1 e = u¯λ− u¯σ− Res1(e ) 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

⋆′ ⋆ k [c, b] k ′ (B Σ ) = C2 C2 C2 2+ 2 c = aσ2 c + uσ2 b ⋆ Here, ⋆ ranges in RO(C ) and to get k (B Σ ) for ⋆ in RO(C ), we have to ′ 2 C2 C4 2+ 4 restrict to RO(C2) representations of the form n + 2mσ2. In this way,

⋆ ⋆′ k (B Σ ) = k (B Σ )[u¯±] C2 C4 2+ C2 C2 2+ σ where ⋆ = n + mσ + kλ in RO(C4) corresponds to ⋆′ = n + m + 2kσ2 in RO(C2). The correspondence of generators is: a e¯ = u¯σ(aσ2 b + bc) u e¯ = u¯σuσ2 c e¯λ = c2 ρ 2 e¯ = u¯σb 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 ⋆ ⋆ hC4 Σ hC4 k (BC4 2+) = k [w] ⋆ hC ⋆ h so that the localization map k (BC Σ2+) k 4 (BC Σ2+) induced by k k and C4 4 → 4 → inverting uσ, uλ is: eu u u w 7→ σ λ eλ u w2 7→ λ ea u u w3 + u a w 7→ σ λ σ λ eρ u u w4 + a u w3 + u a w2 + a a w 7→ σ λ σ λ σ λ σ λ ⋆ Σ ⋆ Proposition 5.3. The module k (BC4 2+) is not flat over k . . ⋆ ⋆ Σ Σ2σ λ Proof Let R = k and M = k (BC4 2+). Consider the map f : R − R 2 → given on top level by multiplication with aσ/aλ and determined on the lower 2 levels by restricting (so it’s multiplication with vu¯σ on the middle level and 0 on the bottom level). If M is a flat R-module then we have an exact sequence

f 2σ λ 0 M ⊠ Ker( f ) M Σ − M → R → −→ 11 4 4 The restriction functor Res2 from R modules to Res2 R modules is exact and 4 4 4 symmetric monoidal, so we replace M, R, Ker( f ) by Res2 M, Res2 R, Res2 Ker( f ) respectively and have an exact sequence of C2 Mackey functors. Using the notation involving the C2 generators c, b and writing a = aσ2 , u = uσ2 , we have M = 2i 2i+1 2i+1 2i+1 2i 2i i 0R b , cb i 0R ab , ub , acb , ucb / . 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}: ∼

vu¯2 0 (M ⊠ Ker( f ))(C /C ) M(C /C ) σ M(C /C ) → R 2 2 → 2 2 −−→ 2 2 and v acts trivially on ab, ub, ac, uc i.e. on M(C2/C2) so we get

(M ⊠R Ker( f ))(C2/C2) = M(C2/C2)

We compute directly from deﬁnition that (M ⊠R Ker( f ))(C2/C2) is isomorphic v to M(C /C ) I where I := Ker(R R). But M(C /C ) I 2 2 ⊗R(C2/C2) −→ 2 2 ⊗R(C2/C2) → M(C2/C2) has image IM(C2/C2) hence

IM(C2/C2) = M(C2/C2) This contradicts that ab = e′λ+1 is not divisible by any element of the ideal I (e′λ+1 is only divisible by u¯ i R which are not in I). σ± ∈ cellular decomposition of Σ 6. A BC4 2

We denote the generators of C4 and Σ2 by g and h respectively; let also τ be the sign representation of Σ2 and ρ = 1 + σ + λ the regular representation of C4. The graph subgroups of C Σ are C = g , C = g2 , ∆ = gh , ∆ = 4 × 2 4 h i 2 h i h i ′ g2h , e. h Wei thus have a model for the universal space: E Σ = S(∞(ρ τ)) C4 2 ⊗ Σ R ∞ and BC4 2 is P with nontrivial C4 action: g(x , x , x , x ,...)=(x , x , x , x ,...) 1 2 3 4 1 − 2 − 4 3 S(∞(ρ τ)) is the space ⊗ 2 S(∞) = (xn) : finitely supported and ∑ xi = 1 { i } with C Σ action 4 × 2 g(x , x , x , x , x ,...)=(x , x , x , x , x ,...) 1 2 3 4 5 1 − 2 − 4 3 5 h(x , x ,...)=( x , x ,...) 1 2 − 1 − 2 We shall use the notation (x ,..., x ) for the point (x ,..., x ,0,0,...) S(∞). 1 n 1 n ∈ Moreover, the subspace of S(∞)+ where only x1,..., xn are allowed to be nonzero shall be denoted by (x1,..., xn) . We now describe a{ cellular decomposition} of E Σ where the orbits are C C4 2+ 4 × Σ /H SV where V is a C representation. 2+ ∧ 4 Start with (x1) the union of two points (1), ( 1) and the basepoint. This is • C Σ /C{ . } − 4 × 2 4+ 12 (x ) includes in (x , x ) = S(1 + σ)+. The cofiber is the wedge of two • { 1 } { 1 2 } circles, corresponding to x2 being positive or negative, and the action is g(x , +)=(x , ) , h(x , +)=( x , ) 1 1 − 1 − 1 − After applying the self equivalence given by f (x1, +)=(x1, +) and f (x1, ) = ( x , ), the action becomes − − 1 − g(x , +)=( x , ) , h(x , +)=(x , ) 1 − 1 − 1 1 − σ This is exactly C Σ /∆+ S . 4 × 2 ∧ (x1, x2) includes in (x1, x2, x3,0), (x1, x2,0, x4) ; 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 g(x, y)i =(x, y)i+1 , h(x, y)i =( x, y)i+2 − − − Applying the self equivalence f (x, y)1 =(x, y)1 , f (x, y)2 =( y, x)2 , f (x, y)3 =( x, y)3 , f (x, y)4 =(y, x)4 − − − − the action becomes g(x, y)i = ( y, x)i+1 and h(x, y)i = (x, y)i+2 i.e. we have − C Σ /∆ Sλ. 4 × 2 +′ ∧ (x1, x2, x3,0), (x1, x2,0, x4) includes in (x1, x2, x3, x4) = S(ρ τ) and the • { } 3 { } ⊗ cofiber is the wedge of four S ’s corresponding to the signs of x3, x4. Analo- gously to the item above, we get the space C Σ /∆ S1+λ. 4 × 2 +′ ∧ The process now repeats: (x1, x2, x3, x4) includes in (x1, x2, x3, x4, x5) and • { 4 } { } the cofiber is the wedge of two S ’s corresponding to the sign of x5 and we get C Σ /C S1+σ+λ. And so on... 4 × 2 4+ ∧ Σ We get the decomposition of BC4 2+ where the associated graded is: jρ gr4j = S jρ+σ gr4j+1 = S jρ+λ gr4j+2 = Σ C4/C2+ jρ+1+λ gr4j+3 = Σ C4/C2+ ⋆ ⋆ Σ This filtration gives a spectral sequence of k modules converging to k (BC4 2+) that we shall analyze in the next section. Σ 6.1. A decomposition using trivial spheres. The cellular decomposition of BC4 2 we just established, consists of one cell in every dimension, whereby ”cell” we V mean a space of the form (C /H)+ S where H is a subgroup of C and V is a 4 ∧ 4 real non-virtual C4-representation; let us call this a ”type I” decomposition. It is also possible to obtain a decomposition using only ”trivial spheres”, namely with n cells of the form (C4/H)+ S ; we shall refer to this as a ”type II” decomposition. A type I decomposition∧ can be used to produce a type II decomposition by V using the type II decompositions of each type I cell (C4/H)+ S . This is use- ful for computer-based calculations, since type II decompositions∧ lead to chain V complexes as opposed to spectral sequences (k ((C4/H)+ S ) 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 13 ⋆ Σ of k (BC4 2+) in a finite range (this can be helpful with our spectral sequence calculations: see Remark 7.6). Σ We note however that the type II decomposition of BC4 2 obtained this way is rather inefficient and not minimal. For example, consider the subspace spanned by homogeneous coordinates (x1 : x2 : x3 : x4) ; this is obtained from the { } 1+λ subspace (x1 : x2 :0: x4), (x1 : x2 : x3 :0) by attaching a cell (C4/C2)+ S . { 1+λ } 3 ∧ The sphere S itself has a top dimensional cell C4+ S ; combining the two 3 3 ∧ we get (C /C )+ C S = (C C ) S i.e. 2 cells in dimension 3. On 4 2 ∧ 4+ ∧ 4+ ∨ 4+ ∧ the other hand, (x1 : x2 : x3 : x4) can also be obtained from the subspace { } 3 (x1 : x2 :0: x4), (x1 : x2 : x3 : 0), (x1 :0: x3 : x4) by attaching C4+ S i.e. 1{ cell in dimension 3. Working out the lower dimensions,} we can see∧ that the R σ+λ type II decomposition of P = (x1 : x2 : x3 : x4) obtained by expanding the type I decomposition has 12 total{ cells, while it is possi} ble to obtain a more efficient type I decomposition with 9 total cells.

he spectral sequence for Σ 7. T BC4 2 ⋆ Σ Applying k on the ﬁltration of BC4 2+ gives a spectral sequence Es,V = kV gr = kV B Σ 1 s ⇒ C4 2+ The differential dr 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 E1 page, we will need some notation: For a G- Mackey functor M and subgroup H G, MG/H denotes the G-Mackey functor deﬁned on orbits as M (G/K) =⊆M(G/H G/K); the restriction, transfer G/H × and Weyl group action in MG/H are induced from those in M. For G = C4 and

H = C2, the bottom level of MC4/C2 is: M (C /e) = M(C /e C /C ) = M(C /e) M(C /e) = M(C /e) x, y C4/C2 4 4 × 4 2 4 ⊕ 4 4 { } where x, y are used to distinguish the two copies of M(C4/e), i.e. so that any element of M (C /e) can be uniquely written as mx + m y for m, m M(C /e). C4/C2 4 ′ ′ ∈ 4 The Weyl group WC4 e = C4 acts as

g(mx + m′y)=(gm)(gx)+(gm′)(gy)=(gm)y +(gm′)x i.e. y = gx for a ﬁxed generator g C . ∈ 4 We can then describe MC4/C2 in terms of M and the computation of the restriction and transfer on x, which are shown in the following diagram:

M(C /C ) x + gx 4 2 { } x+gx x+gx x x+gx 7→ 7→

MC /C2 = M(C /C ) x, gx C4/C2 4 4 2 { } x x x 0 7→ 7→

M(C /e) x, gx C4 4 { }

14 If M = R is a Green functor, then RC4/C2 is an R-module. Its top level, namely R(C /C ) x + gx , is an R(C /C ) module via extension of scalars along the 4 2 { } 4 4 restriction map Res4 : R(C /C ) R(C /C ). 2 4 4 → 4 2

7.1. The E1 page. The rows in the E1 page are:

⋆,4j ⋆ jρ E1 = k − ⋆,4j+1 ⋆ jρ σ E1 = k − − ⋆,4j+2 ⋆ jρ λ E1 =(k − − )C4/C2 ⋆,4j+3 ⋆ jρ λ 1 E1 =(k − − − )C4/C2 We will write ejρ, ejρ+σ, ejρ+λ, ejρ+λ+1 for the unit elements corresponding to the E1 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 e¯V, e¯V for their restrictions to the middle and bottom levels respectively. In this way:

⋆, ⋆ jρ jρ+σ ⋆ jρ+λ jρ+λ+1 E ∗ = k e , e (k ) e , e 1 { } ⊕ C4/C2 { } ⋆, and the three levels of the Mackey functor E1 ∗, from top to bottom, are: ⋆ ⋆ k ejρ, ejρ+σ k ejρ+λ(x + gx), ejρ+λ+1(x + gx) C4 { } ⊕ C2 { } ⋆ ⋆ k e¯jρ, e¯jρ+σ k e¯jρ+λx, e¯jρ+λgx, e¯jρ+λ+1x, e¯jρ+λ+1gx C2 { } ⊕ C2 { } ⋆ ⋆ k e¯jρ, e¯jρ+σ k e¯jρ+λx, e¯jρ+λgx, e¯jρ+λ+1x, e¯jρ+λ+1gx e { } ⊕ e { } ⋆ ⋆ ⋆ ⋆ For the top level, k is a k module through the restriction Res4 : k k : C2 C4 2 C4 → C2 ⋆ k [u 1] ⋆ C4 σ− kC = 1, √a¯λu¯λ 2 aσ { }

C4 It’s important to note that this is not a cyclic k⋆ module.

At this point, the reader may want to look over pictures of the E1 page that we have included in the Appendix A.

7.2. The d1 differentials. In this subsection, we explain how the d1 differentials on each level are computed. We shall need this crucial remark: Σ Remark 7.1. The restriction of the C4 action on BC4 2 to C2 C4 results in a C2 Σ ⊆ space equivalent to BC2 2. The equivariant cohomology of this space is known from subsection 4.1 and we shall use this result to compute the middle level Σ spectral sequence for BC4 2. Further restricting to the trivial group e C4, 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 1 ∞ and the nontrivial d ’s are k x, gx k x, gx , x x + gx (since k∗(R P ) is k in every nonnegative degree).{ } → { } 7→ 15 The d1’s on middle and top level are computed from the fact that they are k⋆ module maps, hence determined on

jρ jρ+σ i ǫ jρ+λ+ǫ′ e , e , u¯σ− √a¯λu¯λ e (x + gx) for the top level (ǫ, ǫ′ = 0,1), and on e¯jρ, e¯jρ+σ, e¯jρ+λx, e¯jρ+λ+1x ⋆ ⋆ for the middle level. We remark that because k is not a cyclic k module, it C2 C4 does not sufﬁce to compute the top level d1 on ejρ, ejρ+σ, ejρ+λ, ejρ+λ+1. The d1 differentials from row 4j to row 4j + 1 are all determined by the dif- 1 jρ 1 σ jρ+σ 1 σ 1 1 ferential d : ke k − e . Note that k − is generated by 0 uσ− u¯σ− (this notation was deﬁned→ in [Geo19] and expresses the generators of all| three| levels from top to bottom separated by vertical columns). The d1 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 d1 differentials from row 4j + 1 to row 4j + 2 are all determined by d1 : kejρ+σ (kσ λ+1) ejρ+λ. Note that (kσ λ+1) is generated by → − C4/C2 − C4/C2 1 vu¯ (x + gx) vu¯ (x, gx) u¯ u¯− (x, gx) σ | σ | σ λ The differential is trivial on the bottom level, but on middle level the C2 computation gives kσ (B Σ ) = 0 forcing the differential to be nontrivial (the only other C2 C4 2+ σ λ+1,4j+2 2 1 way to kill E1 − (C4/C2) = k is for the d differential from row 4j + 2 to 4j + 3 to be the identity k2 k2 on middle level, which can’t happen as we show in the next paragraph). Thus:→ 1 jρ+σ jρ+λ d (e¯ ) = vu¯σe¯ (x + gx) 1 jρ+σ jρ+λ d (e ) = vu¯σe (x + gx) The d1 differentials from row 4j + 2 to row 4j + 3 are determined by 1 i jρ+λ i jρ+λ+1 d : k u¯− e k u¯− e and C4/C2 σ → C4/C2 σ 1 i jρ+λ i jρ+λ+1 d : k u¯− √a¯ u¯ e k u¯− √a¯ u¯ e C4/C2 σ λ λ → C4/C2 σ λ λ On bottom level, these d1’s all are x x + gx and the commutation with restriction and transfer gives: 7→ d1(e¯jρ+λx) = e¯jρ+λ+1(x + gx) 1 i jρ+λ d (u¯σ− e (x + gx)) = 0 1 i jρ+λ d (u¯σ− √a¯λu¯λe (x + gx)) = 0 Finally, the d1 differentials from row 4j + 3 to row 4j + 4 are determined by 1 i jρ+λ+1 1 σ jρ+ρ d : k u¯− e k − e and C4/C2 σ → 1 i jρ+λ+1 2 σ+λ jρ+ρ d : k u¯− √a¯ u¯ e k − e C4/C2 σ λ λ → 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 E1 page computation. 16 7.3. Bottom level computation. We can immediately conclude that the bottom level spectral sequence collapses in E2, giving a single k in every RO(C4) degree. Thus there are no extension problems and the C4 (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 E2 = E∞. s,V t,V If we have a middle level element α E∞ and E∞ = 0 for t > s then α lifts ⋆ ∈ t,V uniquely to k (BC Σ2+). If on the other hand E∞ = 0 for some t > s, then there C2 4 6 are multiple lifts of α. In that case, we pick the lift for which there are no exotic 2 restrictions (if possible). For example, if Res1(α) = 0 in E∞ and there is a unique 2 lift β of α such that Res1(β) = 0, then we use β as our lift of α. With this in mind, and the computation of the E2 terms, we have: The elements e¯jρ survive the spectral sequence and lift uniquely to elements e¯jρ • ⋆ in k (B Σ ). C2 C4 2+ i jρ+σ jρ+σ The elements u¯σe¯ don’t survive, but every other multiple of e¯ does. • These multiples are generated by: jρ+σ jρ+σ jρ+σ a¯λe¯ , √a¯λu¯λe¯ , u¯λe¯ ⋆ – The elements u¯ e¯jρ+σ lift uniquely to elements e¯j,u in k (B Σ ). λ C2 C4 2+ jρ+σ – For each j 0, the element a¯λe¯ has two distinct lifts. On E∞ we have that ≥ ⋆ Res2(a¯ e¯jρ+σ) = 0 and on k (B Σ ) only one of the two lifts has trivial 1 λ C2 C4 2+ restriction. We denote that lift by e˜j,a. jρ+σ – Similarly, the elements √a¯λu¯λe¯ have trivial restriction on E∞ and unique ⋆ lifts with trivial restriction on k (B Σ ), that we denote by ej,au. C2 C4 2+ The elements e¯jρ+λx, e¯jρ+λgx don’t survive while the elements e¯jρ+λ(x + gx) • ⋆ do. They lift uniquely to elements in k (B Σ ) that we denote by e¯jρ+λ. We C2 C4 2+ have the relation ve¯jρ+λ = 0 The elements e¯jρ+λ+1(x + gx) don’t survive while the elements e¯jρ+λ+1x = • ⋆ e¯jρ+λ+1gx do. They lift uniquely to elements in k (B Σ ) that we denote by C2 C4 2+ e′ jρ+λ+1. Remark 7.2. We should explain the notation used for the generators above. First, the elements e¯j,u, e¯jρ+λ will turn out to be the restrictions of top level ele- j,u jρ+λ ⋆ ments e , e respectively, both in E∞ and in k (B Σ +), hence their notation. C4 C4 2 j,au jρ+λ+1 Second, the elements e , e′ are never restrictions, neither in E∞ nor in ⋆ k (B Σ ), so their notation is rather ad-hoc: the au in ej,au serves as a re- C4 C4 2+ j,au jρ+σ jρ+λ+1 minder of the √a¯λu¯λ in e = √a¯λu¯λe¯ , while the prime ′ in e′ is used to distinguish them from the top level generators ejρ+λ+1 that the e′ jρ+λ+1 transfer to. j,a j,a Finally, the elements e˜ are restrictions of top level elements e in E∞, but not ⋆ in k (B Σ ) due to nontrivial Mackey functor extensions (exotic restrictions). C4 C4 2+ That’s why we denote them by e˜j,a as opposed to e¯j,a; the e¯j,a are reserved for 4 j,a j,a ′ jρ+λ+1 Res2(e ) = e˜ + u¯σe (see Lemma 7.12). 17 For convenience, when j = 0 we write e˜a, eau, e¯u in place of e˜0,a, e0,au, e¯0,u respectively.

⋆ ⋆ jρ jρ +σ Now recall that k (BC Σ2+) = k e 2 , e 2 2 (see subsection 4.1). We shall C2 2 C2 { } write our middle level C4 generators in terms of the C2 generators. Proposition 7.3. We have:

jρ j 2jρ2 e¯ = u¯σe

j,u j+1 2jρ2+σ2 e¯ = u¯σ uσ2 e

j,au j+1 2jρ2+σ2 e = u¯σ aσ2 e

j,a j+1 (2j+1)ρ2 e˜ = u¯σ aσ2 e

jρ+λ j 2jρ2+σ2 j (2j+1)ρ2 e¯ = u¯σaσ2 e + u¯σuσ2 e

′ jρ+λ+1 j (2j+1)ρ2+σ2 e = u¯σe . Σ Σ Proof The map f : EC4 2 EC2 2, f (x1, x2, x3, x4,...)=(x1, x3, x2, x4,...) is a C Σ equivariant homeomorphism→ and induces a map on ﬁltrations: 2 × 2 S(1) S(1 + σ) X S(ρ) ···

S(1) S(ρ ) S(1 + ρ ) S(2ρ ) 2 2 2 ··· 1 (the downwards arrows are f while the arrows in the opposite direction are f − ). To keep the notation tidy, we verify the correspondence of generators for j = 0. a 1 ′λ+1 In the C4 spectral sequence, we have e˜ u¯σ− , e in degree λ + 1 and filtrations σ2+1 2σ2+1 1, 3 respectively. In the C2 spectral sequence, we have aσ2 e and e in the same degree and filtrations 2, 3 respectively. The correspondence of filtrations gives:

e′λ+1 = e2σ2+1

a 1 σ2+1 2σ2+1 e˜ u¯σ− = aσ2 e + ǫe where ǫ = 0, 1. Applying restriction on the second equation reveals that ǫ = 0 a 1 σ2+1 and thus e˜ u¯σ− = aσ2 e . The correspondence of ﬁltrations in degrees λ 1, λ gives: −

u 1 σ2 e¯ u¯σ− = ǫ1aσ2 uσ2 + uσ2 e au 1 σ2 σ2+1 e u¯σ− = ǫ2aσ2 e + ǫ3uσ2 e

λ σ2 σ2+1 e¯ = ǫ4aσ2 e + ǫ5uσ2 e where ǫi = 0,1. Looking at degree 2λ + σ 2 in the C4 spectral sequence we see that we have a relation −

u a λ ′λ+1 a¯λe¯ = u¯λe˜ + ǫ6√a¯λu¯λu¯σe¯ + ǫ7u¯σu¯λe where again ǫi = 0, 1. Combining the equations above we conclude that u a λ a¯λe¯ = u¯λe˜ + √a¯λu¯λu¯σe¯ 18 and

u 1 σ2 e¯ u¯σ− = uσ2 e

au 1 σ2 e u¯σ− = aσ2 e

λ σ2 σ2+1 e¯ = aσ2 e + uσ2 e

As a corollary we obtain the relations: j,a j,u jρ+λ u¯λe˜ = a¯λe¯ + √a¯λu¯λu¯σe¯ j,au j,u u¯λe = √a¯λu¯λe¯ j,au j,u √a¯λu¯λe = a¯λe¯ j,au j,a jρ+λ a¯λe = √a¯λu¯λe˜ + a¯λu¯σe¯ v jρ+λ = i e¯ 0 a¯λ ⋆ ⋆ Therefore, k (B Σ ) is spanned as a k module by e¯jρ, e˜j,a, ej,au, e¯j,u, e¯jρ+λ, e′ jρ+λ+1 C2 C4 2+ C2 ⋆ Σ under the relations above. The bottom level ke (BC4 2+) is free on the restrictions of e¯jρ, e¯j,u, e¯jρ+λ, e′ jρ+λ+1.

The C4/C2 (Weyl group) action is trivial: The only extension extensions that j,a j,a jρ+λ+1 j,au j,au jρ+λ may arise are ge˜ = e˜ + ǫe′ and ge = e + ǫ′e¯ where ǫ, ǫ′ = 0,1; applying restriction shows that ǫ = ǫ′ = 0.

The cup product structure can be understood in terms of the C2 generators c, b ⋆ of subsection 4.1. As an algebra, k (B Σ ) is generated by e˜a, eau, e¯u, e¯λ, e′λ+1, e¯ρ C2 C4 2+ under multiplicative relations that are implied by the correspondence of generators: ρ 2 e = u¯σb a e˜ = u¯σaσ2 b au e = u¯σaσ2 c u e¯ = u¯σuσ2 c λ 2 e¯ = c = aσ2 c + uσ2 b e′λ+1 = cb Remark 7.4. The reader may notice that this description of the middle level ⋆ k (B Σ ) is rather different from the one given in Proposition 5.1. Let us C2 C4 2+ now explain this discrepancy. First, the relation au u u¯λe = √a¯λu¯λe¯ allows us to replace eau by the quotient u √a¯λu¯λe¯ u¯λ au u which is why e does not appear in Proposition 5.1 but (√a¯λu¯λe¯ )/u¯λ does. a λ+1 Second, in Lemma 7.12, we shall see that e˜ + u¯σe′ is the restriction of a top 19 level generator ea, which we denote by e¯a. We can replace the generator e˜a by the element e¯a and get the relation:

′λ+1 a u λ u¯σu¯λe = u¯λe¯ + a¯λe¯ + √a¯λu¯λu¯σe¯

Thus we can replace the generator e′λ+1 by the quotient 1 u λ u¯σ− a¯λe¯ + √a¯λu¯λe¯ u¯λ ⋆ which is what we do in the description of the middle level k (B Σ ) found C2 C4 2+ in Proposition 5.1. For our convenience, we shall continue to use the generators e˜j,a, ej,au, e′ jρ+λ+1 in the following subsections instead of their replacements. 7.5. Top level differentials. In this subsection, we compute the top level of the E∞ page. From subsection 7.2, we know that (the top level of) the E2 page is generated by jρ jρ+σ i ǫ jρ+λ+ǫ′ e , αe , u¯σ− √a¯λu¯λ e where i, j 0, ǫ, ǫ = 0,1 and α kC4 um , m 0 . We also have the relation: ≥ ′ ∈ ⋆ \ { σ ≥ } jρ+λ vu¯σe = 0 For degree reasons, the elements ejρ survive the spectral sequence. i jρ+λ+1 i jρ+λ+1 The elements u¯σ− e , u¯σ− √a¯λu¯λe are transfers hence also survive (by the middle level computation of subsection 7.4). If α kC4 is a transfer then so are the elements αejρ+σ and thus they sur- ∈ ⋆ ⋆ m vive. The remaining elements α k uσ , m 0 can be broken into three ∈ C4 \ { ≥ } categories: multiples of a • λ multiples of u /ui • λ σ a ui • σ σ jρ+σ Proposition 7.5. The elements aλe survive the spectral sequence, while the elements i jρ+σ aσuσe support nontrivial differentials: 2 i jρ+σ i+2 jρ+λ+1 d (aσuσe ) = vu¯σ e for i, j 0. ≥ . jρ+σ 3 jρ+σ (j+1)ρ Proof The elements aλe can only support d (aλe ) = e and applying restriction shows that this cannot happen. jρ+σ Fix j 0. For degree reasons, the only differential aσe can support is 2 jρ+≥σ 2 jρ+λ+1 jρ+σ d (aσe ) = vu¯σe . If aσe survives then it lifts to a unique element α of jρ+2σ jρ+σ jρ+2σ 4 jρ+λ+1 k , while a e has two possible lifts to k that differ by Tr (uσe′ ). C4 λ C4 2 Both lifts have the same restriction, which by Lemma 7.12 is computed to be j,a jρ+λ+1 e˜ + u¯σe′ (the proof of the Lemma works regardless of the survival of jρ+σ aσe ). Now one of those lifts, that we shall call β, satisﬁes: 2 aσ β = aσα aλ 20 jρ+3σ in k (B Σ ). Applying Res4 gives that C4 C4 2+ 2 2 4 aσ 4 2 j,a ′ jρ+λ+1 3 ′ jρ+λ+1 Res2 Res2(β) = 0 = vu¯σ(e˜ + u¯σe ) = 0 = vu¯σe = 0 aλ ⇒ ⇒ which contradicts the computation of the module structure of the middle level.

σ 2σ Remark 7.6. The non-survival of aσe is consistent with the computation that k C4 2 0 has dimension 1 (spanned by aσe ) by the computer program of [Geo19].

All the other elements of E2 survive the spectral sequence: i jρ+σ i jρ+λ i jρ+λ Proposition 7.7. The elements (uλ/uσ)e , u¯σ− e , u¯σ− √a¯λu¯λe survive the spectral sequence for i, j 0. ≥ Proof. We work page by page. On E2 we have: θ 2( i jρ+λ) = (j+1)ρ d u¯σ− e ǫ1 i 2 e aσuσ− θ u d2(u¯ i√a¯ u¯ ejρ+λ) = ǫ a e(j+1)ρ + ǫ λ e(j+1)ρ σ− λ λ 2 2 i 3 λ 3 i+1 aσuσ− uσ i jρ+λ where ǫi = 0, 1. Multiplying by aσ and using that aσu¯σ− e = 0 and that i jρ+λ aσu¯σ− √a¯λu¯λe = 0 shows that ǫ1 = ǫ2 = ǫ3 = 0. On E3 we have: θ d3(u¯ iejρ+λ) = ǫ e(j+1)ρ+σ σ− 1 2 i 2 aσuσ− θ d3(u¯ i√a¯ u¯ ejρ+λ) = ǫ a e(j+1)ρ+σ σ− λ λ 2 3 i 3 λ aσuσ− u θ d3 λ ejρ+σ = ǫ e(j+1)ρ i 3 2 i 4 uσ aσuσ− where again ǫi = 0, 1. We see that ǫ1 = ǫ2 = 0 by multiplication with aσ, while 2 ǫ3 = 0 can be seen by multiplying with aσ. The pattern of higher differentials is the same as in E2, E3 and the same arguments show that there are no higher differentials. In conclusion: ⋆ Corollary 7.8. The E∞ page is generated as a k module by C4

jρ jρ+σ i jρ+σ i ǫ jρ+λ+ǫ′ e , aλe , (uλ/uσ)e , u¯σ− √a¯λu¯λ e where i, j 0 and ǫ, ǫ = 0,1. We have relations: ≥ ′ jρ+λ 2 jρ+λ+1 vu¯σe = vu¯σe = 0 s,V t,V 7.6. Coherent lifts. If we have a top level element α E∞ and E∞ = 0 for t > s ⋆ ∈ t,V then α lifts uniquely to k (BC Σ2+). If on the other hand E∞ = 0 for some t > s, C4 4 6 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 σ ”coherent”. Let us explain what that means with an example: The element uλe has a unique lift x while (u /ui )eσ has multiple distinct lifts if i 5. If we 0 λ σ ≥ 21 i σ i choose xi to lift (uλ/uσ)e then it will always be true that uσxi = x0; however, we i shouldn’t write xi = x0/uσ unless we can also guarantee that:

uσxi = xi 1 − This expresses the coherence of fractions (also discussed in subsection 3.2 and Appendix B) which is the cancellation property: u u λ σ = λ σ uσ i e i 1 e uσ uσ− ⋆ This holds on E∞ and we also want it to hold on k (B Σ ). C4 C4 2+ i σ 2 i σ One more property enjoyed by the (uλ/uσ)e is that aσ(uλ/uσ)e = 0; it turns i σ 2 out that there are unique lifts xi of (uλ/uσ)e such that aσxi = 0 and those lifts also satisfy the coherence property uσxi = xi 1: − j,u i jρ+λ i Proposition 7.9. For i, j 0, there are unique lifts e /uσ, e /uσ of the elements i jρ+σ i jρ+λ ≥ (uλ/uσ)e , u¯σ− e respectively that satisfy: ej,u 2 = aσ i 0 uσ ejρ+λ 2 = aσ i 0 uσ These lifts are also coherent. . i jρ+σ Proof Fix i, j 0. We first deal with lifts of (uλ/uσ)e . ≥ i jρ+σ s Existence: Fix ⋆ to be the RO(C4) degree of (uλ/uσ)e and write F for the ⋆ decreasing filtration on k (B Σ ) defining the spectral sequence, namely: C4 C4 2+

s,⋆ s s+1 E∞ = F /F

4j+1 i jρ+σ 2 We start with any random lift α0 F of (uλ/uσ)e ; if aσα0 = 0 then we are ∈ 2 s0 2 i jρ+σ done. Otherwise take s0 maximal with aσα0 F ; since aσ(uλ/uσ)e = 0 we 4j+∈2,⋆ have s0 > 4j + 1. In fact s0 > 4j + 2 since E = 0. 4j+3,⋆ 3 i jρ+λ+1 We now prove that s0 > 4j + 3: E is spanned by u¯σ− e so we need 2 3 i jρ+λ+1 4j+3,⋆ i to investigate the possibility aσα = u¯σ− e on E . Multiplying by uσ i jρ+σ reduces us to the case i = 0, where uσα is the unique lift of uλe . But we can 2 i 2 i see directly that (aσ/aλ)uσα = 0 for degree reasons, hence aσuσα = 0 as well. s s +1 s0,⋆ As s0 > 4j + 3, we can see directly that F 0 /F 0 = E∞ is generated by an V ⋆ V 2 s 2 V element βe where β k − is divisible by a . If α F 0 is a lift of (β/a )e C4 σ ′ σ ∈ i jρ+σ 2 ∈ then α1 = α0 + α′ is a lift of (uλ/uσ)e . If aσα1 = 0 then we are done, otherwise 2 s1 s aσα1 F for s1 > s0 so we get α2 by the same argument as above. Since F = 0 for large∈ enough s, this inductive process will eventually end with the desired lift. i jρ+σ Uniqueness: If α, α′ are two lifts of (uλ/uσ)e then their difference is a ⋆ ﬁnite sum p of elements β eV where each β k is a fraction with a2 in its ′ ′ C4 σ 2 2 2 ∈ 2 denominator. If aσα = aσα′ = 0 then aσ p = 0 = aσβ′ = 0 = β′ = 0 = p = 0. ⇒ ⇒ ⇒ 22 i jρ+σ 2 Coherence: Unﬁx i and let xi be the lift of (uλ/uσ)e with aσxi = 0. Then i 1 jρ+σ 2 uσxi is a lift of (uλ/uσ− )e and aσ(uσxi) = 0 hence by uniqueness:

uσxi = xi 1 − i jρ+λ i jρ+σ The case of u¯σ− e is near identical to what we did above for (uλ/uσ)e . The changes are as follows. First, s0 > 4j + 2 (instead of s0 > 4j + 1). Next we can see that s0 > 4j + 4 if i > 1, and multiplying by uσ also proves the i = 0,1 cases (this replaces the argument that showed s0 > 4j + 3). The rest of the arguments are identical. ⋆ 7.7. Top level generators. The elements ejρ have unique lifts to k (B Σ ) that C4 C4 2+ we continue to denote by ejρ. On the other hand, for each j 0 there are two possible lifts of a ejρ+σ. There ≥ λ is no good way to make a unique choice at this point, so we shall write ej,a for either. In this subsection we shall prove: ⋆ ⋆ Proposition 7.10. The k module k (B Σ ) is generated by C4 C4 C4 2+ j,u jρ+λ jρ j,a e e e , e , i , i uσ uσ where i, j 0. ≥ ⋆ By Corollary 7.8 it sufﬁces to prove that the k -algebra generated by the ele- C4 jρ j,a j,u i jρ+λ i ments e , e , e /uσ, e /uσ contains lifts of the elements i jρ+λ+1 i jρ+λ i jρ+λ+1 u¯− e , √a¯ u¯ u¯− e , √a¯ u¯ u¯− e E∞ σ λ λ σ λ λ σ ∈ Lemma 7.11. The elements ejρ+λ+1 = 4( i ′ jρ+λ+1) i : Tr2 u¯σ− e uσ i jρ+λ+1 are coherent lifts of u¯ e E∞. Furthermore, σ− ∈ ejρ+λ ejρ+λ+1 = aσ i i 1 uσ uσ−

. 4 i ′ jρ+λ+1 i jρ+λ+1 Proof We see directly that Tr2(u¯σ− e ) lift u¯σ− e and coherence follows from the Frobenius relations. jρ+λ Next, we see directly from the E∞ page that e is not in the image of the 4 4 ⋆ transfer Tr . Since Ker(aσ) = Im(Tr ) in k (B Σ ), we must have a module 2 2 C4 C4 2+ extension of the form: jρ+λ jρ+λ+1 aσe = uσe 2 jρ+λ i jρ+λ i By Proposition 7.9, aσe /uσ = 0 hence aσe /uσ is a transfer. The equation above shows that a ejρ+λ/ui = 0 and the only way a ejρ+λ/ui can be a nonzero σ σ 6 σ σ jρ+λ i 4 i+1 ′ jρ+λ+1 transfer is for aσe /uσ = Tr2(u¯σ− e ).

Before we can lift the rest of the E∞ generators, we will need the following exotic restriction: 23 Lemma 7.12. Both choices of ej,a have the same (exotic) restriction:

4 j,a j,a ′ jρ+λ+1 Res2(e ) = e˜ + u¯σe

. j,a jρ+λ+1 4 ′ jρ+λ+1 Proof The two choices of e differ by uσe = Tr2(u¯σe ) hence have the same restriction. From the E∞ page:

4 j,a j,a ′ jρ+λ+1 Res2(e ) = e˜ + ǫu¯σe where ǫ = 0, 1. Transferring this gives 4 j,a jρ+λ+1 Tr2(e˜ ) = ǫuσe Now transferring the middle level relation j,a j,u jρ+λ u¯λe˜ = a¯λe¯ + √a¯λu¯λu¯σe¯ shows that 4 j,a jρ+λ uλ Tr2(e˜ ) = aσuλe and thus Tr4(e˜j,a) = 0 which proves ǫ = 1. 2 6 Lemma 7.13. The elements jρ+λ e u ej,u √ = λ j,a + i : i+1 e aλ i+1 uσ uσ uσ i jρ+λ are coherent lifts of u¯ √a¯ u¯ e E∞. σ− λ λ ∈ Proof. Fix i, j 0 and fix ⋆ to be the degree of the element ≥ u ej,u λ j,a + i e aλ i uσ uσ 4j+1,⋆ This element is by definition in filtration 4j + 1, however its projection to E∞ is: u u λ jρ+σ + λ jρ+σ = i aλe aλ i e 0 uσ uσ 4j+2,⋆ so it is actually in filtration 4j + 2. But observe that E∞ is generated by i+1 jρ+λ u¯σ− √a¯λu¯λe so it suffices to check that u ej,u λ j,a + i e aλ i uσ uσ 4j+2,⋆ i is not 0 when projected to E∞ . Multiplying by uσ reduces us to the case i = 0 and then:

4 j,a j,u j,a ′ jρ+λ+1 j,u jρ+λ ′ jρ+λ+1 Res2(uλe + aλe ) = u¯λe˜ + u¯σu¯λe + a¯λe¯ = u¯σ√a¯λu¯λe¯ + u¯σu¯λe using Lemma 7.12 and the middle level computation of subsection 7.4. Projecting 4j+2,⋆ this restriction to E∞ returns u¯ √a¯ u¯ e¯jρ+λ = 0 σ λ λ 6 as desired. jρ+λ i i j,u i Coherence of e√ /uσ follows from the coherence of uλ/uσ and e /uσ. 24 Lemma 7.14. The elements jρ+λ+1 e√ 4 i√ ′ jρ+λ+1 i := Tr2(u¯σ− a¯λu¯λe ) uσ i jρ+λ+1 are coherent lifts of u¯ √a¯ u¯ e E∞. Furthermore, σ− λ λ ∈ jρ+λ e ejρ+λ+1 √ = aσ i i 1 uσ uσ− Proof. The fact that these transfers are lifts follows from the E∞ page; coherence follows from the Frobenius relations. We check the equality directly: jρ+λ e a u a ej,u √ = σ λ j,a + σ = 4( i√ ) j,a + 4( j,au i) = aσ i e aλ Tr2 u¯σ− a¯λu¯λ e aλ Tr2 e u¯σ− uσ uσ uσ 4 i j,a i+1 ′ jρ+λ+1 j,au i = Tr2(u¯σ− √a¯λu¯λe˜ + √a¯λu¯λu¯σ− e + a¯λe u¯σ− ) =

4 i+1 jρ+λ i+1 ′ jρ+λ+1 = Tr2(a¯λu¯σ− e¯ + √a¯λu¯λu¯σ− e ) =

4 i+1 ′ jρ+λ+1 = Tr2(√a¯λu¯λu¯σ− e ) = jρ+λ+ e 1 = √ i 1 uσ− j,au j,a jρ+λ We used the middle level relation a¯λe = √a¯λu¯λe˜ + a¯λu¯σe¯ and the fact i jρ+λ jρ+λ i that u¯σ− e¯ is the restriction of e /uσ which follows from the same fact on E∞.

Lemmas 7.11, 7.13, 7.14 combined with Corollary 7.8 prove Proposition 7.10.

7.8. Mackey functor structure. ⋆ Σ Proposition 7.15. The Mackey functor structure of k (BC4 2+) is determined by: ej,u ejρ+λ 4( jρ) = jρ 4 = j,u i 4 = jρ+λ i Res2 e e¯ , Res2 i e¯ u¯σ− , Res2 i e¯ u¯σ− uσ uσ j,u jρ+λ 4 j,au i e 4 ′ jρ+λ+1 i e Tr2(e u¯σ− ) = aσ i+1 , Tr2 e u¯σ− = aσ i+1 uσ uσ 4 j,a j,a ′ jρ+λ+1 Res2(e ) = e˜ + u¯σe where i, j 0. ≥ . jρ j,u i Proof We can see directly that there are no Mackey functor extensions for e , e /uσ jρ+λ i and e /uσ. The rest were established in the previous two subsections, apart from: j,u 4 j,au i e Tr2(e u¯σ− ) = aσ i+1 uσ 2 j,u i j,u i To see this, recall that aσ(e /uσ) = 0 hence aσ(e /uσ) is a transfer. Moreover, j,u i j,u i aσ(e /uσ) = 0 which is seen on the E∞ page, and the only way that aσ(e /uσ) 6 j,u i 4 j,au i can be a nonzero transfer is for aσ(e /uσ) = Tr2(e u¯σ− ). 25 7.9. Top level module relations. With the exception of relations expressing co- j,u i j,u i 1 jρ+λ i jρ+λ i 1 herence (uσ(e /uσ) = e /uσ− and uσ(e /uσ) = e /uσ− ), the rest of the module relations are: ⋆ ⋆ Proposition 7.16. The k module k (B Σ ) is generated by: C4 C4 C4 2+ j,u jρ+λ jρ j,a e e e , e , i , i uσ uσ under the relations: 2 j,u aσ e m i = 0 aλ uσ θ 2 jρ+λ a aσ s a e λ ej,a + ej,u = σ i 2 m 1 i 1 m 2 am i uσ− aλ − uσ− aλ − λ uσ for i, j, m 0. ≥ Proof. For m > 0, we have the possible extensions: a2 ej,u θ σ = ρ+ m i ∑ ǫ e∗ ∗ aλ uσ ∗ a∗σu∗σaλ∗ ∗ where each denotes a nonnegative index (with different instances of being ∗ ∗ possibly different indices) and each ǫ = 0, 1. Thus, multiplication by aλ 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 ∗ ∗ a2 ej,u σ = m i 0 aλ uσ Similarly, if m > 0, we have the possible extensions: 2 jρ+λ aσ e s jρ+λ θ ρ+ m i = i 2 m 1 e√ + ∑ ǫ e∗ ∗ aλ uσ uσ− a − ∗ a∗σu∗σaλ∗ λ ∗ m and multiplying with aλ reduces us to: ejρ+λ θ 2 = ρ+ aσ i ∑ ǫ m e∗ ∗ uσ ∗ a∗σu∗σa∗− ∗ λ But ejρ+λ ejρ+λ+1 2 = = 4( ′ jρ+λ+1 i+1) = aσ i aσ i 1 aσ Tr2 e u¯σ− 0 uσ uσ− hence ǫ = 0 for all . Thus ∗ ∗ 2 jρ+λ a e s jρ+λ σ = e am i i 2 m 1 √ λ uσ uσ− aλ − and substituting j,u jρ+λ uλ j,a e e√ = e + aλ uσ uσ 2 jρ+λ+1 gives the desired relation. For i = m = 1 we get (aσ/aλ)(e /uσ) = 0 which jρ+λ 2 jρ+λ+1 lifts the E∞ relations vu¯σe = vu¯σe = 0. 26 As special cases we get the relations: ejρ+λ 2 = aσ i 0 uσ a2 ejρ+λ σ = 0 aλ uσ θ aσ a2 ejρ+λ aλ j,a = σ i e i+2 uσ aλ uσ s a2 ejρ+λ ej,u = σ am m+2 u λ aλ σ for i, j, m 0. ≥ 7.10. Top level cup products. ⋆ ⋆ Proposition 7.17. Asak algebra, k (B Σ ) is generated by ea, eu/ui , eλ/ui , eρ. C4 C4 C4 2+ σ σ Proof. First of all, ejρ = (eρ)j since there are no extensions in degree jρ (to see that (eρ)j = 0 apply restriction). Let A be the algebra span of ea, eu/ui , eλ/ui , eρ. 6 σ σ To see that ej,a A observe: ∈ jρ a jρ j,a jρ+λ+1 e e = ǫaσaλe + e + ǫ′uσe and since jρ+λ+1 4 ′ jρ+λ+1 4 jρ ′λ+1 jρ λ+1 e = Tr2(e ) = Tr2(e e ) = e e j,a we get that e A regardless of the status of ǫ, ǫ′. Now suppose∈ by induction that all elements in filtration 4j are in A. We have that: ≤ eu ej,u θ θ jρ = + + ρ + ,a e i i ∑ ǫ e∗ ∑ ǫ′ e∗ uσ ··· uσ ∗ a∗σu∗σaλ∗ ∗ a∗σu∗σaλ∗ ρ ,a where are in filtration < 4j + 1 hence in A. Since e∗ , e∗ A for any 0, ···j,u i ∈ ∗ ≥ we get e /uσ A. This establishes that everything in filtration 4j + 1 is in A. Finally, ∈ ≤ eλ ejρ+λ θ θ jρ = + + ρ + ,a e i i ∑ ǫ e∗ ∑ ǫ′ e∗ uσ ··· uσ ∗ a∗σu∗σaλ∗ ∗ a∗σu∗σaλ∗ jρ+λ i where are in filtration < 4j + 2, so by the same argument e /uσ A as well. This··· completes the induction step. ∈

Inverting uσ, uλ gives ⋆ khC4 [eρ, ea, eu, eλ] modulo relations, which is isomorphic to hC ⋆ 2 k 4 (B Σ ) = k[a , a , u±, u±, w]/a , w = 1 C4 2+ σ λ σ λ σ | | 1 There are two possible choices for w, differing by aσuσ− , but both work equally well for the following arguments. 27 a u i λ i Proposition 7.18. After potentially replacing the generators e , e /uσ, e /uσ with al- ⋆ gebra generators in the same degrees of k (B Σ ) and satisfying the same already C4 C4 2+ established relations, the localization map

⋆ hC4⋆ k (BC Σ2+) k (BC Σ2+) C4 4 → 4 is given by:

eu u u w 7→ σ λ eλ u w2 7→ λ ea u u w3 + u a w 7→ σ λ σ λ eρ u u w4 + a u w3 + u a w2 + a a w 7→ σ λ σ λ σ λ σ λ

Proof. Using the C2 result (see subsection 4.1), we have the correspondence on the middle level generators:

u e¯ u¯σu¯λw • eλ 7→ u¯ w2 • 7→ λ Res4(ea) u¯ (u¯ w3 + a¯ w) • 2 7→ σ λ λ e¯ρ u¯ (a¯ w2 + u¯ w4) • 7→ σ λ λ from which we can deduce that the correspondence on top level is:

u e uσuλw + ǫ1aσuλ • λ 7→ 2 1 e uλw + ǫ2aσuσ− uλw • a 7→ 3 2 e uσuλw + ǫ3aσuλw + uσaλw • eρ 7→ u u w4 + ǫ a u w3 + u a w2 + ǫ a a w • 7→ σ λ 4 σ λ σ λ 5 σ λ where the ǫi range in 0,1. i u i u i+2 We may add ǫ1aσuλ/uσ to e /uσ to force ǫ1 = 0; we may add ǫ2aσe /uσ to λ i λ a e /uσ to force ǫ2 = 0 and we may add ǫ3aσe to e to force ǫ3 = 0. It remains to prove that ǫ4 = ǫ5 = 1. This is a computation based on the Bockstein ⋆ ⋆ homomorphism β : k (X) k +1(X). For X = S0 we have: C4 → C4

β(aσ) = β(aλ) = β(uλ) = 0

β(uσ) = aσ

ρ+1 For X = B Σ we see that β(eρ) = 0 for degree reasons (k (B Σ ) = 0) C4 2+ C4 C4 2+ and in the homotopy fixed points, β(w) = w2, β(w3) = w4. Thus applying β on eρ u u w4 + ǫ a u w3 + u a w2 + ǫ a a w shows that ǫ = ǫ = 1. 7→ σ λ 4 σ λ σ λ 5 σ λ 4 5 28 ⋆ Proposition 7.19. We have the multiplicative relations in k (B Σ ): C4 C4 2+ u u e e uλ λ i j = i+j 2 e uσ uσ uσ − eλ eu u eu = λ a + i j i+j e aλ i+j uσ uσ uσ uσ u a e uλ ρ uλ a e i = i 1 e + aσ i e uσ uσ− uσ eλ eλ u u eλ = λ ρ + λ a + i j i+j+1 e aσ i+j+2 e aλ i+j uσ uσ uσ uσ uσ eλ eu u a = ρ + λ ρ e i i+1 e aσ i+1 e uσ uσ uσ u a 2 λ ρ e ρ ρ a (e ) = uσe e + aσ e + uσaλe + aσaλe uσ Proof. First, eu eu u u u = ǫ a λ + ǫ a λ eu + ǫ λ eλ + i j 0 λ i+j 2 1 σ i+j 2 i+j 2 ··· uσ uσ uσ − uσ uσ − where ǫi = 0,1 and is the sum of elements mapping to 0 in homotopy ··· 2 fixed points, but all having denominator aσ. Mapping to homotopy fixed points 2 shows ǫ0 = ǫ1 = 0 and ǫ2 = 1, while multiplying by aσ trivializes the LHS (by 2 u i aσ(e /uσ) = 0) and thus shows that = 0. The same argument applied to: ···

eλ eu θa2 u eu u u = λ + λ + + λ a + λ λ + i j ǫ0 i+j 2 ǫ1aσaλ i+j ǫ2aλ i+j ǫ3 i+j e ǫ4aσ i+j e uσ uσ aσuσ − uσ uσ uσ uσ ··· shows eλ eu θa2 eu u = λ + + λ a i j ǫ0 i+j 2 aλ i+j i+j e uσ uσ aσuσ − uσ uσ i+j 2 There are two ways to show that ǫ0 = 0: The first is to multiply with aσuσ − λ u 2 λ 4 ′λ+1 and compute aσe (e /uσ) using aσe = Tr2(e ) 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 ≥ u i a generators done in the proof of Proposition 7.18, e /uσ, e were in filtration 1 λ i ≥ and e /uσ were in filtration 2. Thus, with the original generators, the extension λ u 2 ≥ 2 for e e /uσ does not involve the filtration 0 term aλθ/aσ. 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 ǫ0 = 0. Similarly we have:

eu θa2 θa eu u eλ u a = λ + λ a + + λ a + + λ ρ + e i ǫ0 i 4 ǫ1 i 3 e ǫ2aσaλ i ǫ3aσ i e ǫ4aλ i 2 ǫ5 i 1 e uσ uσ− aσuσ− uσ uσ uσ− uσ− ··· 29 for i 3, and mapping to homotopy ﬁxed points and multiplying by a2 shows ≥ σ eu θa2 θa u u a = λ + λ a + λ a + λ ρ e i ǫ0 i 4 ǫ1 i 3 e aσ i e i 1 e uσ uσ− aσuσ− uσ uσ− u i 4 au i Multiplying by aσ and using that aσ(e /uσ) = Tr2(e u¯σ− ) shows that ǫ1 = 0. To show ǫ0 = 0 we use the ﬁltration argument above. These arguments also work with: eλ eλ θa2 θa eu u eλ = λ + λ a + + λ a + + i j ǫ0 i+j 2 ǫ1 i+j 2 e ǫ2aσaλ i+j 2 ǫ3aσ i+j+2 e ǫ4aλ i+j uσ uσ uσ − aσuσ − uσ − uσ uσ u + ǫ λ eρ + 5 i+j+1 ··· uσ λ λ a e θaλ a e aσuλ ρ θa ρ uλ ρ u e i = ǫ6 i 2 e + ǫ7aσaλ i 1 + ǫ8 i+1 e + ǫ9 i 3 e + ǫ10 i+1 e e + uσ aσuσ− uσ− uσ aσaλuσ− uσ ···

u a 2 2 2 a ρ e ρ λ ρ (e ) = ǫ11aσaλ + ǫ12aσaλe + ǫ13uσaλe + ǫ14aσ e + ǫ16uσe e uσ to complete the proof. We also have the nontrivial Bockstein: u λ β(e /uσ) = e Appendix A. Pictures of the spectral sequence

In this appendix, we have included pictures of the E1 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 eV’s and x, gx’s from the generators. The eV’s can be recovered by looking at the filtration s (e.g. in filtration s = 4j we get ejρ) and to denote the presence of 2-dimensional vector spaces k x, gx we write k2 next to each generator. { } 2 For example, in the very first picture there is an element x0,1/uσ in coordi- 5,5 nates (5,5). This represents the fact that the top level of E1 is generated by 2 ρ+σ (x0,1/uσ)e . In the picture directly below and in the same coordinates we have 2 5,5 2 ρ+σ vu¯σ− meaning that the middle level of E1 is generated by (vu¯σ− )e¯ . We have 4 2 ρ+σ x0,1 ρ+σ Tr2(vu¯σ− e¯ ) = 2 e uσ 2 1 2 In the same picture, if we look at coordinates (2,2) we see v, vk , u¯λ− k in the top, middle and bottom levels respectively. This represents that the three levels 2,2 λ λ λ of E1 are generated by ve (x + gx) for the top, ve¯ x, ve¯ gx for the middle, and 1 λ 1 λ u¯λ− e¯ x, u¯λ− e¯ gx for the bottom level. We have 4 λ λ Tr2(ve¯ x) = ve (x + gx) These pictures are all obtained automatically by the computer program of [Geo19] available here.

30 v u¯ 1 1 v u¯ 1 6 a¯λ σ− s¯u¯σ− u¯λ σ−

θ x0,1 x0,1 5 2 aλ aσuσ uσ

x0,1 x0,1 4 aσ uσ

3 v

2 v

1 0

0 1

0 1 2 3 4 5 6

v u¯ 1k2 1 2 v u¯ 1k2 6 a¯λ σ− s¯u¯σ− k u¯λ σ−

2 5 vu¯σ−

1 4 vu¯σ−

3 vk2

2 vk2

1 1 u¯σ−

0 1

0 1 2 3 4 5 6

1 2 2 6 u¯σ− u¯λ− k

2 1 5 u¯σ− u¯λ−

1 2 4 u¯σ− u¯σ−

1 2 3 u¯λ− k

1 2 2 u¯λ− k

1 1 u¯σ−

0 1

0 1 2 3 4 5 6 v v 6 a¯λ s¯ u¯λ

x0,1 x0,1 5 aσ uσ

4 x0,1

3 vu¯σ

2 vu¯σ

1 1

0 uσ aσ

σ 1 σ σ + 1 σ + 2 σ + 3 σ + 4 σ + 5 − v k2 2 v k2 6 a¯λ sk¯ u¯λ

1 5 vu¯σ−

4 v

2 3 vu¯σk

2 2 vu¯σk

1 1

0 u¯σ

σ 1 σ σ + 1 σ + 2 σ + 3 σ + 4 σ + 5 − 2 2 6 u¯λ− k

1 1 5 u¯σ− u¯λ−

1 4 u¯λ−

1 2 3 u¯σu¯λ− k

1 2 2 u¯σu¯λ− k

1 1

0 u¯σ

σ 1 σ σ + 1 σ + 2 σ + 3 σ + 4 σ + 5 − 1 6 vu¯σ−

5 θ

4 0

3 1

2 1

uλ aσuλ 2 1 uσ uσ

u aσuλ a 0 λ uσ λ

λ 2 λ 1 λ λ + 1 λ + 2 λ + 3 λ + 4 − − 1 2 6 vu¯σ− k

2 5 u¯σ−

1 4 u¯σ−

3 1k2

2 1k2

1 1 1 1 u¯λu¯σ− √a¯λu¯λu¯σ− a¯λu¯σ−

0 u¯λ √a¯λu¯λ a¯λ

λ 2 λ 1 λ λ + 1 λ + 2 λ + 3 λ + 4 − − 1 1 2 6 u¯σ− u¯λ− k

2 5 u¯σ−

1 4 u¯σ−

3 1k2

2 1k2

1 1 u¯λu¯σ−

0 u¯λ

λ 2 λ 1 λ λ + 1 λ + 2 λ + 3 λ + 4 − − 6 v

5 0

4 1

3 u¯σ

2 u¯σ

u aσuλ a 1 λ uσ λ

0 uσuλ aσuλ uσaλ aσaλ

ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ + 1 ρ + 2 − − − − 6 vk2

1 5 u¯σ−

4 1

2 3 u¯σk

2 2 u¯σk

1 u¯λ √a¯λu¯λ a¯λ

0 u¯σu¯λ u¯σ√a¯λu¯λ u¯σa¯λ

ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ + 1 ρ + 2 − − − − 1 2 6 u¯λ− k

1 5 u¯σ−

4 1

2 3 u¯σk

2 2 u¯σk

1 u¯λ

0 u¯σu¯λ

ρ 4 ρ 3 ρ 2 ρ 1 ρ ρ + 1 ρ + 2 − − − − vu¯σ vu¯σ vu¯σ 6 a¯λ √a¯λu¯λ u¯λ

5 x0,1

4 0

2 3 vu¯σ

2 2 vu¯σ

1 uσ aσ

2 2 0 uσ aσuσ aσ

2σ 2 2σ 1 2σ 2σ + 1 2σ + 2 2σ + 3 2σ + 4 − − vu¯σ k2 vu¯σ k2 vu¯σ k2 6 a¯λ √a¯λu¯λ u¯λ

5 v

4 vu¯σ

2 2 3 vu¯σk

2 2 2 vu¯σk

1 u¯σ

2 0 u¯σ

2σ 2 2σ 1 2σ 2σ + 1 2σ + 2 2σ + 3 2σ + 4 − − 2 2 6 u¯σu¯λ− k

1 5 u¯λ−

1 4 u¯σu¯λ−

2 1 2 3 u¯σu¯λ− k

2 1 2 2 u¯σu¯λ− k

1 u¯σ

2 0 u¯σ

2σ 2 2σ 1 2σ 2σ + 1 2σ + 2 2σ + 3 2σ + 4 − − F Appendix B. The RO(C4) homology of a point in 2 coefficients

In this appendix, we write down the detailed computation of k⋆ for ⋆ ∈ RO(C4). We use the following notation for Mackey functors (compare with [Geo19]).

k 0 k 0 1 0 k = k = k = k = k − k h i 0 h i k 1 0 1 0 k k 0 0

k k k k 0 1 0 1 0 1 1 0 ♯ L = k p∗L = k Q = k Q = k 0 1 1 0 k k 0 0

k 0 1 0 L♯ = k♭ = k − k 0 1 0 1 k k 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].

nσ+mλ B.1. k∗S . k if = n + 2m ∗  ♯ nσ+mλ Q if n < n + 2m and n is even k (S ) =  ≤ ∗ ∗ − ∗  Q if n + 1 < n + 2m and n is odd ≤ ∗ ∗ − k if 0 < n  h i ≤ ∗  unum u¯nu¯m u¯nu¯m generates k = k • σ λ | σ λ | σ λ n+2m n m i i n m i i ♯ u a − u u¯ a¯ − u¯ 0 generates k = Q for 0 i < m • σ λ λ| σ λ λ| n+2i ≤ n 1 m i i n m i i 1 aσuσ− a − u u¯σa¯ − u¯ − √a¯λu¯λ 0 generates kn+2i 1 = Q for 1 i m, n > 0 • λ λ| λ λ | − ≤ ≤ m i i aσaλ − uλ m i i 1 a¯λ − u¯λ− √a¯λu¯λ 0 generates k2i 1 = Q for 1 i m, n = 0 • uσ | | − ≤ ≤ an iui am 0 0 generates k = Z /2 for 0 i < n • σ− σ λ | | i h i ≤ 36 −nσ−mλ B.2. k∗S . If n, m are not both 0,

L if = n 2m and m = 0 ∗ − − 6  p∗L if = n 2m and n > 1, m = 0  ∗ − − k if = 1 and n = 1, m = 0  nσ mλ  −♯ ∗ − k (S− − ) =  Q if n 2m < < n 1 and + n is odd ∗  − − ∗ − − ∗ Q if n 2m < < n 1 and + n is even − − ∗ − − ∗   k if n 1 < 1 and m = 0 h i − − ≤ ∗ − 6  k if n + 1 < 1 and m = 0 h i − ≤ ∗ − 

4 1 2 1 1 Tr1 n m Tr1 n m n m generates k n 2m = L for m = 0 • u¯σu¯ u¯σu¯ u¯σu¯ − − 6 λ λ λ

θ n ¯ n n 2 u¯σ− u¯σ− generates k n = p∗L for m = 0, n 2 • u − − ≥ σ

1 1 0 u¯σ− u¯σ− generates k 1 = k for n = 1, m = 0 • | | − −

s s¯ 0 generates k = Q♯ for 2 i m n i 2 m i n i 2 m i n 2m+2i 3 • u a − u − u¯ a¯ − u¯ − − − − ≤ ≤ σ λ λ σ λ λ

x v 0,1 0 generates k = Q for 1 i < m n m i i 1 n m i i 1 n 2i • u a − u − u¯ a¯ − u¯ − − − ≤ σ λ λ σ λ λ

x 0,1 0 0 generates k = k for 0 i n 1, m = 0 n i i m 1 i 2 • a − u a − − − h i ≤ ≤ − 6 σ σ λ

θ = < = n i n 2 0 0 generates k i k for 2 i n, m 0 • a − u − − h i ≤ σ σ

mλ−nσ B.3. k∗S . If n, m are both nonzero,

k if 2m n < 2 h i − ∗ ≤ − k if = 2m n 1  ∗ − ≥ −  k k if = 2m n 2 h i ⊕ ∗ − ≤ −  Q if 1 < 2m n and + n is odd  mλ nσ  ♯ − ≤ ∗ − ∗ k (S − ) =  Q if 1 < 2m n and + n is even ∗  − ≤ ∗ − ∗ k Q if n + 1 < 2m n and + n is odd and 2 h i ⊕ − ≤ ∗ − ∗ ∗ ≤ −  ♯  k Q if n + 2 < 2m n and + n is even and 2 h i ⊕ − ≤ ∗ − ∗ ∗ ≤ −  Q if = n and n 2  ∗ − ≥  k if = 1 and n = 1  h i ∗ −  37 m m ¯m uλ u¯λ u¯λ n n n generates the k in k2m n • uσ u¯σ u¯σ −

i m i i m i aλuλ − a¯λu¯λ − ♯ n n 0 generates the Q in k2m n 2i for 0 < i < m • uσ u¯σ − −

a ai um i √a¯ u¯ a¯i u¯m i 1 σ λ λ − λ λ λ λ − − < n+1 n 0 generates the Q in k2m n 2i 1 for 0 i m • u u¯σ − − − ≤ σ

θam λ < n i i 2 0 0 generates the k in k i for 2 i n • a − u − h i − ≤ σ σ m θaλ m n n 2 a¯λ u¯σ− 0 generates k n = Q for n 2 • u − − ≥ σ

1 m 0 u¯σ− a¯λ 0 generates k 1 = k for n = 1 • − h i

nσ−mλ B.4. k∗S . If n, m are both nonzero,

Q♯ if = n 2 and n, m 2 ∗ − ≥  k if = 1 and n = 1, m 2 h i ∗ − ≥   k Q if n 2m < < n 2 and + n is even and 0 h i ⊕ ♯ − ∗ − ∗ ∗ ≥  k Q if n 2m < < n 2 and + n is odd and 0 h i ⊕ − ∗ − ∗ ∗ ≥  Q if n 2m < < n 2 and + n is even and < 0  nσ mλ  ♯ − ∗ − ∗ ∗ k (S − ) =  Q if n 2m < < n 2 and + n is odd and < 0 ∗  − ∗ − ∗ ∗ L k if = n 2m and n 2m 0 and m 2 ⊕h i ∗ − − ≥ ≥  L if = n 2m and n 2m < 0 and m 2  ∗ − − ≥  ♯ >  L if = n 2 and n 1 and m = 1  ♭ ∗ − k if = 1 and n = 1 and m = 1  − ∗ −  k if 0 < n 2m  h i ≤ ∗ −  38 2 n 2 n aσuσ− vu¯σ ♯ m m 1 0 generates kn 2 = Q for n, m 2 • a a¯ − − ≥ λ λ

vu¯σ 0 m 1 0 generates kn 2 = k for n = 1, m 2 • a¯ − − h i ≥ λ

n n x0,2uσ vu¯σ i 1 m i 1 i 1 m i 0 generates the Q in kn 2m+2i 2 for 2 i m 1 • a − u − − a¯ − u¯ − − − ≤ ≤ − λ λ λ λ

n n suσ s¯u¯σ ♯ i 2 m i i 2 m i 0 generates the Q in kn 2m+2i 3 for 2 i m • a − u − a¯ − u¯ − − − ≤ ≤ λ λ λ λ

x un vu¯n 0,2 σ σ ¯n ¯ m m 2 m 1 u¯σu¯λ− generates the L in kn 2m for m 2 • u − u¯ − − ≥ λ λ

a2 un 2 σ σ− n ¯n ¯ 1 ♯ vu¯σ u¯σu¯λ− generates kn 2 = L for n > 1, m = 1 • aλ −

¯ ¯ 1 ♭ 0 vu¯σ u¯σu¯λ− generates k 1 = k for n = m = 1 • − −

i n i aσuσ− m 0 0 generates the k in kn i for 2 < i n • a h i − ≤ λ B.5. Subtleties about quotients. In this subsection, we investigate the subtleties regarding quotients y/x, similar to what we did in [Geo19] for the integer coefficient case. C4 The crux of the matter is as follows: If we have ax = y in k⋆ then we can immediately conclude that a = y/x as long as a is the unique element in its RO(C4) degree satisfying ax = y. Unfortunately, as we can see from the detailed C4 C4 description of k⋆ , there are degrees ⋆ for which k⋆ 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. C4 2 For a concrete example, take k 2+4σ λ which is k with generators a, b such that − − uλ uσa = uσb = 3 uσ 4 so both a, b are candidates for uλ/uσ (for degree reasons, there is a unique choice 3 2 for uλ/uσ). To distinguish a, b, we use multiplication by aσ: for one generator, say 2 2 a, we have aσa = 0 while for the other generator we get aσb = θaλ. So now 2 2 aσ(a + b) = aσb = θaλ 2 2 and both a + b, b are candidates for (θaλ)/aσ. However, θ/aσ is defined uniquely and we insist x y xy = z w zw 39 whenever xy = 0, thus (θa )/a2 is uniquely defined from: 6 λ σ θaλ θ 2 = 2 aλ aσ aσ Multiplying with u returns 0 and as u b = 0, we conclude that σ σ 6 θaλ a + b = 2 aσ 2 2 Since aσa = 0 and aσb = θaλ we conclude that: uλ uλ θaλ a = 4 , b = 4 + 2 uσ uσ aσ More generally, we can use uσ and aσ multiplication to distinguish >0 >0 >0 >0 aλ∗ u∗λ θa∗λ aλ∗ u∗λ θa∗λ , 2 , + 2 u∗σ aσ∗≥ u∗σ u∗σ aσ∗≥ u∗σ 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(C4) degree. We can also distinguish between >0 >0 >0 >0 aσaλ∗ u∗λ θa∗λ aσaλ∗ u∗λ θa∗λ , 2 , + 2 u∗σ aσ∗≥ u∗σ u∗σ aσ∗≥ u∗σ by uσ and aσ multiplication, although it’s easier to use that only the first of the three elements is a transfer. We distinguish 2 2 a∗≥ u x u a∗≥ u x u σ σ∗ , 0,2 σ∗ , σ σ∗ + 0,2 σ∗ aλ∗ aλ∗ uλ∗ aλ∗ aλ∗ uλ∗ i by aλ multiplication (which for large enough i annihilates only the second term) and aσ multiplication (which annihilates only the first term). We similarly distinguish 2 2 a∗≥ u su a∗≥ u su σ σ∗ , σ∗ , σ σ∗ + σ∗ aλ∗ aλ∗ uλ∗ aλ∗ aλ∗ uλ∗ i 2 by aλ and aσ multiplication. References

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Department of Mathematics, University of Chicago E-mail: [email protected] Website: math.uchicago.edu/ nickg ∼