THE SLICE SPECTRAL SEQUENCE OF MACKEY FUNCTORS FOR THE C 4 ANALOG OF REAL K-THEORY

Transcription

1 THE SLICE SPECTRAL SEQUENCE OF MACKEY FUNCTORS FOR THE C ANALOG OF REAL K-THEORY M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL WORK IN PROGRESS Contents List of Tables. General nonsense about equivariant stable homotopy theory. The RO(G)-graded homotopy of H 3 3. The case G = C. Some chain complexes of Mackey functors 3 5. The C -spectrum k H 6. Slices for k H and K H 3 7. Generalities on differentials 7 8. k H as a C -spectrum 9 9. The first differentials over C 3. Higher differentials and exotic Mackey functor extensions References 6 List of Tables Some C -Mackey functors Some C -Mackey functors 3 3 Some elements of filtration in the homotopy of and slice spectral sequence for k H. Some elements of positive filtration in the homotopy of and slice spectral sequence for k H. 5 The purpose of this paper is to describe the slice spectral sequence of a 3- periodic C -spectrum K H (to be defined in 5) related to the C norm MU ((C)) = N MU R of the real cobordism spectrum MU R. Part of this spectral sequence is illustrated in an unpublished poster produced in late 8 and shown at the end of this paper. It shows the spectral sequence convering to the homotopy of the fixed point spectrum K C H. Here we will describe the corresponding spectral sequence of Mackey functors converging to the graded Mackey functor π K H. The C 8 analog Date: September 6,. The authors were supported by DARPA Grant FA and NSF Grants

2 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL of K H is 56-periodic and detects the Kervaire invariant classes θ j. The C analog is the real K-theory spectrum K R. We will rely extensively on the results, methods and terminology of [HHR].. General nonsense about equivariant stable homotopy theory We first discuss some structure on the homotopy groups of a G-spectrum X. For each representation V we get a Mackey functor π G V X = πg Σ V X; we will often suppress G from the notation when it is clear from the context. Its components are the ordinary homotopy groups of various fixed point sets. In [HHR,..5] the group π G k X(G/H) = π k (X H ) (for an integer k) is denoted by π H k X = [Sk, X] H. Here S k has the trivial group action, so an H-equivariant map to X must land in the fixed point spectrum X H. Thus π H k X = π k (X H ), the ordinary kth homotopy group of the ordinary spectrum X H. Since the Weyl group of H acts on X H, this group is a module over it. For a representation V of G, the group π G V X(G/H) = πh V X = [S V, X] H is isomorphic to [S, S V X] H = π (S V X) H. However fixed points do not respect smash products, so we cannot equate this group with π (S V H X H ) = [S V H, X H ] = π V H X H = π G V H X(G/H). Conversely a G-equivariant map S V X represents an element in For K H G we have maps [S V, X] G = π G V X = π G V X(G/G). Tr H K π G V X(G/H) Res H K π (Σ V X) H π G V X(G/K) π (Σ V X) K which we call the fixed point restriction and transfer maps. spectrum, we have the fixed point Frobenius relation When X is a ring () Tr H K(Res H K(a)b) = atr H K(b) for a π X(G/H) and b π X(G/K). In particular this means that () atr H K(b) = when Res H K(a) =. We can also regard X as an H-spectrum for any subgroup H of G; we will not make a notational distinction between these two structures on X. As such it has an RO(H)-graded Mackey functor over H of homotopy groups. For simplicity we assume now that G is abelian. Recall that for a Mackey functor M over H, the abelian group M(H/K) (for a subgroup K of H) is a module over [H/K]. For

3 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 3 the RO(H)-graded abelian group π H X(H/K) for a G-spectrum X, this module structure extends to one over [G/K]. We will define maps relating these Mackey functors over the various subgroups and call them group action restriction and transfer maps, denoted by r H K and th K. The map r G H is induced by the forgetful functor from G-spectra to H-spectra denoted in [HHR,..] by i H ; for trivial H it is denoted by i. Given a representation V of G restricting to W on H G and a G-spectrum X, we have maps of G-spectra (3) S V pinch fold Since for each subgroup L H, the pinch and fold maps induce (G/H) + S V π H W X(H/L) = πh F (SW, X)(H/L) = = π G F (S W, X)(G/L) G + H S W =: S W = π G F ((G/H) + S V, X)(G/L), π G V X t G H r G H π H W X π G V X(G/L) t G H (H/L) πh W X(H/L) π G V X(G/K) π H W X(H/H K) r G H (G/K) where the X on the right is the restriction of the X on the left to an H-spectrum, K G and L H. We can conjugate elements on the right by elements of G with the subgroup H acting trivially. The group action transfer nominally depends on the choice of V that restricts to W, but two such choices V and V lead to canonically isomorphic groups. For L H G we have a diagram π G V X(G/L) π G V X(G/L) t G H (H/L) t G H (H/L) π H W X(H/L) The groups on the left are isomorphic because they depend only on the restrictions of V or V to L. By assumption they have the same restrictions to H.. The RO(G)-graded homotopy of H We describe part of the RO(G)-graded Green functor π (H), where H is the integer Eilenberg-Mac Lane spectrum H in the G-equivariant category, for some

4 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL cyclic -groups G. For each actual (as opposed to virtual) G-representation V we have an equivariant reduced cellular chain complex C V for the space S V. It is a complex of [G]-modules with H (C V ) = H (S V ). One can convert such a chain complex C V of [G]-modules to one of Mackey functors as follows. Given a [G]-module M, we get a Mackey functor M defined by () M(G/H) = M H for each subgroup H G. We call this a fixed point Mackey functor. When M is a permutation module, meaning the free abelian group on a G-set B, we call M a permutation Mackey functor [HHR,.5]. Given a finite G-CW spectrum X, meaning one built out of cells of the form G + e n, we get a reduced cellular chain complex of [G]-modules H C X, leading to a chain complex of fixed point Mackey functors C X. Its homology is a graded Mackey functor H X with H X(G/H) = π (X H)(G/H) = π (X H) H. In particular H X(G/e) = H X, the underlying homology of X. In general H X(G/H) is not the same as H (X H ) because fixed points do not commute with smash products. We will see an illustration of this below in Example 5, where we will also see that H X need not be a graded fixed point Mackey functor. For a finite cyclic -group G = C k, the irreducible representations are the -dimensional ones λ(m) corresponding to rotation through an angle of πm/ k for < m < k, the sign representation σ and the trivial one of degree one, which we denote by. The -local homotopy type of S λ(m) depends only on the -adic valuation of m, so we will only consider λ( j ) for j k. The planar rotation λ( k ) though angle π is the same representation as σ. We will describe the chain complex C V for V = a + bσ + c j λ( k j ). j k for nonnegative integers a, b and c j. The isotropy group of V (the largest subgroup fixing all of V ) is C k = G for b = c = = c k = G V = C k =: G for b > and c = = c k = C k l for c l > and c +l = = c k = The sphere S V has a G-CW structure with reduced cellular chain complex C V of the form for n = d Cn V [G/G = ] for d < n d [G/C k j ] for d j < n d j and j l otherwise. where so d l = V. a for j = d j = a + b for j = a + b + c + + c j for j l,

5 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 5 The boundary map n : Cn V Cn V is determined by the fact that H (C V ) = H (S V ). More explicitly, let γ be a generator of G and θ j = γ t for j k. t< j Then we have for n = + d ( γ)x n = n for n d even and + d n d n x n for n d odd + d n d n otherwise, where is the fold map sending γ. We will use the same symbol below for the quotient map [G/H] [G/K] for H K G. The elements x n [G] for + d n V are determined recursively by x +d = and x n x n = θ j for + d j < n + d j. It follows that H V C V = generated by either x + V or its product with γ, depending on the parity of b. This complex is C V = Σ V C V/V where V = V G. This means we can assume without loss of generality that V =. An element x H n C V (G/H) = H n S V (G/H) corresponds to an element x π n V H(G/H). We will denote the dual complex Hom (C V, ) by C V. Its chains lie in dimensions n for n V. An element x H n ( V )(G/H) corresponds to an element x π V n H(G/H). The method we have just described determines only a portion of the RO(G)- graded Mackey functor π H, namely the groups in which the index differs by an integer from an actual representation V or its negative. For example it does not give us π σ λ() H for G. Example 5. The case G = C 8 and V = σ + λ(). The representation V is not orientable since it involves an odd multiple of σ. Its unit sphere S(V ) is S with the following action of G. There is a generator γ which rotates the equator though an angle of π/ while reflecting through the equatorial plane. Thus the poles are fixed by each proper subgroup, and no other point is fixed by a nontrivial subgroup. It follows that in the one point compactification S V of V we have S V for H = e (S V ) H = S for H = C or C S for H = G S(V ) has a G-CW structure with two -cells (the north and south poles) interchanged by γ, eight -cells (equally spaced longitudinal lines joining the two poles with alternating orientations) cyclically permuted by γ and eight -cells (regions between two adjacent longitudinal lines) cyclically permuted by γ.

6 6 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL This means that S V has a similar G-CW structure with two fxied -cells in which each positive dimensioanl cell is the double cone on a cell in S(V ). The reduced cellular chain complex C V is (6) C V C V C V C V 3 [G/G ] γ [G] +γ 7 [G] where x =, x 3 = + γ and x = ( + γ )( + γ ). The number beneath each arrow indicates its rank as a homomorphism. H 3 C 3 is the subgroup generated by ( γ)x = ( γ)( + γ )( + γ ). The corresponding chain complex of fixed point Mackey functors is C V C V C V C V 3 +γ [G/G ] [G/G ] [G/G ] +γ ( γ) +γ ( γ) γ [G/G ] [G/C ] +γ [G] +γ +γ 3 +γ 7 +γ [G/G ] +γ [G/C ] +γ [G] and its homology is H S V H S V H S V H 3 S V / / / / where denotes [G/H]/( + γ) for the appropriate proper subgroup H. In these diagrams of Mackey functors M, the top and bottom groups are M(G/G) and M(G/e) with the values of M on intermediate groups in between. Downward and upward pointing arrows are restrictions and transfers respectively. Note that these homology groups are not fixed point Mackey functors, and H (G/H) is not the same as H (S V ) H for any nontrival subgroup H.

7 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 7 For the dual spectrum S V we apply the functor Hom (, ) to (6). The resulting chain complex of fixed point Mackey functors is C V C V C V C V 3 +γ +γ +γ +γ [G/G ] [G/G ] [G/G ] γ +γ [G/G ] +γ ( γ)(+γ ) [G/C ] ( γ)x +γ [G] +γ +γ +γ +γ [G/G ] +γ [G/C ] +γ [G] with homology H S V H S V H S V H 3 S V / Notice that H 3 S V is quite different from H 3 S V. Example 5 illustrates the nonoriented case of the following, whose proof we leave as an exercise. Proposition 7. The top homology group. Let G be a finite cyclic -group and V a nontrivial representation of G of degree d with V G = and isotropy group G V. Then Cd V = [G/G V ] and = C V d (i) If V is oriented then H d S V =, the constant -valued Mackey functor in which each restriction map is an isomorphism and each transfer Tr K H is multiplication by K/H. H d S V = (G, G V ), the constant -valued Mackey functor in which { Res K for K GV H = K/H for G V H and Tr K H = { K/H for K GV for G V H.

8 8 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL (These determine all restrictions and transfers.) The functor (G, e) is also known as the dual. These isomorphisms are induced by the maps H d S V H d S V [G/G V ] (G, G V ) (ii) If V is not oriented then H d S V =, where { for H = G (G/H) = otherwise where each restriction map Res K H is an isomorphism and each transfer Tr K H is multiplication by K/H for each proper subgroup K. We also have H d S V = (G, G V ), where for H = G and V = σ (G, G V ) (G/H) = / for H = G and V σ otherwise with the same restrictions and transfers as (G, G V ). These isomorphisms are induced by the evident maps H d S V H d S V [G/G V ] (G, G V ) Definition 8. Three elements in π G (H). Let V be an actual (as opposed to virtual) representation of the finite cyclic -group G with V G = and isotropy group G V. (i) The equivariant inclusion S S V defines an element in π V S (G/G) via the isomophisms π V S (G/G) = π S V (G/G) = π S V G = π S =, and we will use the symbol a V to denote its image in π V H(G/G). (ii) The underlying equivalence S V S V defines an element in π V S V (G/G V ) = π V V S (G/G V ) and we will use the symbol e V to denote its image in π V V H(G/G V ). (iii) If W is oriented, there is a map : C W W as in Proposition 7 giving an element u W H W S W (G/G) = π W W H(G/G). For nonoriented V Proposition 7 gives a map and an element : C V V u V H V S V (G/G ) = π V V H(G/G ).

9 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 9 Note that a V and e V are induced by maps to equivariant spheres while u W is not. This means that in any spectral sequence based on a filtration where the subquotients are equivariant H-modules, elements defined in terms of a V and e V will be permanent cycles, while mulitples of u W can support differentials. Note also that a = e = u =. The trivial representations contribute nothing to π (H). We can limit our attention to representations V with V G =. Among such representations of cyclic -groups, the oriented ones are precisely the ones of even degree. Lemma 9. Properties of a V, e V and u W. The elements a V π V H(G/G), e V π V V H(G/G V ) and u W π W W H(G/G) for W oriented satisfy the following. (i) a V +W = a V a W and u V +W = u V u W. (ii) G/G V a V = where G V is the isotropy group of V. (iii) For oriented V, Tr G G V (e V ) and Tr G G V (e V +σ ) have infinite order while Tr G G V (e V +σ ) has order if V >, and Tr G G V (e σ ) =. (iv) For oriented W, Tr G G V (e W )u W = G/G W π H(G/G) =. (v) a V +W Tr G G V (e V +U ) = if V >. (vi) For V and W oriented, u W Tr G G V (e V +W ) = G V /G V +W Tr G G V (e V ). (vii) The au relation. For V and W oriented representations of degree with G V G W, a W u V = G W /G V a V u W. For nonoriented W similar statements hold in π H(G/G ). W is oriented and u W is defined in π W W H(G/G) with Res G G (u W ) = u W. Proof. (i) This follows from the existence of the pairing C V C W C V +W. It induces an isomorphism in H and (when both V and W are oriented) in H V +W. (ii) This holds because H (V ) is killed by G/G V. (iii) This follows from Proposition 7. (iv) Using the Frobenius relation we have Tr G G V (e W )u W = Tr G G V (e W Res G (u W )) = Tr G G V ( G/G V ) = G/G V. (v) We have a V +W Tr G G V (e V +U ) : S V U S W U. It is null because the bottom cell of S W U is in dimension U. (vi) Since V is oriented, then we are computing in a torsion free group so we can tensor with the rationals. It follows from (iv) that Tr G G V +W (e V +W ) = G/G V +W u V u W and Tr G G V (e V ) = G/G V u V so u W Tr G G V +W (e V +W ) = G/G V +W u V = G V /G V +W Tr G G V (e V ).

10 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL (vii) The relevant chain complexes are 3 γ C V : V [G/G V ] [G/G V ] γ C W : W [G/G W ] [G/G W ] γ C V +W W θ : [G/G W ] [G/G W ] W [G/G V ] [G/G V ] m C V C W : m [G/G V ] [G/G W ] m [G/G V ] T (V, W ) [G/G W ] 3 m 3 T (V, W ) T (V, W ) where V and W are fold or reduction maps sending each power of γ to, θ W = γ i and i< G/G W T (V, W ) = [G/G V ] [G/G W ] = G/G W [G/G V ]. To describe the maps m i and i we use left matrix multiplication on column vectors. We have = [ ] V W m = [ V,W ] [ ] γ? = m? γ = [ V,W? ] where V,W is the reduction map and the unidentified maps from T (V, W ), namely 3,, m 3 and m, are not relevant here. We have a noncommuting diagram u V u W C V C V a W C V C W Σ m C V +W m C W a V C W C V C W where the maps to the relevant summands are γ m T (V, W ) [G/G V ] V [G/G V ] V,W [G/G W ] W [G/G W ] [G/G W ]

11 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY where V = W = i< G/G V i< G/G W γ i γ i. The upper composite is G W /G V times the lower one, so a W u V = G W /G V a V u W as claimed. 3. The case G = C Now let G = C with generator γ, and let G G be its index subgroup. Then the above discussion leads at a diagram RO(G) RO(G ) π G X(G/G) Res Tr () [G/G ] π G (X)(G/G ) t π G X(G /G ) r Res Tr Res Tr [G] π G (X)(G/e) t π G X(G /e) t π X r r Here the homotopy groups are modules over the rings shown on the left and graded over the indexing groups shown above. The group action transfers t and t are defined only on groups indexed by representations the smaller group which extend to representations of the larger group. We will make no use of them in this paper. In the bottom row each homotopy group is an underlying homotopy group of X depending only on the degree of the indexing representation. The group action restriction maps are isomorphisms. The group action transfers are t (r (x)) = ( + γ )x and t (r (y)) = ( + γ)y. In the middle row each homotopy group depends only on the restriction of the representation to G. The restriction r is an isomorphism in each RO(G)-graded degree, but it misses half of the RO(G )-graded degrees. The transfer is multiplication by + γ when the representation of G is the restriction of one of G. We need some notation for Mackey functors to be used in spectral sequence charts. The first four in Table are fixed point Mackey functors (), meaning they

12 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Table. Some C -Mackey functors / / [G/G ] /[G/G ] [G/G ] [G/G ] / / /[G/G ] [G/G ] / / / / / / / / [G/G ] [G/G ] [G] / [G/G ] [G/G ] are fixed points of an underlying [G]-module M, such as = [G]/(γ ) [G/G ] = [G]/(γ ) = [G]/(γ + ) [G/G ] = [G]/(γ + ). There are short exact sequences Here the hat symbol is used for a Mackey functor induced up from C, for which our notation is shown in Table, where, the dual of, is the kernel of the surjective map.

13 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 3 Table. Some C -Mackey functors / / / / / / [G] We have short exact sequences () () Proposition 3. Exactness of Mackey functor induction. The induction functor above is exact. It sends a C -Mackey functor M of the form M(C /C ) Res to the C -Mackey functor M of the form Tr M(C /e) M(C /C ) = M(C /C ) M(C /C ) M(C /C ) M(C /C ) = [C ] [C] M(C /C ) [C ] [C ]Res [C ] [C ]Tr M(C /e) = [C ] [C] M(C /e) The same holds for induction up to G of a Mackey functor defined for a subgroup H of G for any finite G. Definition. A [C ]-enriched C -Mackey functor. For a C -Mackey functor M as above, M will denote the C -Mackey functor enriched over [C ] defined by M(C /H) = [C ] [C] M(C /H) for H = C or e with structure maps as above.. Some chain complexes of Mackey functors As noted above, a G-CW complex X, meaning one built out of cells of the form G + H e n, has a reduced cellular chain complex of [G]-modules C X, leading to a chain complex of fixed point Mackey functors (see ()) C X. When X = S V for a representation V, we will denote this complexes C V. Its homology is the graded Mackey functor H X. Here we will apply the methods of to three examples.

14 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL (i) Let G = C with generator γ, and X = S nρ for n >, where ρ denotes the regular representation. We have seen before [HHR, 3.6] that it has a reduced cellular chain complex C with [G]/(γ ) for i = n (5) C nρ i = [G] for n < i n otherwise. Let c i denote a generator of C i. The boundary operator d is given by c i for i = n (6) d(c i+ ) = γ i+ n (c i ) for n < i n otherwise where γ i = ( ) i γ. For future reference, let { ɛ i = ( ) i for i even = for i odd. This chain complex has the form n n + n + n + 3 n γ γ 3 γ n [G] γ γ 3 [G] [G] ɛ n γ n [G] Passing to homology we get n n + n + n + 3 n H n / / H n (G/G) [G]/(γ n+ ) where H n (G/G) = { for n even for n odd { for n even and H n = for n odd Here and are fixed point Mackey functors but is not. Similar calculations can be made for S nρ for n <. The results are indicated in Figure.

15 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY Figure. The (collapsing) Mackey functor slice spectral sequence for H n Snρ. The symbols are defined in Table. In other words the RO(G)-graded Mackey functor valued homotopy of H is as follows. For n > 3/ we have π i Σ nρ H = π i nρ H = for n even and i = n for n even and i = n j with < j n/ for n odd and i = n for n odd and i = n + j with < j (n + )/ otherwise For n < 3/ we have π i Σ nρ H = π i nρ H = for n even and i = n for n even and i = n + j with < j ( 3 n)/ for n odd and i = n for n odd and i = n + j with < j ( 3 n)/ otherwise We can use Definition 8 to name some elements of these groups. Note that H is a commutative ring spectrum, so there is a commutative multiplication in π H, making it a commutative Green functor. For such a functor M on a general group G, the restriction maps are a ring homomorphisms while the transfer maps satisfy the Frobenius relations ().

16 6 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Figure. The Mackey functor slice spectral sequence for G + G n H, for G = C Σnρ and G = C. The symbols are for C -Mackey functors defined in Table. (7) Then -slice: a = a σ π Σ ρ H(G/G) = π σ H(G/G) x = u σ π Σ ρ H(G/e) = π σ H(G/e) with γ(x) = x -slice: u = u σ π Σ ρ H(G/G) = π σ H(G/G) with Res(u) = x negative slices: z n = e nρ π n Σ nρ H(G/e) = π n(σ ) H(G/e) for n > a i Tr(x n ) π n i Σ (n++i)ρ H(G/G) = π (n+)(σ )+iσ H(G/G) for n > and i are the generators of their respective groups. We have relations a = Res(a) = z n = x n Tr(x n ) = u n/ for n even and n Tr(z n/ ) for n even and n < for n odd and n > 3. (ii) Let G = C with generator γ, G = C G, the subgroup generated by γ, and Ŝ(n, G ) = G + G Snρ. Thus we have C (Ŝ(n, G )) = [G] [G ] C nρ with C nρ as in (5). The calculations of the previous example carry over verbatim by the exactness of 3.

17 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 7 The results are indicated in Figure, which is obtained from the Figure by putting a hat over each symbol. We will name elements in the groups shown here as follows. For an element α π G i V H(G /K) = π G i Σ V H(G /K), such as those listed in (7), we denote the two corresponding elements in π G i (G + ΣV H)(G/K) = [G] G [G ] π G i Σ V H(G /K) = π G i V H(G /K) π G i V H(G /K) by α and γ( α). We have γ ( α) = ± α, and there is no canonical choice of α. When the representation V of G is the restriction of a representation W of G, then this group is π G i W (G + G H)(G/K) When K = G, the two elements have the same image in π G i (G + G ΣV H)(G/G) = π i (G + G ΣV H) G = π i (Σ V H) G = π i (Σ V H)(G /G ) under the transfer, namely the element corresponding to α under the evident isomorphisms. Hence if α (or a set of such elements) generates the group π G i Σ V H(G /G ), then t ( α) (or the corresponding set) generates π G i (G + ΣV H)(G/G). G with (iii) Let G = C and X = S nρ. Then the reduced cellular chain complex is for i = n [G/G C nρ i = ] for n < i n [G] for n < i n otherwise where d(c i+ ) = c i for i = n γ i+ n c i for n < i n θ i+ n c i for n < i < n and i even γ i+ n c i for n < i < n and i odd otherwise, θ i = γ i ( + γ ) = ( ( ) i γ)( + γ ). The fixed point Mackey functors for = [G/G], [G/G ] and [G] = [G/e], are, and. In low dimensions the chain complex of Mackey functors is n n + n + n + 3 In homology this gives γ +γ n n + n + n + 3

18 8 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Figure 3. The Mackey functor slice spectral sequence for n Σnρ H. The symbols are defined in Table. The Mackey functor at position (n s, s) is π n( ρ) sh. In dimensions near n we have n n + n + n + 3 γ n γ n+ γ n+ θ n+3 γ n+ ɛ n γ n ɛ n+ ɛ n+ ɛ n+3 [G/G ] [G/G ] [G/G ] [G/G ] γ n [G/G ] γ n+ γ n+ [G] γ n+ γ n+ γ n+3 θ n+3 ɛ n+ γ n+ γ n+ [G] [G] The homology is for n and i even and i < n for n and i odd and i < n for n odd and i even and i < n H n+i = for n even and i = n for n odd and i = n otherwise Again similar calculations can be made for S nρ for n <. The results are indicated in Figure 3. The Mackey functors in filtration (the horizontal axis) are the ones described in Proposition 7.

19 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 9 As in (i), we name some of these elements. Let G = C and G = C G. Recall that the regular representation ρ is + σ + λ where σ is the sign representation and λ is the -dimensional representation given by a rotation of order. Note that while Figure shows all of π H for G = C, Figure 3 shows only a bigraded portion of this trigraded Mackey functor for G = C, namely the groups for which the index differs by an integer from a multiple of ρ. We will need to refer to some elements not shown in the latter chart, namely a σ π σ H(G/G) a λ π λ H(G/G) u σ π σ H(G/G ) u λ π λ H(G/G) u σ π σ H(G/G) with a λ u σ = a σu λ and Res (u σ ) = u σ; see Definition 8 and Lemma 9. We will denote the generator of E s,t (G/H) by x t s,s, y t s,s and z t s,s for H = G, G and e respectively. Then the generators for the groups in the -slice are y, = u ρ = u σ Res (u λ ) π Σ ρ H(G/G ) = π 3 σ λ H(G/G ) with γ(x, ) = x, x 3, = a σ u λ π 3 Σ ρ H(G/G) = π σ λ H(G/G) y, = Res (a λ )u σ π Σ ρ H(G/G ) = π σ λ H(G/G ) x,3 = a ρ = a σ a λ π Σ ρ H(G/G) = π σ λ H(G/G) and the ones for the 8-slice are x 8, = u λ+σ = u ρ π 8 Σ ρ H(G/G) = π 6 σ λ H(G/G) with y, = y 8, = Res (x 8, ) x 6, = a λ u λ+σ π 6 Σ ρ H(G/G) = π σ λ H(G/G) with x 3, = x 6, and y, y, = y 6, = Res (x 6, ) x, = a λ u σ π Σ ρ H(G/G) = π σ λ H(G/G) with y, = y, = Res (x, ) and x,3 x 3, = x, x 6, = x 3, π Σ ρ H(G/G) = π σ λ H(G/G). These elements and their restrictions generate π Σ mρ H for m = and. For m > the groups are generated by products of these elements. There are relations The element a σ = Res a σ = a λ = Res a λ = Res a λ = z, = Res (y, ) = Res (u ρ ) π Σ ρ H(G/e) is invertible with γ(y) = y, z, = z 8, = Res (x 8, ) and z m, := z m, = e mρ π m Σ mρ H(G/e) for m >. These elements and their transfers generate the groups in π m Σ mρ H for m >.

20 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Proof??? Theorem 8. Divisibilities in the negative regular slices for C. There are the following infinite divisibilities in the third quadrant of the spectral sequence in Figure 3. x, = Tr (z, ) is infinitely divisible by x, and x,3, meaning that x j, xk,3x j k, j 3k = x, for j, k. x 7, is infinitely divisible by x,, x 6, and x 8,, meaning that x i,x j 6, xk 8,x 7 i 6j 8k, i j = x 7, for i, j, k, subject to the relation x 6, = x 8, x,. x, is infinitely divisible by x,, x 6, and x 8,, meaning that x i,x j 6, xk 8,x i 6j 8k, i j = x, for i, j, k with x 7 i 6j 8k, i j = x 3, x i 6j 8k, i j = x,3 x 8 i 8j 6k, i j for i, j, k. y 7, = Res (x 7, ) is infinitely divisible by y, and y,, meaning that y j, yk,y 7 j k, j = y 7, for j, k. 5. The C -spectrum k H Before defining our spectrum we need to recall some formulas from [HHR]. Let G be a finite cyclic -group with generator γ. In [HHR, 5.7] we defined generators (9) r k = r G k π C kρ MU ((G)) (C /C ) (note that this group is a module over G/C ) and r k = r Res (r k ) π e nρ MU ((G)) (e/e) = π u nmu ((G)). These are defined in terms of the coefficients m k π C kρ H () MU ((G)) (C /C ) of the logarithm of the formal group associated with the left unit map from MU to MU ((G)). For small k we have r = ( γ)(m ) r = m γ(m )( γ)(m ) r 3 = ( γ)(m 3 ) γ(m )(m + m γ(m ) 3γ(m ) m ) Now let G = C and G = C G. The generators r G k are the r k defined above. We also have generators r G k defined by similar formulas with γ replaced by γ ; recall that γ (m k ) = ( ) k m k. Thus we have r G = m r G = m + m r G 3 = m 3 m m m 3

21 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY If we set r = and r 3 =, we get () r G 3 γ(r G r G = ( + γ)(r ) r G = 3r γ(r ) + γ(r ) r G 3 = 5r γ(r ) + 5r γ(r ) + γ(r ) 3 3 ) = r γ(r ) ( ) ( 5γ(r ) 5r γ(r ) + r γ(r ) + 5r γ(r ) + 5r) = 5r 5 γ(r ) + r γ(r ) r 3 γ(r ) 3 r γ(r ) 5r γ(r ) 5 Let k H be the G-spectrum obtained from MU ((G)) by killing the r n s and their conjugates for n. We will often use a (second) subscript ɛ to indicate the action of γ, so γ(x ɛ ) = x +ɛ and x +ɛ = ±x ɛ. Then we have () π u k H = π k H (G/e) = [r, γ(r )] = [r,, r, ] where γ (r,ɛ ) = r,ɛ. Here we use r,ɛ and r,ɛ to denote the images of elements of the same name in the homotopy of MU ((G)). The Periodicity Theorem [HHR, 9.] states that inverting a class D π ρ k H (G/G) whose image under r Res is divisible by r G a permanent cycle. Let 3,r G D = (r Res ) ( r G,r G,r G 3 γ(r G 3 ) 3, (see ()) and r, r, makes u 8ρ ) π ρ k H (G/G) (see Table 3 for a more explicit description) and K H = D k H. Then we know that Σ 3 K H is equivalent to K H. The Slice and Reduction Theorems [HHR, 6. and 6.5] imply that the kth slice of k H is the kth wedge summand of H N S iρ. i It follows that over G the kth slice is a wedge of k + copies of H S kρ. The group π G ρ k H (G /e) is not in the image of the group action restriction r because ρ is not the restriction of a representation of G. However, π u k H is refined (in the sense of [HHR, 5.9]) by a map from () S ρ = G + G Sρ s k H. The reduction theorem implies that the -slice P k H is S ρ H. We know that π (S ρ H) =. We use the symbols r and γ(r ) to denote the generators of the underlying abelian group of (G/e) = [G/G ]. These elements have trivial fixed point transfers and π (S ρ H)(G/G ) =.

22 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Tables 3 and describe some elements in the low dimensional homotopy of k H, which we now discuss. Given an element in π MU ((G)), we will often use the same symbol to denote its image in π k H. For example, in [HHR, 9.] (3) d k π G ( k )ρ MU ((G)) = π G ( k )ρ MU ((G)) (G/G) was defined to be the composite S (k )ρ N S (k )ρ N r k N MU ((G)) MU ((G)). We will use the same symbol to denote its image in π G ρ k H (G/G). The element η π S (coming from the Hopf map S 3 S ) has image a σ r π G k R(G /G ). There are two corresponding elements η ɛ π G k H (G /G ) for ɛ =,. We use the same symbol for their preimages under r. We denote by η again the image of either under the transfer Tr. It is the image of the Hopf map in π k H (G/G), and Res (η) = η + η. Its cube is killed by a d 3 in the slice spectral sequence, as is the sum of any two monomials of degree 3 in the η ɛ. It follows that in E each such monomial is equal to η. 3 It has a nontrivial transfer, which we denote by x 3. In [HHR, 5.5] we defined () f k = a k ρn g r k π k MU ((G)) (G/G) Proof??? for a finite cyclic -group G. Its slice filtration is k(g ) and we conjecture that (5) Tr G G (u σres G G (x)) = a σf x. Proof??? Somehow this is related to the first slice differential, d + G (u σ ) = a σf. In particular, for G = C we have f = a σ a λ d with Tr (u σ Res (x)) = a σ f x. For example Tr (η η ) = Tr (u σ Res (a λ d )) = a σ f a λ d = f. The Hopf element ν π 3 S has image a σ u λ d π 3 k H (G/G), so we also denote the latter by ν. It has an exotic restriction η 3 (filtration jump two), which implies that ν = Tr (Res (ν)) = Tr (η) 3 = x 3.

23 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 3 One way to see this is to use the Periodicity Theorem to equate π 3 k H with π 9 k H, which can be shown to be the Mackey functor in slice filtration 3. Another argument not relying on periodicity is given below in (39). The exotic restriction on ν implies with filtration jump. Res (ν ) = η 6, Theorem 6. The Hurewicz image The elements η π k H (G/G), ν π 3 k H (G/G), ɛ π 8 k H (G/G), κ π k H (G/G), and κ π k H (G/G) are the images of elements of the same names in π S. We refer the reader to [Rav86, Table A3.3] for more information about these elements. Proof. Suppose we know this for ν and κ. Then ν is represented by an element of filtration 3 whose product with ν is nontrivial. This implies that ν 3 has nontrivial image in π 9 k H (G/G). This is a nontrivial multiplicative extension in the first quadrant, but not in the third. Since ν 3 = ηɛ in π S, this implies that η and ɛ are both detected and have the images stated in Table. It follows that ɛκ has nontrivial image here. Since κ = ɛκ in π S, κ must also be detected. Its only possible image is the one indicated. Both ν and κ have images of order 8 in π tmf and its K() localization. The latter is the homotopy fixed point set of an action of the binary tetrahedral group G acting on E. This in turn is a retract of the homotopy fixed point set of the quaternion group Q 8. A restriction and transfer argument shows that both elements have order at least in the homotopy fixed point set of C Q 8. WE NEED TO RELATE THIS C ACTION TO THE ONE WE ARE STUDYING. More details needed here. Let so δ = Res ( ). Hence we have 6. Slices for k H and K H = u ρ d π 8 k H (G/G) δ = u ρ Res ( d ) π k H (G/G ), δ m = m [m/] Res ( [m/] ) δ { = u σ Res (u [m/] σ u m d λ m ) for m odd Res (u m/ σ um d λ m ) for m even. Theorem 7. The slice E -term for k H. The slices of k H are { Ps s m s/ k H = X m,s/ m for s even and s otherwise where { Σ mρ H for m = n X m,n = G + H for m < n G Σ(m+n)ρ

24 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Table 3. Some elements of filtration in the homotopy of and slice spectral sequence for k H. Refer to Table for targets of differentials and transfers. Element Description r,ɛ π G ρ k H (G /G ) with Images from (9) defined in [HHR, 5.7] r, = r, r,ɛ π G k H(G /e) u σ Res (r,ɛ ) r,ɛ π G k H(G/e) with Preimages of the above under r, r, = r, generating = π G k H/torsion s,ɛ π G ρ k H (G/G ) Preimages of r,ɛ under r d π G ρ k H (G/G) with Image from (3) defined in [HHR, 9.] r Res ( d ) = r, r, and Tr (Res ( d )) = t π G ρ k H (G/G) with ( ) ɛ Tr (s,ɛ ) Res (t ) = s, s, D π ρ k H (G/G), the periodicity d ( 5t + t d + 9 d ) element u σ π σ k H (G/G ) with Isomorphic image of π k H (G/G ) γ(u σ ) = u σ and Tr (u σ ) = a σ f (exotic transfer). Σ,ɛ E, k H(G/G ) with ( ) ɛ u ρ s,ɛ Σ, = Σ, and d 3 (Σ,ɛ ) = ηɛ (η + η ) = ( ) ɛ u σ Res (u λ )s,ɛ T E, k H(G/G) with Tr (Σ,ɛ ) = ( ) ɛ u λ Tr (u σ s,ɛ ) Res (T ) = Σ, + Σ, and d 3 (T ) = η 3 T E,8 k H(G/G) with ( ) ɛ Tr (Σ,ɛ δ ) = u σ u T = (T λ t d ), Res (T ) = (Σ, Σ, )δ and d 3 (T ) = δ E, k H(G/G ) with u ρ Res ( d ) = u σ Res (u λ d ) γ(δ ) = δ, Tr (δ ) = and d 3 (δ ) = η η (η + η ) E,8 k H(G/G) with u ρ d Res ( ) = δ = u σ u λ d and d 5 ( ) = νx The structure of π u k H as a [G]-module (see ()) leads to four types of orbits and slices: () { (r, r, ) l} leading to X l,l for l ; see the leftmost diagonal in Figure 5. On the -line we have a copy of (see Table ) generated under restrictions by d l l = u lρ d l = u l σu l λ E,8l (G/G). In positive filtrations we have E j,8l generated by

25 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 5 Table. Some elements of positive filtration in the homotopy of and slice spectral sequence for k H. Element Description η ɛ π G k H (G /G ) with η ɛ = a σ r,ɛ η ɛ π G k H (G/G ) Preimage of the above under r, generating the summand of π k H f π k H (G/G) a σ a λ d, generating the summand of π k H η π G k H (G/G) with Tr (η ɛ ) + f Res (η) = η + η π G k H (G/G ) ηɛ, η η π G k H(G/G ) with Tr (ηɛ ) = η and u σ Res (a λ )s,ɛ and u σ Res (a λ d ), generating the torsion in π G k H Tr (η η ) = f (exotic transfer) η 3 = ηη = η η = η 3 π G 3 k H(G/G ) η ɛ u σ Res (a λ d ) = η ɛ u σ Res (a λ s,ɛ ) x 3 π 3 k H (G/G) with Res (x 3 ) = ν π 3 k H (G/G) with Res (ν) = η 3 and Tr (η) 3 a σ u λ d, generating = π 3 k H ν = x 3 (exotic restriction and group extension) x E,8 (G/G) with d 5(x ) = f 3, a λ u σ d Res (x ) = (η η ) = η and x = f ν ν π 6 k H (G/G) a λ u λ u σ d =, η, f, f ɛ π 8 k H (G/G) Represents x E 8,6 (G/G) ν 3 = ηɛ π 9 k H (G/G) Represents f x E 8,6 (G/G) κ π k H (G/G) a λ u σu 3 λ d κ π k H (G/G) a λ u3 σu λ d 6 a j λ ul σu l j λ d l E j,8l (G/G) for < j l and E k+l,8l generated by a k σ a l λ u l k l σ d E k+l,8l (G/G) for < k l. () { (r, r, ) l+} leading to X l+,l+ for l ; see the leftmost diagonal in Figure 6. On the -line we have a copy of generated under restrictions by u l+ σ δ l+ = u l+ σ Res (u λ d ) l+ E,8l+ (G/G ). In positive filtrations we have E j,8l+ generated by Res (a j λ ul+ j λ d l+ ) E j,8l+ (G/G ) for < j l +, E j+,8l+ generated by a σ a j λ ul σu l+ j λ d l+ E j+,8l+ (G/G) for j l + and E k+l+3,8l+ generated by a k+ σ a l+ λ u l k l+ σ d E k+l+3,8l+ (G/G) for < k l.

26 6 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL (3) { r i,r l i,, rl i, ri,} leading to Xi,l i for i < l; see other diagonals in Figure 5. On the -line we have a copy of generated (under Tr, Res and the group action) by u l σs l i Res (u l λ d i ) E,l (G/G ) In positive filtrations we have E j,l generated by u l σs l i Res (a j λ ul j d i λ ) = η j E j,l (G/G ) for < j l ɛ u l j σ s l i j Res (u l j λ d i ) for < j < l i. () { r i,r l+ i,, r l+ i, r i,} leading to Xi,l+ i for i l; see other diagonals in Figure 6. On the -line we have a copy of generated (under transfers and the group action) by r, Res (u l σs l i )Res (u l λ d i ) E,l+ (G/e) In positive filtrations we have E j+,l+ generated by η ɛ u l σs l i Res (a j λ ul j d i λ ) E j+,l+ (G/G ) for j l = η j+ ɛ u l j σ s l i j Res (u l j d i λ ) for j l i. Corollary 8. A subring of the slice E -term. The ring E k H (G/G ) is (see Tables 3 and ) [δ, Σ,ɛ, η ɛ : ɛ =, ]/ ( η ɛ, δ Σ, Σ,, η ɛ Σ,ɛ+ + η +ɛ δ ). In particular the elements η and η are algebraically independent mod with γ ɛ (η m η n ) π m+n X m,n (G/G ) for m n. The element (η η ) is the fixed point restriction of u σ a λ d E,8 k H(G/G), which has order, and the transfer of the former is twice the latter. The element η η is not in the image of Res and has trivial transfer in E. Proof. We detect this subring with the monomorphism E k H (G/G ) r E k H (G /G ) η ɛ a σ r,ɛ Σ,ɛ u σ r,ɛ δ u σ r, r,, in which all the relations are transparent.

27 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 7 Corollary 9. Slices for K H. The slices of k H are { Ps s m s/ k H = X m,s/ m for s even and s otherwise where X m,n is as in Theorem 7. Here m can be any integer, and we still require that m n. Proof. Recall that K H is obtained from k H by inverting a certain element D π ρ k H described in Table 3. Thus K H is the homotopy colimit of the diagram k H D Σ ρ k H D Σ 8ρ k H D Desuspending by ρ converts slices to slices, so for even s we have P s s K H = lim P s+6k k Σ kρ s+6k k H = lim k Σ kρ = lim m s/+8k X m,s/+8k m X m k,s/+k m k m s/+k = lim = X m,s/ m k k m s/ m s/ X m,s/ m. 7. Generalities on differentials Now we turn to differentials. Our starting point is the Slice Differentials Theorem of [HHR, 9.9], which says that in the slice spectral sequence for MU ((G)) for an arbitrary finite cyclic -group G of order g, the first nontrivial differential on various powers of u σ is (3) d r (u k σ ) = a k σ a k ρ N g (rg k ) ( σ) Er,r+k r MU ((G)) (G/G), where r = + ( k )g and ρ is the reduced regular representation of G. particular d 3 (u σ ) = a 3 σr E 3, σ 3 MU R (G/G) for G = C (3) d 5 (u σ ) = a 3 σa λ d E 5,6 σ 5 MU ((G)) (G/G) for G = C d 7 (u σ) = a 7 σr 3 E 7, σ 3 MU R (G/G) for G = C. In Now, as before, let G = C and G = C G. We need to translate the d 3 above in the slice spectral sequence for MU R into a statement about the one for k H. We

28 8 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL have an equivariant multiplication map m of G -spectra MU ((G)) MU R η L MU R MU R m MU R r G r G, + r G, r G r G 3 a 3 σ(r G, + r G,) a 3 σr G ( 5r G, r G,(r G, + r G,) +(r G,) 3 mod (r G, r G 3 ) ) r G 3 where the elements lie in π G ρ ( )(G /G ). In the slice spectral sequence for MU ((G)), d 3 (u σ ) and d 7 (u σ) must be G-invariant since u σ is, and they must map respectively to a 3 σr G and a 7 σr G 3, so we have d 3 (u σ ) = a 3 σ (r G, + r G,) = a σ (η + η ) d 7 (u σ ) = a 7 σ ( 5r G, r G,(r G, + r G,) + (r G,) 3 + ) = a 7 σ (r G,) 3 + since a 3 σ (r G, + r G,) = in E and similarly for k H where the missing terms in d 7 (u σ ) vanish. Pulling back along the isomorphism r and the monomorphism Res leads to the following. Proposition 3. The differentials on u λ and u σ. The following differentials occur in the slice spectral sequence for k H. d 3 (u λ ) = a λ η d 3 (Res (u λ )) = Res (a λ )(η + η ) d 3 (u σ Res (u λ )) = u σ Res (a λ )(η + η ) = ηη + η η d 3 (u λ η) = a λ η = a λtr (u σ s,ɛ ) d 3 (Res (u λ )η ɛ ) = Res (a λ )(η + η )η ɛ d 3 (u σ Res (u λ )η ɛ ) = u σ Res (a λ )(η + η )η ɛ = (η η ) + η η η ɛ d 5 (u σ ) = a 3 σa λ d d 5 (u λ) = a λa σ u λ d = a λ u λ f d 7 (Res (u λ ) ) = Res (a λ)η 3. The elements u σ, u σ and Res (u λ ) are permanent cycles. The first satisfies Tr (u σ Res (x)) = a σ f x π + x k H (G/G). Since the slice filtrations of u σ Res (x) and its transfer exceed those of x by and respectively, we have an exotic Mackey functor extension. Proof. The differentials were established above. Note that u σ E, σ (G/G ) since the maximal subgroup for which the sign representation σ is oriented is G, on which it is restricts to the trivial representation of degree. This group depends only on the restriction of the RO(G)-grading to

29 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 9 Proof needed for Tr (u σ ) G, and the isomorphism extends to differentials as well. This means that u σ is a place holder corresponding to the permanent cycle E, (G/G ). As remarked above, we lose no information by inverting the class D, which is divisible by d. It is shown in [HHR, 9.] that inverting the latter makes u σ a permanent cycle. 8. k H as a C -spectrum It is helpful to explore the restriction of the slice spectral sequence to G, for which the -brigraded portion E (G /G ) is the isomorphic image of the ring of Corollary 8. In the following we identify Σ, δ and r with their images under r. From the differentials of (3) we get d 3 (Σ,ɛ ) = η 3 ɛ + η ɛ η +ɛ d 3 (δ ) = η η + η η d 7 (δ ) = d 7 (u σ)r,r, = a 7 σr G 3 r,r, = a 7 σ(5r γ(r ) + 5r, r, + γ(r ) 3 )r,r,. The d 3 s above make all monomials in η and η of any given degree 3 the same in E (G/G ) and E (G /G ), so d 7 (δ ) = η 7. Similar calculations show that d 7 (Σ,ɛ) = η 7. This leads to the following, for which Figure is a visual aid. Theorem 33. The slice spectral sequence for k H as a C -spectrum. Using the notation of Table and Definition, we have E, (G /e) = [r,, r, ] with r,ɛ E, (G /e) E, (G /G ) = [δ, Σ,ɛ, η ɛ : ɛ =, ]/ ( ηɛ, δ ) Σ, Σ,, η ɛ Σ,ɛ+ + η +ɛ δ, so E s,t = l for (s, t) = (, l) with l l+ for (s, t) = (, l + ) with l u+l for (s, t) = (u, l + u) with l and u > u+l for (s, t) = (u, l + u ) with l and u > otherwise. The first differentials are determined by d 3 (Σ,ɛ ) = η ɛ (η + η ) and d 3 (δ ) = η η (η + η )

30 3 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL resulting in l E s,t = for (s, t) = (, l) with l and l even l for (s, t) = (, l) with l and l odd l+ for (s, t) = (, l + ) with l +l for (s, t) = (, l + ) with l even +l for (s, t) = (, l + ) with l even for (s, t) = (s, l + s) with s 3 and l even otherwise. There is a second set of differentials determined by resulting in E s,t 8 = E s,t = d 7 (Σ,ɛ ) = d 7 (δ ) = η 7 l for (s, t) = (, l) with l and l divisible by l for (s, t) = (, l) with l and l mod l for (s, t) = (, l) with l and l odd l+ for (s, t) = (, l + ) with l +l for (s, t) = (, l + ) with l divisible by +l for (s, t) = (, l + ) with l and l mod +l for (s, t) = (, l + ) with l divisble by l for (s, t) = (, l + ) with l and l mod for (s, t) = (s, l + s) with 3 s 6 and l otherwise. divisible by Corollary 3. Some nontrivial permanent cycles. The following elements in E s,8i+s k H (G/G ) and their transfers are nontrivial permanent cycles: Σ i j,ɛ δ j for j i (i + elements of infinite order including δi ), i even and s =. η ɛ Σ i j,ɛ δ j for j < i and η ɛδ i and s =. ηɛ Σ i j,ɛ δ j ) for i even and s =. (i + elements or order ) for i even { } for j < i and δi η, η η, η (i + 3 elements or order ηδ s i for 3 s 6 ( elements or order ) and i even. Σ i j,ɛ δ j + δi for j i (i + elements of infinite order including δ i ), i odd and s =. η ɛ Σ i j,ɛ δ j +δi for j i and η δ i (Σ, +δ ) = η δ i (Σ, +δ ) (i + elements of order ), i odd and s =. ηɛ Σ i j,ɛ δ j +δi for j i, ηδ i (Σ, +δ ) = η η δ i (Σ, +δ ) and η η δ i (Σ, + δ ) = ηδ i (Σ, + δ ) (i + elements of order ) for i odd and s =.

31 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY Figure. The slice spectral sequence for k H as a C -spectrum. The Mackey functor symbols are as in Table, The C -structure of the Mackey functors is not indicated here. In each bidegree we have a direct sum of the indicated number of the indicated Mackey functor. Each d 3 has maximal rank, leaving a cokernel of rank, and each d 7 has rank. Blue lines indicate exotic transfers, which also have maximal rank. In E,8i+ k H (G/G ) we have Σ i+ j,ɛ δ j for j i and δj, i + 3 elements of infinite order, each in the image of the transfer Tr. 9. The first differentials over C Theorem 7 lists elements in the slice spectral sequence for k H over C in terms of r, s, d ; η, a σ, a λ ; u λ, u σ, and u σ. All but the u s are permanent cycles, and the action of d 3 on u λ, u σ and u σ is described above in Proposition 3. Proposition 35. d 3 on elements in Theorem 7. We have the following d 3 s, subject to the conditions on i, j, k and l of Theorem 7:

32 3 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL On X l,l : d 3 (a j λ ul σu l j λ d l ) = d 3 (a k σ a l On X l+,l+ : d 3 (a σ a j λ ul d 3 (a k+ λ u l k l σ d σ u l+ j λ σ a l+ λ ) = a j+ λ ηu l σu l j λ d l π X l,l+ (G/G) for j odd for j even d 3 (δ l+ ) = ηu l+ σ Res (a λ u l l+ λ d ) π X l+,l+ (G/G ) d 3 (u l+ σ Res (a j λ ul+ j = d l+ ) = u l k l+ σ d ) = λ ηu l+ d l+ )) d l+ ) σ Res (a j+ λ u l j λ π X l+,l+ (G/G ) for j even for j odd d l+ ηa σ a j+ λ u l σ u l j λ π X l+,l+ (G/G) for j even for j odd On X i,l i : d 3 (u l σs l i Res (u l λ d i )) = d 3 (η j u l j σ s l i j Res (u l j d i λ )) = η 3 u l σ s l i Res (u l λ d i ) π X i,l+ i (G/G ) for l odd for l even η j+ u l j σ s l i j Res (a λ u l j λ d i ) π X i,l+ i (G/G ) for l j odd for l j even On X i,l+ i : d 3 (r Res (u l σs l i )Res (u l λ d i )) = d 3 (η j+ u l j σ s l i j Res (u l j d i λ )) = η j+ u l j σ s l i j Res (a λ u l j λ d i ) π X i,l+ i (G/G ) for l j odd for l j even Note that in each case the first index of X is unchanged by the differential, and the second one is increased by one. Since X m,n is a summand of the (m + n)th slice, each d 3 raises the slice degree by as expected.

33 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY Figure 5. The d 3 s on the slice summands X,n for n. The symbols are defined in Table Figure 6. The d 3 s on the slice summands X 5,n for n 5. These differentials are illustrated in Figures 5 and 6. In order to pass to E we need the following exact sequences of Mackey functors. d 3 d3 d 3 d3 The resulting subquotients of E are shown in Figures 7 and 8 and described below in Theorem 36. In the latter the slice summands are organized as shown in the Figures rather than by orbit type as in Theorem 7. Theorem 36. The slice E -term for k H. The elements of Theorem 7 surviving to E, which live in the appropriate subquotients of π X m,n, are as follows.

34 3 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Figure 7. The subquotient of the slice E -term for k H for the slice summands X,n for n. Exotic transfers are shown in blue Figure 8. The subquotient of the slice E -term for k H for the slice summands X 5,n for n 5. Exotic restrictions and transfers are shown in green and blue respectively. () In π X l,l (see the leftmost diagonal in Figure 7), on the -line we still have a copy of generated under fixed point restrictions by l E,8l. In positive filtrations we have E j,8l generated by a j λ ul σu l j λ d l E j,8l a j λ ul σu l j λ a k σ a l λ u l k l σ d d l = a σa j (G/G) for j even and < j l, λ u l+ σ ul j λ d l E j,8l (G/G) for j odd and < j l and E k+l,8l generated by E j+k,8l (G/G) for < k l.

35 THE SLICE SPECTRAL SEQUENCE FOR THE C ANALOG OF REAL K-THEORY 35 () In π X l,l+ (see the second leftmost diagonal in Figure 7), in filtration we have, generated (under transfers and the group action) by d l r Res (u l σ Res (u l λ ) E,8l+ (G/e). In positive filtrations we have E,8l+ generated (under transfers and ηu l σ Res (u λ d ) l = E,8l+ (G/G ) the group action) by E k+,8l+ for < k l generated by x = η k+ u l k σ Res (u λ d ) l k E k+,8l+ (G/G ) with ( γ)x = Tr (x) =. (3) In π X l+,l+ (see the leftmost diagonal in Figure 8), on the -line we have a copy of generated under fixed point (l+)/ E,8l+. In positive filtrations we have a σ a j λ ul u l+ σ E j,8l+ generated by Res (a j λ ul+ j λ d l+ ) E j,8l+ (G/G ) for < j l +, E j+,8l+ generated by d l+ E j+k,8l+ (G/G) for j l + and E k+l+3,8l+ generated by u l k l+ σ d E k+l+,8l+ (G/G) for < k l +. σ u l+ j λ a k+ σ a l+ λ () In π X l+,l+ (see the second leftmost diagonal in Figure 8), in filtration we have, generated (under transfers and the group action) by λ d l+ r Res (u l+ σ Res (u l+ ) E,8l+6 (G/e). In positive filtrations we have E k+3,8l+6 for k l generated under transfer by x = η k+3 l k E k+3,8l+6 (G/G ) with ( γ)x =. (5) In π X m,m+i for i (see the rest of Figures 7 and 8), in filtration we have E,m+j+ generated under transfers and group action by r Res (u m+j σ s j )Res (u m+j λ d m ) E,m+j+ (G/e) for j E,8l+ generated under transfers and group action by Res (u l+ σ s l+ m )Res (u l+ d m ) λ

36 36 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL E,8l+ (G/e) for l m/ E,8l generated under transfers and restriction x 8l,m = Σ l m, δ m + lδ l and group action by where Σ,ɛ = u ρ s,ɛ and δ = u ρ Res ( d ) E,8l (G/G ) for m l. In positive filtrations we have E,8l+ generated under transfers and group action by η Res ( l ) = η δ l = η u l σ Res (u λ d ) l E,8l+ (G/G ) and E s,8l+s generated under transfers and group action by η s ɛ x 8l,m E s,8l+s (G/G ) for s =, and m l. Proposition 37. Some nontrivial permanent cycles. The elements listed in Theorem 36(5) other than η ɛ δ l are all nontrivial permanent cycles. Proof. Each such element is either in the image of E, (G/e) under the transfer and therefore a nontrivial permanent cyle, or it is one of the ones listed in Corollary 3. In subsequent discussions and charts, starting with Figure 3, we will omit the elements Proposition 37. Analogous statements can be made about the slice spectral sequence for K H. Each of its slices is a certain infinite wedge spelled out in Corollary 9. Their homotopy groups are determined by the chain complex calculations of Section and illustrated in Figures and 3. Analogs of Figures 5 8 are shown in Figures 9. In each figure, exotic transfers and restrictions are indicated by blue and green lines respsctively. As in the k H case, most of the elements shown in this chart can be ignored for the purpose of calculating higher differentials. The resulting reduced E for K H is shown in Figure. The information shown there is very useful for computing differentials and extensions. The periodicity theorem tells us that π n K H and π n 3 K H are isomorphic. For n < 3 these groups appear in the first and third quadrants respectively, and the information visible in the spectral sequence can be quite different. For example, we see that π K H = while π 3 K H has a subgroup isomorphic to. The quotient / is isomorphic to. This leads to the exotic restrictions and transfer in dimension 3 shown in Figure. Information that is transparent in dimension implies subtle information in dimension 3. Conversely, we see easily that π K H = while π 8 K H has a quotient isomorphic to. This leads to the long transfer in dimension 8.