THE SLICE SPECTRAL SEQUENCE FOR RO(C p n)-graded SUSPENSIONS OF HZ I

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1 THE SLICE SPECTRAL SEQUENCE FOR RO(C n)-graded SUSPENSIONS OF H I M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL WORK IN PROGRESS Contents. Introduction.. Names of reresentations 3.2. Reresentation Sheres 3.3. The slice filtration 4 2. Dramatis ersonae: some secial -modules Forms of Forms of B 7 3. The RO(C n)-graded homotoy of H Realizing forms of toologically The gold (or au) relation 3.3. Fiber sequences realizing HB k,j Bredon homology comutations Bredon homology and B k,j A criterion for slice codimension 5 5. The slices of S λ H A sequence of reresentations Secial slices The 2ρ-eriodic slice tower 2 6. The slices of S mλ H 22 References 24. Introduction A key ingredient to our solution to the Kervaire invariant one roblem [HHR] was a new equivariant tool: the slice filtration. Generalizing the C 2 -equivariant work of Dugger [Dug5] and modeled on Voevodsky s motivic slice filtration [Voe2], this is an exhaustive, strongly convergent filtration on genuine G-sectra for a finite grou G. While quite owerful, the filtration quotients are quite difficult to determine, and much of our solution revolved around determining them for various sectra built out of MU. Date: February 8, 24. The authors were suorted by DARPA Grant FA and NSF Grants DMS-956, DMS and the Sloan Foundation.

2 2 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Classically one has a ma from a sectrum X to its n th Postinikov section P n X. The latter is obtained from X by attaching cells to kill off all homotoy grous above dimension n, and the tower can be thought of as assembling X by utting in the homotoy grous one at a time. This rocess can also be described as a localization or nullification functor, where we localize by killing the subcategory of n-connected sectra. Though this is a very big subcategory, it is generated by a much smaller set: the sheres of dimension at least (n + ). In the equivariant context, there are two axes along which we can vary this construction: () there are more sheres, namely reresentation sheres for real reresentations of G and (2) there are subgrous of G, the behavior for each of which can be controlled. The equivariant Postnikov filtration largely ignores the first otion, choosing instead to build a localization tower killing the subcategories generated by all sheres (with a trivial action!) of dimension at least (n + ) for all subgrous of G. Equivalently, we nullify the subcategory generated by all sectra of the form G + H S m for m > n. The slice filtration takes more seriously the otion of blending the reresentation theory and the homotoy theory. Definition.. For each integer n, let τ n denote the localizing subcategory of G-sectra generated by G + H S kρ H ɛ, where H ranges over all subgrous of G, ρ H is the regular reresentation of H, k H ɛ n and ɛ =,. If X is an object of τ n, we say that X is slice greater than or equal to n and that X is slice (n )-connected. The associated localization tower: is the slice tower of X. X P X The slice filtration and the slice tower refine the non-equivariant Postnikov tower, in the sense that forgetting the G-action takes the slice tower to the Postnikov tower, but the equivariant layers of the slice tower are in general much more comlicated. In articular, they need not be Eilenberg-Mac Lane sectra. This aer is the start of a short series analyzing several curious oints of the slice filtration for cyclic -grous alied to a very simle yet interesting family: the susensions of the Eilenberg-Mac Lane sectrum H by virtual reresentation sheres. This aer will discuss the susensions by multiles of the irreducible faithful reresentation λ; subsequent ones building on the thesis work of Yarnall will address the remaining cases. We make a huge, blanket assumtion and a slight abuse of notation. Everything we consider is localized at, where is the rime dividing the order of the grou. Thus when we write things like, we actually mean (). This being said, we have tried as much as ossible to make integral statements which are rime indeendent

3 SLICES AND RO(G)-GRADED SUSPENSIONS I 3 (secifying actual reresentations, rather than J-equivalence classes whenever ossible). Additionally, essentially everything we say holds for odd rimes. The case of = 2 behaves quite differently, given the existence of a non-trivial -dimensional real reresentation, the sign reresentation... Names of reresentations. Fix an identification of C n with the n th roots of unity µ n. For all k, let λ(k) denote the comosite of the inclusion of these roots of unity with the degree k-ma on S. This is a reresentation of comlex dimension. The real regular reresentation will be denoted ρ. We have a slitting ρ = + n 2 j= λ(j) for > 2. We will also let ρ denote the reduced regular reresentation, the quotient of the regular reresentation by the trivial summand..2. Reresentation Sheres. Reresentation sheres for C n have an excetionally simle cell structure. This renders comutations much more tractable than one might exect. The key feature is that the subgrous of C n are linearly ordered, and we can use this to build significantly smaller cell structures that might be exected. Our analysis follows [HHR] and [HHR]. We first consider the shere S λ(k). A cell structure is given by rays from the origin through the roots of unity, together with the sectors between these one cells. This gives a cell structure S C n/c k + e C n/c k + e 2. A icture is given in Figure for λ() and C 8. Figure. We can smash these together to get a cell structure on S V for any [virtual] reresentation V. The simlification arises from the different stabilizers which arise. For all m k, the restriction i C m λ( k ) is trivial. Thus S λ(m )+λ( k ) = S λ( m) S λ(k) = S λ(m) S λ(k) = S λ(k) C k/c m + e 3 C k/c m + e 4. By induction, this shows how to build any reresentation shere: decomose V into irreducibles, sort these in order of decreasing stabilizer subgrou, and reeat

4 4 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL the above trick. As an aside, this also allows a determination of a cell structure for virtual reresentation sheres, where we aroach them the same way. Our discussion of cell structure focused rimarily on that of λ( k ). This is because equivariant homotoy grous are actually more honestly called JO(G)-graded: they deend not on the reresentation but rather on the homotoy tye of the associated shere. In the -local context, most of the irreducible reresentations for C n have equivalent one oint comactifications: if r is rime to, then S λ(rk) S λ(k). For this reason, we can single out a single JO-equivalence class: let λ k = λ( k ). We will denote λ simly by λ. As k varies, the associated reresentation sheres hit every JO-equivalence class. We summarize the above discussion in a simle roosition. Proosition.2. If then S V = S kn C n/c n + e kn+ V = k n + k n λ n + + k λ, C n/c n + e kn+2kn where γ is a generator of C n. C n/c n γ + e kn+2 C n/c n γ + e kn+2kn C n/c n 2 + e kn+2kn +... γ C n + e dim V,.3. The slice filtration. We quickly recall several imortant facts about the slice filtration. This section contains no new results, and all roofs can be found in [Hil] and in [HHR, 3-4]. We first connect the slice filtration and the Postnikov filtration. Proosition.3. If X is (n )-connected for n, then X is in τ n. The converse of this is visibly not true, as generators of the form S kρ G are not (k G )-connected for G {e}. A slight elaboration on this lets us generalize this for smash roducts. Proosition.4. If X is ( )-connected and Y is in τ n, then X Y is in τ n. Unfortunately, smashing reduced regular reresentation sheres for G shows that results of this form are the best ossible. The form of the generating sectra for τ n shows that the slice tower commutes with susensions by regular reresentation sheres. Proosition.5. For any sectrum X, and identically for slices. P k Σ ρ X = Σ ρ P k G X, The final feature we will need is a recie for determining for which n a articular reresentation shere S V is in τ n. This is restatement of [Hil, Corollary 3.9], secializing the Corollary there to the case Y = S.

5 SLICES AND RO(G)-GRADED SUSPENSIONS I 5 Proosition.6. If W is a reresentation of G such that S W is in τ dim W and V is a sub reresentation such that () the inclusion induces an equality V G = W G and (2) for all roer subgrous H, the restriction i H SV is in τ dim V, then S V is in τ dim V. The conditions are very easy to check, by induction on the order of the grou. In general, we will alway choose W to be a regular reresentation or a reduced regular reresentation, as these are guaranteed to be in the correct localizing categories. 2. Dramatis ersonae: some secial -modules A large number of Mackey functors will show u in our analysis. They are all variants of the constant Mackey functor and the Mackey functor B that is the Bredon homology Mackey functor. Many of our discussions are facilitated by ictures of Mackey functors, and following Lewis, we draw them vertically. A generic Mackey functor M for C will be drawn res M(G/G) M(G/e) tr γ where res is the restriction ma, tr is the transfer, and γ generates the Weyl grou. For larger cyclic grous, we will use the obvious extension of this notation. Moreover, if the Weyl action is trivial or obvious, then we will suress the ma γ. 2.. Forms of. The constant Mackey functor (in which all restriction mas are the identity are the transfers are multilication by the index) and its dual (in which the roles of restriction and transfer are reversed) are just two members of a family of distinct Mackey functors which take the value on each orbit G/H. For C n, there are 2 n distinct Mackey functors, corresonding to the n choices is the restriction or for adjacent subgrous. A large subfamily of these occur in our discussion, and we give some notation here. Definition 2.. Let j < k n be integers. For each air, let (k, j) denote the Mackey functor with constant value for which the restriction mas are s < j, res s+ = s resc s+ C = s j s < k, k s. From now on, when a cyclic grou aears as an index, we will abbreviate it by the order of the grou as we did above. Thus the restriction of (k, j) to C j is just the constant Mackey functor. For subgrous of C k which roerly contain C j, it looks like the dual to the constant Mackey functor, and for those which roerly contain C k, it looks again like.

6 6 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Equivalently, (k, j) is the unique -valued Mackey functor C n in which res j, tr k and res n are isomorhisms. j k While = (n, ), does not occur in our list. If we allow j = k, then = (j, j) for any choice of j. For the reader s convenience, we draw out the four variants of that occur for C 2. (, ) (2, ) (2, ) In our analysis of the slices, we will need some elementary comutations with (k, j). We will eventually roduce a rojective resolution of all of these, but for now, we shall content ourselves to (n, k). First a brief digression on induced and restricted Mackey functors. Definition 2.2. Let M and N be Mackey functors for abelian grous G and H G resectively, and let i H denote the forgetful functor from G-sets to H-sets. Then the induced Mackey functor G H (N) on G and the restricted Mackey functor G H (M) on H are given by G H N(G/K) = N(i HG/K) { [G] [H] N(H/K) for K H = ([G] [H] N(H/H)) K for H K G M(H/K) = M(G/K) for K H. H The Weyl action of G in G N is induced by the H-action on N and the G-action H on G/K. The Weyl action of H in G M is the restriction of the the G-action on H M. Note that the second descrition of G N is not comlete for general G since H there are subgrous K that neither contain nor are contained in H. However it is comlete when G is a cyclic -grou, the case of interest here. In articular if N is the fixed oint Mackey functor for a [H]-module N defined by N(H/K) = N K, then G H N = [G] [H] N, the fixed oint Mackey functor for the [G]-module [G] [H] N. We now return to cyclic -grous. For the constant Mackey functor on G = C n, the comosite of induction and the restriction to C k is the fixed oint Mackey functor for the C n-module [C n/c k]: n n ( ) k = [C n/c k k].

7 SLICES AND RO(G)-GRADED SUSPENSIONS I 7 As such, it is very easy to describe mas out of this Mackey functor in the category of -modules: Hom ( n n (), M) = Hom k k n ( n, n M) = M(C k k k n/c k) for a Mackey functor M on G. In articular, these are all rojective objects in the category of -modules. These observations determine the mas in the following roosition. Proosition 2.3. The sequence [C n/c k] γ [C n/c k] (n, k) is exact, where γ denotes a generator of C n. The ma [C n/c k] (n, k) is left-adjoint to the identity ma from to the restriction to C k of (n, k). The first three terms are a rojective resolution of (n, k) in the category of -modules. Proof. It is obvious that is the kernel of the ma γ. For subgrous of C k, the fixed oint Mackey functors are a direct sum of coies of, and γ acts by ermuting the summands. Thus the quotient Mackey functor is also the constant Mackey functor for subgrous of C k. When we look at subgrous of C n which contain C k then we are actually looking at the Mackey functor for the integral regular reresentation of the grou C n/c k. The fixed oints are given by various transfers, and the fixed oints are the obvious inclusions. Passing to the quotient by ( γ) then sets all these inclusion mas to multilication by the index. The transfer mas are the identity. A useful way to understand these is that we have for C k the value n k. The restriction mas are diagonals, and the transfers are fold mas. Then the statement about assing to a quotient is obvious Forms of B. The Hom grous between various (k, j) are all easy to work out (the associated Hom grous are all ). We shall encounter mas from (k, j) to, and it is not difficult to see that the mas are arameterized by where the element in (k, j)(g/e) goes. Definition 2.4. Let B k,j denote the quotient Mackey functor associated to the inclusion (k + j, j) which is an isomorhism when evaluated on G/e. Equivalently, B k,j is the quotient of the unique ma (k + j, ) (j, ) which is the identity when restricted to C j. Let B k,j be the quotient of the unique ma (n, j) (n, k + j) which is the identity when restricted to C j. It will be helful to allow k = in the above definition, in which case B,j is the zero Mackey functor.

8 8 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL The Mackey functor B k,j is simle to describe: / k m k + j, B k,j (C n/c m) = / m j j < m < k + j, m j. All restriction mas are the canonical quotients, and the transfers are multilication by. Thus j refers to the largest subgrou for which B k,j restricts to, while k indicates the order of the maximal -torsion resent. Here is a artial Lewis diagram illustrating the definitions of B k,j and B k,j for G = C n, where m = n k j, s k and s = k s. The horizontal sequences are short exact. M (k + j, ) (j, ) B k,j B k,j (n, k + j) (n, j) M(G/G) M(G/C k+j ) M(G/C j+s) M(G/C j ) M(G/e) s s j k m k s j m s s / k / k / s m s m / k / k / s s m k m k s s s s s j j We will also need the following hybrid of B k,j and B k,j. Definition 2.5. For l, let B l k,j be the Mackey functor which agrees with B k,j when restricted to C l+k+j and which for grous containing C l+k+j, agrees with B k,j. For k l, let B l k,j be B k+l,j. In other words, B l k,j becomes the dual once we reach C n/c l+k+j u the Lewis diagram. In articular, while moving B k,j = B k,j and B m k,j = B k,j, where again m = n k j. For < l < m, B l k,j is the quotient of a ma between -valued Mackey functors, but not between the ones defined above. Here is another artial Lewis diagram illustrating this definition.

9 SLICES AND RO(G)-GRADED SUSPENSIONS I 9 M B k,j = B m k,j B l k,j B k,j = B k,j M(G/G) / k M(G/C l+k+j ) / k M(G/C k+j ) / k M(G/C j+s) / s M(G/C j ) m l m l l k s / k / k / k / s l k s m l l / k / k / k / s k s M(G/e) In our analysis of S kλ H, we will only need those B k,j with j =. To simlify notation, let B k be the Mackey functor B k,. 3. The RO(C n)-graded homotoy of H 3.. Realizing forms of toologically. A nice feature of the Mackey functor (k, j) is that H(k, j) is a virtual reresentation shere smashed with H. We will see this via a direct chain comlex comutation. Theorem 3.. For j < k, we have an equivalence Σ λ k λ j H H(k, j). Proof. We use the simlified cell structures described above. If we smash S λ k with the cell structure for S λj, then we have a comlex of the form ( ) C n/c j + S ( ) C n/c j + e e S λ k. Smashing with H and alying π gives a chain comlex of Mackey functors [C n/c k] γ [C n/c k] n k resk j [C n/c j ] γ n k resk j [C n/c j ] The ma labeled n res k k is actually the canonical inclusion of fixed oints for the j subgrous. Thus there is no homology in the to degree. Similarly, the kernel of the left most γ is, which mas isomorhically to the kernel of the second γ. Thus the homology of our chain comlex is the homology of the simler comlex: (n, k) n k resk j (n, j).

10 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Now we make a few observations. For subgrous of C k, the ma (n, k) is an isomorhism, and therefore the ushout looks like (n, j). For subgrous of C n which contain C k, the ma (n, k) (n, j) is an equivalence, and therefore the ushout looks like. This is the descrition of (k, j). Corollary 3.2. As a Mackey functor, π λk S λj H = (k, j). Using the multilicative structure of H, we can extend this to a ma of H-module sectra. Let v j/k : S λ k H S λj H be a generator of (k, j)(c n/c n) which restricts to a chosen orientation of the underlying 2-sheres. Since we are working at an odd rime, all reresentations of C n are orientable. In the equivariant context, this means that π dim V (S V H) = H dim V (S V ; ) =. A choice of generator for (C n/c n) determines one for all other values of the Mackey functor and, since all restriction and transfer mas are injective, this is detected in the underlying homotoy. Let u V : S dim V S V H be such a choice of generator. We will refer to these as orientation classes. By choosing one for each irreducible reresentation of C n, we can ensure this is comatible with the multilicative structure in the sense that Proosition 3.3. The comosite S 2 is homotoic to u λj. Thus u V W = u V u W. u λ k S λ k H v j/k S λj H, v j/k = u λ j u λk. Proof. The comosite is detected in π 2 S λj H =, and it is determined by the underlying ma. By assumtion, this is a ma of degree The gold (or au) relation. The orientation classes u λk carve out a ortion of the RO(C n)-graded homotoy of H. They do not tell a comlete icture, however. We need some classes in the Hurewicz image. Definition 3.4. For a reresentation V, let a V : S S V denote the inclusion of the origin and the oint at infinity into S V.

11 SLICES AND RO(G)-GRADED SUSPENSIONS I We will often refer to the classes a V as Euler classes. It is obvious that if V G {}, then a V is null-homotoic (as the ma factors through the inclusion S V G S V ). It follows that the restriction of a V to any subgrou for which V H {} is also null-homotoic. For grous for which V H = {}, the class a V and all of its susensions and owers are essential. We will also let a V denote the Hurewicz image. Our chain comlex models show that this generates H (S V ; ). The classes a λk can be generalized in an interesting way. If j < k < n, there are unstable mas a λk : S λj S λ k. a λj This can be formed most easily by observing that the k j th ower ma extends to the one oint comactifications. Comosing with the Euler class a λj obviously results in S λ k, and by our assumtions on k, this ma is essential (and in fact, torsion free). Smashing this with H roduces an essential ma which we will also call by the same name a λk : S λj H S λ k H. a λj Our chain comlex models show that the classes u λ,..., u λn and a λ,..., a λn generate as an algebra the ortion of the RO(C n)-graded homotoy of H in dimensions of the form k V. There is a relation which is immediate from our fractional classes. Theorem 3.5. In the RO(C n)-graded homotoy of H, we have a relation a λk u λj = k j a λj u λk. Proof. If we comose a λk /a λj with the ma u λj /u λk, then we get a ma u λj a λk : S λj H S λj H. u λk a λj This ma is comletely determined by the effect in underlying homotoy, and this is obviously the ma of degree k j. Clearing denominators gives the desired result. Remark 3.6. There is only one other kind of relation in this sector of the RO(C n)-graded homotoy: n j a λj =. This follows from the above remarks, as the restriction to C j of a λj is nullhomotoic. Comosing with the transfer yields multilication by n j in homology. The cellular models for reresentation sheres show that no other relations are ossible: all homology grous are cyclic. H k V (S ; )

12 2 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL 3.3. Fiber sequences realizing HB k,j. If we aly the Eilenberg-Mac Lane functor to the short exact sequence (k + j, j) B k,j, then we get a fiber sequence of sectra. Combing this with equivalence from Theorem 3. yields a fiber sequence Σ λ k λ j H H HB k,j. Since B k,j is zero on cells induced u from subgrous of C j, for all i j, the ma a λi induces an equivalence HB k,j Σ λi HB k,j. Smashing our fiber sequence with S λj then gives a more readily interreted form. Proosition 3.7. There is a fiber sequence Σ λ k H Σ λj H HB k,j. Proosition 3.7 is the heart of our analysis of the slice filtration for these RO(G)-graded susensions: we can stri away coies of S λj, relacing them with coies of S λ k for k > j. Before continuing, we note two more natural fiber sequences which tie in the dual Mackey functors. Proosition 3.8. We have fiber sequences and Σ λ k λ H Σ λj λ H B k,j Σ λn λj H Σ λn λ k H B k,j. 4. Bredon homology comutations We will see in the 5 below that each slice for s λ H has the smash roduct of a certain reresentation shere with some HB k. The relevant homotoy grous will be given in Theorem gives some rearatory calculations for this result. 4.2 gives some results that will hel us determine the slices in Bredon homology and B k,j Exact Sequences involving B. A small initial observation is needed. The Mackey functors B (k,j do not really deend on C n in the sense that once n is at least (k + j), the value for the Mackey functors on orbits stabilizes. Thus in the short exact sequences that follow, it is helful to retend they are written as Mackey functors for the Prüfer grou C and then restrict the result to any chosen C n. The net result is that if any index is greater than n, just set it equal to n in the formulas. To comute the E 2 -age of the slice sectral sequence for S mλ, we need to determine H (S V ; B k,j ) for various V. For this, we make several additional observations. If we let φ j denote the quotient ma C n C n/c j, then there is an associated inflation (aka ullback) functor φ j : C n/c j -Mackey C n-mackey. The Mackey functors B k,j are all in the image of φ j, and this is an exact functor. For us, it simly inserts zeros at the bottom of a Lewis diagram.

13 SLICES AND RO(G)-GRADED SUSPENSIONS I 3 Proosition 4.. We have natural isomorhisms φ j B k,m = B k,m+j and φ j Bl k,m = B l k,m+j. It therefore suffices to consider the case of j =. We then have the following generalization of Proosition 2.3, and the roof is a direct comutation. Proosition 4.2. We have an exact sequence B min(l,k), n n γ B l l k, n n B l l k, B l k k,. Proosition 4.2 is the initial inut for comuting the Mackey functor cellular homology. The ma γ is the attaching ma of the even cells in S V to the odd cells when we use our minimal cell model. Thus the cellular homology is comletely determined by looking at the mas from the cokernels in Proosition 4.2 to the kernels as l varies. We have two intermediate lemmata which serve as the work-horse for any comutations. These are immediate comutations, and in both cases, the middle ma is an isomorhism when we restrict to the subgrou for which both middle Mackey functors coincide. Lemma 4.3. For l k, we have an exact sequence B min(k,k+l),max(k,k+l) B l k, B k, B k,k+l in which the middle ma is the identity when restricted to C k+l. Pulling this back gives what haens for a general air (k, j): Corollary 4.4. For any j and for l k, we have an exact sequence B min(k,k+l),max(k,k+l)+j B l k,j B k,j B k,k+l+j in which the middle ma is the identity when restricted to C k+l+j The Bredon homology of S V with coefficients in B k,j. We can slice all of the exact sequences from the revious subsection to determine the Bredon homology of any reresentation shere with coefficients in B k,j. This section is only relevant for readers who wish to run the slice sectral sequence, as it exlains how to comletely determine the slice E 2 -term for S λ H and for S mλ H. First, we reduce to the cases most of interest. Proosition 4.5. Let W be a reresentation of C n, let W j = W C j+, and let U j be the orthogonal comlement of W j in W. Then the natural inclusion a Uj : S Wj S W induces an equivalence uon smashing with HB k,j. Proof. By assumtion, the stabilizer of every oint of U j {} is roerly contained in C j+. We therefore conclude that we can choose a cellular model for S Uj such that the zero skeleton is S and every cell of dimension greater than zero is induced from a roer subgrou of C j+. Since the restriction of B k,j to any roer subgrou of C j+ is zero, we conclude that the Euler class a Uj is an equivalence uon smashing with HB k,j. Thus the Bredon homology comutation we care about cannot distinguish between S W and its C j+-fixed oints. It is also immediate that the chain comlex

14 4 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL comuting the Bredon homology is in the image of the ullback functor, so without loss of generality, we may assume that j =. It is also clear that we may, without loss of generality, restrict attention to fixed oint free reresentations of C n, since the inclusion of a trivial summand serves only to shift the homology grous around. Notation 4.6. Let V be a fixed-oint free reresentation of C n that restricts trivially to C. Then find a JO-equivalent reresentation W, and write W = <r n Define natural numbers K i for i > by K i = 2 k r λ n r, i k r, and let K =. Finally, let < i < < i m denote those integers i such that k i, and let h r = n i r. Since the homology grous of any sace are bound between its connectivity and its dimension, we need only determine the homology grous between dimensions and K m. Theorem 4.7. The Mackey functor valued Bredon homology grous of S V coefficients in B k are given by s <, B k,h s = B min(h,k),max(h,k) s = H s (S V ; B k ) = B min(k,n i),n i K i + 2 s K i 2, i even B min (k,n i),n i K i + 3 s K i, i odd B min (k,hr),h r+ s = K ir B min (k,hr+),max(min(k,h r),h r+) s = K ir +. Proof. This follows immediately from our cell structure of Proosition.2 and the short exact sequences above. Alying the Mackey functor B k to the cellular comlex results in a chain comlex of Mackey functors: r= with B k h h B k γ h h B k γ h h B h k h B k hm hm B k (for ease of exosition, we have assumed that k i 2). By our assumtions on V, we know that the sole instance of B k lies in dimension, and the comlex is homologically graded. In articular, there is the ma γ from every ositive even degree to the adjacent odd degree, u through dim V. Proosition 4.2 allows us to identify the kernels and cokernels of the mas labeled γ, letting us relace the

15 SLICES AND RO(G)-GRADED SUSPENSIONS I 5 cellular chains with a simler comlex: B k h h B k γ h h B k h h B k γ h h B k... B h k k, B min(h,k), B h k k, B min(h,k),... Having controlled the differential from even degrees to odd degrees, it remains only to understand the mas from the cokernels of the various γs to the kernels. This is exactly what Lemma 4.3 records. The result is then a simle alication of this lemma and counting A criterion for slice codimension. In determining slice dimensions, we will need a fact about the Bredon cohomology for a few reresentation sheres. In what follows, let H k (X; M) denote the Mackey functor valued Bredon cohomology of X with coefficients in M. This can be comuted by choosing an equivariant cellular model for X and building the cellular co-chain comlex which assigns to an n-cell with stabilizer H the Mackey functor G G H H M. The mas are induced by the cellular attaching mas in the standard way. Equivalently, we can simly build a cellular model for DX, the Sanier-Whitehead dual of X, and comute the negative Bredon homology of DX. We need this only for X a reresentation shere and M a -module, and even there, we need very few secific comutations. Lemma 4.8. Let V be a reresentation of C n. If λ m is a summand of V, then for every M for which res l m is an isomorhism for l m, H (S V ; M) = H (S V ; M) =. Proof. If λ n is a summand, then connectivity imlies the result. The stabilizers of the comlement of the origin in λ j grows as j does, and our cell models show that the bottom skeleton of S V is that of S λj for j maximal amongst those summands that occur in V. Without loss of generality, we may assume j = m, and we need only comute H ɛ (S λm ; M). Our standard model for D(S λm ) = S λm is (C n/c m) + S 2 (C n/c m) + e e. with coef- This gives us the following cochain comlex for the cohomology of S λm ficients in any Mackey functor M: C (S λm ; M) : M res n m n m n m (M) γ n m n m (M), dim 2 and the first ma is () the comosite of the restriction ma to C m and the diagonal for subgrous of C n which contain C m, and (2) the diagonal for subgrous of C m.

16 6 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL For subgrous of C m and for any M, the sequence is obviously exact. This is really a restatement of the fact that λ m restricts to the trivial reresentation for C m. For the remaining subgrous, we observe that the kernel of the ma denoted ( γ) looks like the constant Mackey functor M(C n/c m) (the value for subgrous of C m is of course a riori different, but those are already understood). Thus we have an exact sequence for the cohomology of S λm with coefficients in M evaluated on C n/c l for l m: H (S λm ; M)(C n/c l) M(C n/c l) res l m M(C n/c m) Corollary 4.9. If λ m is a summand of V for m k, then H (S V ; B k ) = H (S V ; B k ) =. H (S λm ; M)(C n/c l). 5. The slices of S λ H Let L = S λ H. We will see that its slice tower is determined by a sequence of reresentations. The cofiber sequences in the revious section, together with a surrisingly simle induction argument, show that the naturally occurring tower is the slice tower. From this section on, let > A sequence of reresentations. The basic argument is that we will stri away coies of λ from L, relacing them with other irreducible reresentations until we roduce a coy of 2ρ. Since the slice tower commutes with ρ-susensions, this will give us an iterative, eriodic aroach. The sequence of reresentations is curiously simle. Definition 5.. For all j, let and let V =. V j = j λ(2m ), m= The first thing to observe is that V n = 2ρ (we run through a comlete set of coset reresentatives). Since λ(k + n ) = λ(k) as reresentations of C n, we therefore conclude that V j+ n = 2ρ + V j for all j n. We also have V (n ±)/2 = ρ ± and V n j = 2ρ V j for < j < n. The following formula for V j may be useful. Theorem 5.2. Floor function formula for V j. Let our grou be C n for an odd rime. For l, let c l = ( l )/2. Then V j = j + l+ c l l t j + cn + 2. l<n λ l t t c l+ n

17 SLICES AND RO(G)-GRADED SUSPENSIONS I 7 Proof. By definition, V = and V j = V j + λ(2j ) = V j + λ v(2j ) We will illustrate with the case = 3, for which c =. The l = term in our sum is λ j + 3 c t t 2 t 3 ( j + 2 = λ + 3 ) j 3 Increasing j by increases the coefficient of λ when j is congruent to or 3 mod 3, namely the times when v 3 (2j ) = and we are adding a coy of λ to V j. Similarly the l = term λ j + 9 c 3t t 2 t 9 ( j + 7 = λ + 9 ) j + 9 Increasing j by increases the coefficient of λ when j is congruent to 2 or 8 mod 9, namely the times when v 3 (2j ) = and we are adding a coy of λ to V j. The same thing haens for larger l and larger rimes. When we get to l = n, we have λ n = 2 by definition. It will be added to V j when ever 2j is divisible by n, which is equivalent to j being congruent to c n modulo n. The final coefficient, (j + c n )/ n increases for recisely such j. Corollary 5.3. Coefficients in cases of interest. Let j = (a )/2 for odd a >, and let a = 2b +. Then the coeffcient k n l of λ l in the formula of Theorem 5.2 is k n l = ( )b + c for l = b + l c l l t for < l < n t t c b + c + c n n l for l = n. In the case < l < n we have b ( ) l k n l ( ) b + l Note that k is the coefficient of λ n, which by definition is the trivial reresentation of degree two, l.

18 8 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Proof. The coefficient k n of λ for j = (a )/2 with a odd is k n = j + t t t c = t t c = t t c = t t c a t ( a 2 ( b ) 3 3 2t 2 ) 3 2t 2 = ( )(b + ) c = ( )b + c = c a. That of λ l for < l < n is k n l = (a )/2 + l+ c l l t t t c = t t c = t t c l+ b + c + l+ c l l t l+ b + l c l l t l since c c l = c l. Since we have < + c l + l t < l for each t, b ( ) l k n l ( ) We leave the case of k to the reader. b + l l. This will let us simlify several arguments Secial slices. We are now in a osition where we can show that all of the RO(G)-graded susensions of HB j which arise are in fact slices. We begin with several simle theorems about V j -fold susensions. Theorem 5.4. For any Y which is ( )-connected, S Vj Y is slice greater than or equal to (2j). Proof. Since ( )-connected sectra are slice non-negative and since smashing with a slice non-negative sectrum at worst reserves slice connectivity, it suffices to show that S Vj is slice greater than or equal to (2j). Here we can invoke Proosition.6. We first observe that the restriction of V j to any subgrou of C n is the corresonding V j for that subgrou. Thus we may use induction on the order of the grou in

19 SLICES AND RO(G)-GRADED SUSPENSIONS I 9 a very simle away. For j < n let ρ j n W j = 2 3ρ j n +, 2 then V j W j and Vj G = Wj G. By induction, for all roer subgrous H, i H SVj is slice greater than or equal to (2j), and therefore Proosition.6 imlies that desired result. Corollary 5.5. For any Mackey functor M, S Vj HM is slice greater than or equal to (2j). As is often the case, showing that the desired sectra are slice less than or equal to (2j) is by direct comutation. We will reduce the comutation to Corollary 4.9. Theorem 5.6. For every M in which res l m is an isomorhism for m k := v (2j + ), S Vj HM is slice less than or equal to (2j). Here v (2j + ) denotes the number of owers of dividing 2j +. Proof. We need to show that for all triles (r, H, ɛ) such that r H ɛ > 2j, we have [G + H S rρh ɛ, Σ Vj HM] G = [S rρh ɛ, Σ Vj HM] H =. We can again aeal to the linear ordering of the subgrous, breaking them into two cases. First, observe that if 2j + mod k, then j k 2 mod k. This means that i C V k j = bρ + ρ, for some b. Hence for any Mackey functor M, i C k ΣVj HM = Σ bρ+ ρ Hi C k M is a 2j-slice for C k. This is the essential feature of the argument, as it shows that the slice uer bound on this sectrum is determined by the slice uer bound for those subgrous of G which roerly contain C k. The condition that all restriction mas are injections is the same for all subgrous in this range, so without loss of generality, we need only show [S rρ G ɛ, Σ Vj HM] G =. For the rest of this roof we will denote ρ G by simly ρ. It suffices to show this for j n, as larger values of j result in ρ-fold susensions and these commute with slice dimension. We therefore only have to consider slice cells S rρg ɛ, where r n ɛ > 2j. There are two cases, deending on whether or not we have assed the secial n 2. value of j: For j n 2, we know that V j S ρ, with equality iff j = n 2. We therefore must comute [S rρ ɛ, Σ Vj HM] = [S (r ) ρ+v j +(r ɛ), HM], where Vj is the orthogonal comlement of V j in ρ. If r ɛ >, then the connectivity of the domain exceeds the coconnectivity of the range, and therefore all homotoy classes are zero. We ause here to note that this includes the excetional value of j, as here r = ɛ = results in S rρ ɛ being a 2j-slice cell.

20 2 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL For the remaining cases, we assume that r = ɛ = and j < n 2. By assumtion on j, now, the reresentation λ(2j + ) Vj. The corresonding shere is JO-equivalent to the shere of λ v(2j+). We are therefore comuting H ( S V j ; M ), and by Lemma 4.8, this grou is zero. For n 2 < j < n, we have a similar analysis. Here ρ V j 2ρ. Assume that r n ɛ > 2j, as before. In articular, we see that r 2. We again consider [S rρ ɛ, Σ Vj HM] = [S (r 2) ρ+v j +(r 2) ɛ, HM], where Vj is the orthogonal comlement of V j in 2ρ. If (r 2) ɛ is ositive, then connectivity finishes the roof. We need to consider (r 2) ɛ being or (the latter only occurring for j < n ), and this reduces the comutation to that of H ɛ (S V j +(r 2) ρ ; M). If r 3, then all λ i are summands of ρ, and Lemma 4.8 immediately imlies these grous are zero. If r = 2, then just as before, we observe that Vj contains the reresentation λ(2j + ), and so Lemma 4.8 again imlies that these grous are zero. Finally we observe that since v (2 n + ) =, the conditions of the theorem require that all restrictions mas be injections. This means that HM is a zero slice, and hence Σ Vj HM = Σ 2ρ HM is a 2 n -slice. Combining these theorems yields a number of slices. Corollary 5.7. For all j, the sectrum Σ Vj HB v(2j+) is a (2j)-slice. Corollary 5.8. For all j, the sectra Σ Vj H and Σ Vj H(k, l) for k v (2j + ) or l v (2j + ) are (2j)-slices. The condition on k and l above is that v (2j +) / (l, k), the oen interval from l to k The 2ρ-eriodic slice tower. The results of the revious section actually determine for us the slice tower. It is easiest to observe this via the 2ρ-susensions that showed u. As a corollary to the revious section, we have a tower Σ 2ρ L Σ 2ρ λ L Σ 2ρ λ HB.. Σ Vj L. Σ Vj HB v(2j+) L Σ HB

21 SLICES AND RO(G)-GRADED SUSPENSIONS I 2 Recall that HB is contractible, V n = 2ρ and V n = 2ρ λ. All of the layers on the right-hand side of the tower are slices by Corollary 5.7. In articular, they are all simultaneously less than or equal to and greater than or equal to the dimension of the associated reresentation shere. The final one (the one in the uer right corner) is then less than or equal to (2 n 2). Now the 2ρth susension of L is greater than or equal to 2 n, and we therefore conclude that all of the cofiber sequences are exactly the cofiber sequences P n (X) X P n (X) for various sectra X of the form S V L. Slicing them all together, using the obvious 2ρ-eriodicity of the tower, we see that we have determined the slice co-tower of L. We grou this together in the following theorem. Theorem 5.9. All odd slices and all slices in dimensions not congruent to modulo of L are contractible. The (a k )-slice, where a is odd and rime to, is given by Σ Vj HB k where j = (a k )/2, and the mas in the tower are all determined by the cofiber sequences Σ Vj+ H u λ/u λ(2j+) Σ Vj+λ H Σ Vj HB v(2j+). We leave it to the interested reader to state a corollary to Theorem 4.7 as it alies to the Σ Vj HB k here. In articular it would say that the to dimension for the homology of the a k -slice is a k. This means that in the usual slice sectral sequence chart all nontrivial elements occur between lines of sloes and n meeting at (s, t) = (, ). Remark 5.. Yarnall s thesis shows that the slice sections of S n H are all of the form S V H. This result is of a different flavor. Our result here is that the slice connective covers P n L are all reresentations sheres smashed with H. This is a curious and confusing fact. We will now aly Theorem 4.7 to determine the homotoy grous of the slices described by Theorem 5.9. Figure 2 shows the sectral sequence for C 27, subect to the following regrading convention. Under the usual convention, meaning the oint (s, t) shows the Mackey functor E s,t+s 2, all nontrivial elements would lie between lines of sloes 2 and 26 intersecting at (s, t) = (, ), meaning (x, y) = (, ). In order to save sace we rescale in such a way that the vanishing lines have sloe and 8 and the horizontal coordinate is unchanged. In the general case this means [ [ ] x y] = s t s (t + ) This change rescales differentials as well, converting d +2r for r > (the only ones that can occur dues to sarseness) to d +2r. The figure makes use of Mackey functor symbols indicated in Table. The dashed lines are non-trivial extensions determined by hidden transfers. The reader may construct similar charts for other cyclic grous using the information in Corollary 5.3..

22 22 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL Table. Mackey functors aearing in Figures 2 and 3. B = B, B, B, B,2 = B,2 = /3 /3 /3 /3 /3 /3 B 2 = B 2, B 2, B 3 = B 3, /9 /9 /27 /9 /3 3 3 /3 3 /3 /9 /3 3 3 / The slices of S mλ H The slice tower for L = S λ H almost comletely determines the one for S mλ H. We first bound the slice tower of S mλ H from above. Theorem 6.. For all m, S mλ H is less than or equal to 2m. Proof. Since S mλ H restricts to the same sectrum for any subgrou, by induction on the order of G, we know that the restriction is less than or equal to 2m. This means that we need only check that for all k and ɛ such that k n ɛ > 2m, [S kρ ɛ, S mλ H] = [S, S mλ kρ+ɛ H] =. We again can show this using a cell decomosition of S mλ and then smashing it with S kρ+ɛ. This gives ( S (C n + e ) (C n + e 2m ) ) S kρ+ɛ H = ( S kρ+ɛ (C n + e kn +ɛ ) (C n + e 2m kn +ɛ ) ) H. The only ways this could have homotoy in dimension are ossibly from the first term (when k = ɛ = ) or from the final term in the decomosition. Since 2m k n + ɛ < by assumtion, all cells excet the first are -coconnected. The standard comutations show that the first term also contributes no π. We therefore know we need only determine the slices u to the 2m th slice. Our comutation for L actually does most of this. Let F m be the fiber of the natural inclusion S mλ H L. Theorem 6.2. The sectrum F m is greater than or equal to 2m.

23 SLICES AND RO(G)-GRADED SUSPENSIONS I Figure 2. The slice sectral sequence for S λ H for G = C 27. Proof. The long exact sequence in homotoy for the fiber sequence shows that F m S mλ H L s < 2m, s = 2m, π s (F m ) = s = 2k > 2m, B(n, ) s = 2k + > 2m. Since the sectrum F m is (2m )-connected, we know that it is greater than or equal to 2m. Corollary 6.3. The natural ma S mλ H S λ H induces an equivalence P 2m ( S mλ H ) P 2m ( L ). We therefore know all of the slices of S mλ H below the 2m th slice, and we also know that we have a single remaining slice. Moreover, we know the mas down to the slice sections, since they are the comosites S mλ H L P j( L ).

24 24 M. A. HILL, M. J. HOPKINS, AND D. C. RAVENEL We therefore have the slice tower. The fiber of the ma S mλ H P 2m ( S mλ H ) is the (2m)-slice, since it is simultaneously greater than or equal to 2m and less than or equal to 2m. The analysis of the slice tower for L also identifies this with a reresentation shere for us, as determined by the analysis in the revious section. Considering the actual fiber sequences for the slice cotower for L shows us what the (2m)-slice is for S mλ H. The determination of the ma is immediate from the consideration of the layers of the tower (and the descrition of striing off coies of λ and relacing them with λ(2j ) for j at most m). We simlify the resentation by taking advantage of the fact that H is a commutative ring sectrum. Theorem 6.4. For all m, the (2m)-slice of S mλ H is S Vm H. The ma from S Vm H to S mλ H is simly m j= u λ(2j ) u λ = u V m u mλ As a closing remark, we note a fact we found initially quite curious. If is large relative to m, then S mλ H is in fact a (2m)-slice. It takes a while before the slice tower of S mλ H becomes more comlicated. To illustrate how this lays out, we show in Table 2 the J-equivalence classes of V m for = 3, n = 2 and m. Table 2. V j for C 27 and j 27. Note that V 27 j = 2ρ V j. j V j j V j j V j λ 2 λ + λ 3 2λ + λ 4 3λ + λ 5 3λ + λ + λ 2 6 4λ + λ + λ 2 7 5λ + λ + λ 2 8 5λ + 2λ + λ 2 9 6λ + 2λ + λ 2 7λ + 2λ + λ 2 7λ + 3λ + λ 2 2 8λ + 3λ + λ 2 3 9λ + 3λ + λ 2 4 9λ + 3λ + λ λ + 3λ + λ = ρ = ρ + = ρ + λ + 6 λ + 3λ + λ λ + 4λ + λ λ + 4λ + λ λ + 4λ + λ λ + 5λ + λ λ + 5λ + λ λ + 5λ + λ λ + 5λ + 2λ λ + 5λ + 2λ λ + 5λ + 2λ λ + 6λ + 2λ λ + 6λ + 2λ = 2ρ λ λ = 2ρ λ = 2ρ As an examle, we resent the slice sectral sequence for S 8λ H in Figure 3. In this, as before, a dot reresents a coy of B,, a circle indicates B 2,, and underlining increases the second subscrit. Here we also have circled circles, denoting B 3,, and the box indicates a coy of. Asterisks reresent the dual to the named Mackey functor. References [Dug5] Daniel Dugger. An Atiyah-Hirzebruch sectral sequence for KR-theory. K-Theory, 35(3-4): (26), 25.

25 SLICES AND RO(G)-GRADED SUSPENSIONS I Figure 3. The slice sectral sequence for S 8λ H and C 27. [HH] Michael A. Hill and Michael J. Hokins. Equivariant multilicative closure. Online at htt://arxiv.org/abs/arxiv: v2 and on the first author s home age. [HHR] Michael A. Hill, Michael J. Hokins, and Douglas C. Ravenel. The non-existence of elements of Kervaire invariant one. Online at htt://arxiv.org/abs/arxiv: v2 and on the third author s home age. [HHR] Michael A. Hill, Michael J. Hokins, and Douglas C. Ravenel. The Arf-Kervaire roblem in algebraic toology: sketch of the roof. In Current develoments in mathematics, 2, ages 43. Int. Press, Somerville, MA, 2. [Hil] Michael A. Hill. The equivariant slice filtration: A rimer. Online at htt://arxiv.org/abs/arxiv:7.3582v2 and on the author s home age. [Voe2] Vladimir Voevodsky. Oen roblems in the motivic stable homotoy theory. I. In Motives, olylogarithms and Hodge theory, Part I (Irvine, CA, 998), volume 3 of Int. Press Lect. Ser., ages Int. Press, Somerville, MA, 22. Deartment of Mathematics, University of Virginia, Charlottesville, VA address: mikehill@virginia.edu Deartment of Mathematics, Harvard University, Cambridge, MA 238 address: mjh@math.harvard.edu Deartment of Mathematics, University of Rochester, Rochester, NY address: doug@math.rochester.edu