Topological resolutions in K(2)-local homotopy theory at the prime 2
TTOPOLOGICAL RESOLUTIONS IN K (2) -LOCAL HOMOTOPYTHEORY AT THE PRIME IRINA BOBKOVA AND PAUL G. GOERSS
Abstract.
We provide a topological duality resolution for the spectrum E h S ,which itself can be used to build the K (2)-local sphere. The resolution is builtfrom spectra of the form E hF where E is the Morava spectrum for the formalgroup of a supersingular curve at the prime 2 and F is a finite subgroup of theautomorphisms of that formal group. The results are in complete analogy withthe resolutions of [GHMR05] at the prime 3, but the methods are of necessityvery different. As in the prime 3 case, the main difficulty is in identifying thetop fiber; to do this, we make calculations using Henn’s centralizer resolution. Chromatic stable homotopy theory uses the algebraic geometry of smooth one-parameter formal groups to organize calculations and the search for large scalephenomena. In particular, the chromatic filtration on the category of the p -localfinite spectra corresponds to the height filtration for formal groups. The layers of thechromatic filtration are given by localization with respect to the Morava K -theories K ( n ), with n ≥ . Thus, to understand the homotopy type of a finite spectrum X we begin by addressing L K ( n ) X for all prime numbers p and all 0 ≤ n < ∞ . A useful and inspirational guide to this point of view can be found in the table insection 2 of [HG94].If n = 0 , K (0) = H Q and L X is the rational homotopy type of X. For n ≥ , thebasic computational tool in K ( n )-local homotopy theory is the K ( n )-local Adams-Novikov Spectral Sequence H s ( G n , ( E n ) t X ) = ⇒ π t − s L K ( n ) X. Here G n is the automorphism group of a pair ( F q , Γ n ) where F q is a finite field ofcharacteristic p and Γ n is a chosen formal group of height n over F q . Then E n isthe Morava (or Lubin-Tate) E -theory defined by ( F q , Γ n ). We will give more detailsand make precise choices in section 1.First suppose p is large with respect to n . (To be precise, we may take 2 p − > max { n , n + 2 } or p > n =2.) Then the Adams-Novikov Spectral Sequence for X = S collapses and will have no extensions, so the problem becomes algebraic,although by no means easy. See for example, [SY95], [Beh12], or [Lad13], for thecase n = 2 and p >
3. However, if p is small with respect to n , the group G n has finite subgroups of p -power order and the spectral sequence will usually havedifferentials and extensions, so the problem is no longer purely algebraic. At this The second author was partially supported by the National Science Foundation. A portionof the work was done at the Hausdorff Institute of Mathematics, during the Trimester Program“Homotopy Theory, Manifolds, and Field Theories.” a r X i v : . [ m a t h . A T ] J a n IRINA BOBKOVA AND PAUL G. GOERSS point, topological resolutions become a useful way to organize the contributions ofthe finite subgroups. The key to unlocking this idea is the Hopkins-Miller theorem,which implies that G n , and hence all of its finite subgroups, act on E n and that L K ( n ) S (cid:39) E h G n n . See [DH04] for this and more.The prototypical example is at n = 1 and p = 2. Adams and Baird [Bou79], andRavenel [Rav84] showed that here we have a fiber sequence L K (1) S → KO ψ − −−−→ KO where KO is 2-complete real K -theory. For a suitable choice of a height one formalgroup Γ we can take E = K , where K is 2-complete complex K -theory. Then G = Aut (Γ , F ) ∼ = Z × is the units in the 2-adic integers, and C = {± } ⊆ Z × acts through complex conjugation. We can then rewrite this fiber sequence as L K (1) S = E h G −→ E hC ψ − −−−→ E hC , and ψ is a topological generator of Z × / {± } (cid:39) Z .For higher heights the topological resolutions will not be simple fiber sequences,but finite towers of fibrations with the successive fibers built from E hF i where F i runs over various finite subgroups of G n .In [GHMR05] the authors generalized the fiber sequence of the K (1)-local caseto the case n = 2 and p = 3. One way to say what they proved is the following. Letus write E = E to simplify notation. First we have a split short exact sequence ofgroups { } → G (cid:47) (cid:47) G N (cid:47) (cid:47) Z → { } where N is obtained from a determinant (See (1.7)) and hence a fiber sequence L K (2) S −→ E h G ψ − −−−→ E h G , where ψ is any element of G which maps by N to a topological generator of Z .Then, second, there exists a resolution of E h G E h G → E hG → Σ E hSD → Σ E hSD → Σ E hG Here resolution means each successive composition is null-homotopic and all possibleToda brackets are zero modulo indeterminacy; thus, the sequence refines to a towerof fibrations E h G (cid:47) (cid:47) X (cid:47) (cid:47) X (cid:47) (cid:47) E hG Σ E hG (cid:79) (cid:79) Σ E hSD (cid:79) (cid:79) Σ E hSD (cid:79) (cid:79) The maximal finite subgroup of G of 3-power order is a cyclic group C oforder 3; it is unique up to conjugation in G . The group G is the maximal finitesubgroup of G containing C . The subgroup SD is the semidihedral group oforder 16. Because of the symmetry of this resolution, and because G is a virtualPoincar´e duality group of dimension 3, this is called a duality resolution . Thisresolution and related resolutions were instrumental in exploring the K (2)-localcategory at primes p >
2. See [GHM04], [HKM13], [Beh06], [GHM14], [GHMR15],
OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 3 [GH16], and [Lad13]. The latter paper makes a thorough exploration of whathappens at p > n = 2 and p = 2. The prime 2 is much harder for a number of reasons. First, themaximal finite 2-subgroup of G is not cyclic, but isomorphic to Q , the quaterniongroup of order 8. Second, every finite subgroup of G that we consider containsthe central C = {± } of G ; therefore, the homotopy groups of the relevant fixedpoint spectra E hF are much more complicated. This means that the strategy ofproof of [GHMR05] won’t work and we need to find another way.We now state our main result. As our chosen formal group Γ we will use theformal group of a supersingular elliptic curve over F . The curve will be definedover F , so that G ∼ = S (cid:111) Gal( F / F ) where S = Aut(Γ / F ) is the group ofautomorphisms of Γ over F . Once again we have fiber sequence L K (2) S −→ E h G ψ − −−−→ E h G . We have E h G = ( E h S ) h Gal where Gal = Gal( F / F ) and S = S ∩ G . For manycomputational applications, the difference between E h G and E h S is innocuous.See Lemma 1.37. Then our main result is this: Theorem 0.1.
There exists a resolution of E h S in the K (2) -local category at theprime 2 E h S → E hG → E hC → Σ E hC → Σ E hG . The spectrum E hC is 48-periodic, so E hC (cid:39) Σ E hC ; this suspension is thereonly to emphasize the symmetry in the resolution.Once again resolution means each successive composition is null-homotopic andall possible Toda brackets are zero modulo indeterminacy; thus, the sequence againrefines to a tower of fibrations. This result has already had applications: it is aningredient in Agn`es Beaudry’s analysis of the Chromatic Splitting Conjecture at p = n = 2. See [Bea17].The apparent similarity of Theorem 0.1 with the prime 3 analog is tantalizing,especially the suspension factor on the last term, but we as yet have no conceptualexplanation. The proof at the prime 2 can be adapted to the prime 3 but in bothcases it comes down to a very specific, prime dependent calculation.A very satisfying feature of chromatic height 2 is the connection with the theoryof elliptic curves. The subgroup G ⊆ S is the automorphism group of our chosensupersingular curve inside the automorphisms of its formal group. Appealing toStrickland [Str] we can use this to get formulas for the action of G on E ∗ , anecessary beginning to group cohomology calculations. Furthermore, we know that E hG (cid:39) ( E hG ) h Gal (cid:39) L K (2) T mf where
T mf is the global sections of the sheaf of E ∞ -ring spectra on the com-pactified stack of generalized elliptic curves provided by Hopkins and Miller. See[DFHH14]. Similarly, E hC is the localization of global sections of the similar sheaffor elliptic curves with a level 3 structure. See [MR09]. We won’t need anything IRINA BOBKOVA AND PAUL G. GOERSS like the full power of the Hopkins-Miller theory here, although we do use some ofthe calculations that arise from this point of view. See Section 2.We will prove Theorem 0.1 in three steps.First we prove, in Section 3, that there exists a resolution E h S → E hG → E hC → E hC → X where E ∗ X ∼ = E ∗ E hG as twisted G -modules. This resolution and the necessary algebraic preliminarieswere announced in [Hen07], and grew out of the work surrounding [GHMR05]. Thealgebraic preliminaries are discussed in detail in [Bea15].Second, in section 4 we examine the Adams-Novikov Spectral Sequence H ∗ ( G , E ∗ ) = ⇒ π ∗ X. Using a comparison of our resolution with a second resolution, due to Henn, weshow, roughly, that certain classes ∆ k +2 ∈ H ( G , E k +48 ) are permanentcycles—which would certainly be necessary if our main result is true. The ex-act result is in Corollary 4.7. These calculations were among the main results inthe first author’s thesis [Bob14] and the key ideas for the entire project can befound there. This ratifies a comment of Mark Mahowald that there is a class in π L K (2) S which supports non-zero multiplications by η and κ and the only waythis could happen is if ∆ is a permanent cycle. This insight was, we think, the re-sult of long hours of contemplation of the results of Shimomura and Wang [SW02]and, indeed, one reason for this entire project is to find some way to catch upwith those amazing calculations. We will have more to say about this element ofMahowald’s in Remark 4.8.Third and finally, in section 5 we use a variation of this same comparison argu-ment to produce the equivalence Σ E hG (cid:39) X . See Theorem 5.8. At the very endwe add a remark about the possibility or impossibility of resolutions for L K (2) S itself. See Remark 5.10.There is a number of sections of preliminaries. Section 1 provides the usualbackground on the K ( n )-local category as well as some more specific informationon mapping spaces. Section 2 pulls together what we need about the homotopygroups of various fixed point spectra. This draws from many sources, and we tryto be complete there.An essential ingredient in our argument is the existence of the other topologicalresolution for E h S , Henn’s centralizer resolution from § OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 5 Acknowledgements.
There is a wonderful group of people working on K (2)-local homotopy theory and this paper is part of an on-going dialog within thatcommunity. We are grateful in particular to Agn`es Beaudry, Charles Rezk, andespecially Hans-Werner Henn. Of course, others have helped as well. In particular,the key idea—to look for homotopy classes with particularly robust multiplicativeproperties—is due to Mark Mahowald. We miss his insight, irreplaceable knowl-edge, and friendship every day. Finally, we would like to give a heartfelt thanks tothe referee, whose thorough and thoughtful report made this a better paper. Contents
1. Recollections on the K ( n )-local category 52. The homotopy groups of homotopy fixed point spectra 173. Algebraic and topological resolutions 274. Constructing elements in π k +48 X Recollections on the K ( n ) -local category We begin with the standard material on the K ( n )-local category, the Moravastabilizer group, and Morava E -theory, also known as Lubin-Tate theory. We thenget specific at n = 2 and p = 2, discussing the role of formal groups arising fromsupersingular elliptic curves. We review some material on the homotopy type ofthe spectrum of maps between various fixed point spectra derived from Morava E -theory and then spell out some details of the K ( n )-local Adams-Novikov SpectralSequence. Finally, we discuss, the role of the Galois group.1.1. The K ( n ) -local category. Fix a prime p and let n ≥ . Let Γ n be a formalgroup of height n over the finite field F p of p elements. Then for any finite extension i : F p ⊆ F q of F p , we form the group Aut(Γ n / F q ) of the automorphisms of i ∗ Γ n over F q . We fix a choice of Γ n with the property that any extension F p n ⊆ F q givesan isomorphism(1.1) Aut(Γ n / F p n ) ∼ = −→ Aut(Γ n / F q ) . The usual Honda formal group satisfies these criteria: this has a formal group lawwhich is p -typical and with p -series [ p ]( x ) = x p n . However, if n = 2, then the formalgroup of a supersingular elliptic curve defined over F p will also do, and this will beour preferred choice at p = 2. Define(1.2) S n = Aut(Γ n / F p n ) . IRINA BOBKOVA AND PAUL G. GOERSS
If we choose a coordinate for Γ n , then any element of S n defines a power series φ ( x ) ∈ x F p n [[ x ]] , invertible under composition, and the assignment φ ( x ) (cid:55)→ φ (cid:48) (0)defines a map S n −→ F × p n . This is a split surjection and we define S n to be the kernel of this map; this isthe p -Sylow subgroup of the profinite group S n . We then get an isomorphism S n (cid:111) F × p n ∼ = S n .Define the (big) Morava stabilizer group G n as the automorphism group of thepair ( F p n , Γ n ). Since Γ n is defined over F p , there is an isomorphism(1.3) G n ∼ = Aut(Γ / F p n ) (cid:111) Gal( F p n / F p ) = S n (cid:111) Gal( F p n / F p ) . We will often write Gal = Gal( F p n / F p ) when the field extension is understood.We next must define Morava K -theory. There are many variants, all of whichhave the same Bousfield class and define the same localization; we will choose avariant which works well with Morava E -theory. Let K ( n ) = K ( F p n , Γ n ) be the2-periodic ring spectrum with homotopy groups K ( n ) ∗ = F p n [ u ± ]and with associated formal group Γ n . Here the class u is in degree −
2. The group F = F × p n (cid:111) Gal( F p n / F p ) acts on K ( n ) and F p [ v ± n ] ∼ = ( K ( n ) hF ) ∗ = K ( n ) F ∗ where v n = u − ( p n − . The spectrum K ( n ) hF is thus a more classical version ofMorava K -theory.We will spend a great deal of time working in the K ( n )-local category and, whendoing so, all our spectra will implicitly be localized. In particular, we emphasizethat we often write X ∧ Y for L K ( n ) ( X ∧ Y ), as this is the smash product internalto the K ( n )-local category.We now define the Morava spectrum E = E n = E ( F p n , Γ n ). (We will suppressthe n on E n whenever possible to help ease the notation.) This is the complexoriented, Landweber exact, 2-periodic, E ∞ -ring spectrum with(1.4) E ∗ = ( E n ) ∗ ∼ = W [[ u , · · · , u n − ]][ u ± ]with u i in degree 0 and u in degree −
2. Here W = W ( F p n ) is the ring of Wittvectors on F p n . Note that E is a complete local ring with residue field F p n ; theformal group over E is a choice of universal deformation of the formal group Γ n over F p n . (We will be specific about this choice at n = p = 2 below in subsection1.2.) The group G n = Aut(Γ n ) (cid:111) Gal( F p n / F p ) acts on E = E n , by the Hopkins-Miller theorem [GH04] and we have (see Section 1.5) a spectral sequence for anyclosed subgroup F ⊆ G n ,(1.5) H s ( F, E t ) = ⇒ π t − s E hF . We will collectively call these by the name Adams-Novikov Spectral Sequence. Seealso Lemma 1.29 below. If F = G n itself, then E h G n (cid:39) L K ( n ) S and we arecomputing the homotopy groups of the K ( n )-local sphere. OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 7 Various subgroups of G n will play a role in this paper, especially at n = 2and p = 2. The right action of Aut(Γ n ) on End(Γ n ) defines a determinant mapdet : S n = Aut(Γ n / F p n ) → Z × p which extends to a determinant map(1.6) G n ∼ = S n (cid:111) Gal( F p n / F p ) det × (cid:47) (cid:47) Z × p × Gal( F p n / F p ) p (cid:47) (cid:47) Z × p . Define the reduced determinant (or reduced norm ) N to be the composition(1.7) G n N (cid:50) (cid:50) det (cid:47) (cid:47) Z × p (cid:47) (cid:47) Z × p /C ∼ = Z p . where C ⊆ Z × p is the maximal finite subgroup. For example, C = {± } if p = 2.There are isomorphisms Z × p /C ∼ = Z p and we choose one. Write G n for the kernel of N , S n = S n ∩ G n , and S n = S n ∩ G n . The map N : S n → Z p is split surjective andwe have semi-direct product decompositions for each of the groups G n , S n , and S n ;for example, there is an isomorphism S n (cid:111) Z p ∼ = S n . If n is prime to p , we can choose a central splitting and the semi-direct product isactually a product, but that is not the case of interest here.1.2. Deformations from elliptic curves.
Here we spell out what we need fromthe theory of elliptic curves at p = 2; this will give us a preferred formal group anda preferred universal deformation. Choose Γ to be the formal group obtained fromthe elliptic curve C over F defined by the Weierstrass equation(1.8) y + y = x . This is a standard representative for the unique isomorphism class of supersingularcurves over F ; see [Sil09], Appendix A. Because C is supersingular, Γ has height 2,as the notation indicates. Following [Str] let C be the elliptic curve over W ( F )[[ u ]]defined by the Weierstrass equation(1.9) y + 3 u xy + ( u − y = x . This reduces to C modulo the maximal ideal m = (2 , u ); the formal group G of C is a choice of the universal deformation of Γ .Again turning to [Sil09], Appendix A, we have(1.10) G
24 def = Aut( C / F ) ∼ = Q (cid:111) F × where F × ∼ = C acts on Q as the 3-Sylow subgroup of Aut( Q ). Define(1.11) G
48 def = Aut( F , C ) ∼ = Aut( C / F ) (cid:111) Gal( F / F ) . Since any automorphism of the pair ( F , C ) induces an automorphism of the pair( F , Γ ) we get a map G → G . This map is an injection and we identify G with its image. Remark 1.12.
Let C [3] be the subgroup scheme of C consisting of points oforder 3; over F , this becomes abstractly isomorphic to Z / × Z /
3. The group G acts linearly on C [3] and choosing a basis for the F -points of C [3] determinesan isomorphism G ∼ = GL ( Z / G ∼ = Sl ( Z / IRINA BOBKOVA AND PAUL G. GOERSS
Remark 1.13.
The following subgroups will play an important role in this paper.(1) C = {± } ⊆ Q ;(2) C = C × F × ;(3) C , any of the subgroups of order 4 in Q ;(4) G and G themselves.The subgroup C is not unique, but it is unique up to conjugation in G and in G . In particular, the homotopy type of E hC is well-defined. Remark 1.14.
We have been discussing G as a subgroup of S , but it can also bethought of as a quotient. Inside of S there is a normal torsion-free pro-2-subgroup K which has the property that the composition G −→ S −→ S /K is an isomorphism. Thus we have a decomposition K (cid:111) G ∼ = S . See [Bea15] fordetails. The group K is a Poincar´e duality group of dimension 4.1.3. The functor E ∗ X . We define E ∗ X = π ∗ L K ( n ) ( E ∧ X ) . Despite the notation, E ∗ ( − ) is not a homology theory, as it does not take arbitrarywedges to sums, but it is our most sensitive algebraic invariant on the K ( n )-localcategory.Here are some properties of E ∗ ( − ). See [HS99], § m ⊆ E be the maximal ideal and L = L be the 0th derived functorof completion at m . Recall that an E -module is L -complete if the natural map M → L ( M ) is an isomorphism. If M = L ( N ), then M is L -complete; that is, L ( N ) → L ( N ) is an isomorphism for all N . Thus the full subcategory of L -complete modules is a reflexive sub-category of all continuous E -modules. ByProposition 8.4 of [HS99], E ∗ X is L -complete for all X .If N is an E -module, there is a short exact sequence(1.15) 0 → lim k Tor E ( E /m k , N ) → L ( N ) → lim k N/m k N → . Hence if M is L -complete, then M is m -complete, but it will be complete andseparated only if the right map is an isomorphism. See [GHMR05] § E ∗ X is complete and separated. All ofthe spectra in this paper will meet these assumptions.Since G n acts on E , it acts on E ∗ X in the category of L -complete modules. Thisaction is twisted because it is compatible with the action of G n on the coefficientring E ∗ . We will call L -complete E -modules with such a G n -action either twisted G n -modules , or Morava modules.
For example, let F ⊆ G n be a closed subgroup. Then there is an isomorphismof twisted G n -modules(1.16) E ∗ E hF ∼ = map( G n /F, E ∗ ) OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 9 where map( − , − ) denotes the set of continuous maps. On the right hand side ofthis equation, E ∗ acts on the target and the G n -action is diagonal. This needs abit of care, as the proof of this result given in [DH04] is given for the Honda formalgroup and it may not be true in general. However, in analyzing the proof of thecrucial Theorem 2 of [DH04] we see that the isomorphism of (1.16) follows from themore basic isomorphism E E ∼ = map( G n , E ) . The standard proof of this isomorphism for the Honda formal group (see Theorem12 of [Str00], for example) requires only that our formal group be defined over F p and satisfy the stabilization requirement of (1.1). Remark 1.17.
We add here that Theorem 8.9 of [HS99] implies that the functor X (cid:55)→ E ∗ X = π ∗ L K ( n ) ( E n ∧ X )detects weak equivalences in the K ( n )-local category. Given a map f : X → Y ,the class of spectra Z so that L K ( n ) ( Z ∧ f ) is a weak equivalence is closed undercofibrations and retracts; hence if E is in this class, then L K ( n ) S is in this class.1.4. Mapping spectra.
We collect here some basics about the mapping spectra F ( E hF , E hF ) for various subgroups F and F of G n . Remark 1.18.
Let F ⊆ G n be a closed subgroup. We begin with the equivalence(1.19) E ∧ E hF (cid:39) map( G /F, E )from the local smash product to the localized spectrum of continuous maps. It ishelpful to visualize this map as sending x ∧ y to the function gF (cid:55)→ x ( gy ). We willcontinue this mnemonic below: using point-wise defined functions to indicate mapsof spectra which cannot be defined that way. We hope the readers can fill in thedetails themselves; if not, complete details can be found in [GHMR05] § G n on E in (1.19) defines the Morava module structure of E ∗ E hF ;under the isomorphism of (1.19) this maps to the diagonal action on the functions( hφ )( g ) = hφ ( h − g ) . Note that ( E ∗ E hF ) G n = map G n ( G n /F, E ∗ ) ∼ = ( E ∗ ) F . By [DH04], Theorem 2, this extends to an isomorphism H ∗ ( G n , E ∗ E hF ) ∼ = H ∗ ( F, E ∗ ) . See also Lemma 1.29 below.
Remark 1.20.
We now recall some results from [GHMR05]. If X = lim X i is aprofinite set, let E [[ X ]] = lim E ∧ X + i where the + indicates a disjoint basepoint.Then if F is a closed subgroup of G n we have an equivalence(1.21) E [[ G n /F ]] (cid:39) F ( E hF , E )defined as follows. Let F E denote the function spectrum in E -modules. Then E [[ G n /F ]] (cid:39) F E (map( G n /F , E ) , E ) (cid:39) F E ( E ∧ E hF , E ) (cid:39) F ( E hF , E ) . Next note that the equivalence of (1.21) is G n -equivariant with the following actions:in F ( E hF , E ) we act on the target and in E [[ G n /F ]] we act as follows: h ( (cid:88) a g gF ) = (cid:88) h ( a g ) h − gF . We can now make the following deductions. First suppose F = U is open (andhence closed), so that G n /F = G n /U is finite. Let F be finite. Then we haveequivalences (cid:89) F \ G n /U E hF x (cid:39) E [[ G n /U ]] hF (1.22) (cid:39) F ( E hU , E ) hF (cid:39) F ( E hU , E hF ) . The product in the source is over the double coset space, and for a double coset F xU , F x = F ∩ xU x − ⊆ F . Since it depends on a choice of x , the group F x is defined only up to conjugation,but the fixed point spectrum E hF x is well-defined up to weak equivalence.The first map of (1.22) sends a ∈ E hF x to the sum (cid:88) gF x ∈ F /F x ( g − a ) gxU. We say a word about the naturality of the equivalence of (1.22). Suppose U ⊆ V ⊆ G n is a nested pair of open subgroups. Then for each double coset F xU weget a double coset F xV , a nested pair of subgroups F x = F ∩ xU x − ⊆ F ∩ xV x − = G x , and a transfer map tr x : E hF x −→ E hG x associated to this inclusion. Then we have a commutative diagram(1.23) (cid:81) F \ G n /U E hF x tr (cid:15) (cid:15) (cid:39) (cid:47) (cid:47) E [[ G n /U ]] hF g (cid:15) (cid:15) (cid:81) F \ G n /V E hG x (cid:39) (cid:47) (cid:47) E [[ G n /V ]] hF where the map tr is the sum of the transfer maps and the map g induced by thequotient G n /U → G n /V .For a more general closed subgroup F write F = ∩ i U i where U i ⊆ G n is open.Then, for F finite we get a weak equivalence(1.24) holim i (cid:89) F \ G n /U i E hF x (cid:39) F ( E hF , E hF )where the product is as before and F x = F ∩ xU i x − ⊆ F K (2)-LOCAL HOMOTOPY THEORY 11 depends on i and, as in (1.23), there may be transfer maps in the transition mapsfor the homotopy limit. As F is finite and the product in (1.24) is finite, thisimplies that the image of (cid:89) F \ G n /U j π ∗ E hF x −→ (cid:89) F \ G n /U i π ∗ E hF x is independent of j for large j . It follows that there will be no lim term for thehomotopy groups of the inverse limit and hence there is an isomorphism(1.25) lim i (cid:89) F \ G n /U i π ∗ E hF x ∼ = π ∗ F ( E hF , E hF )1.5. The K ( n ) -local Adams-Novikov Spectral Sequence. This is the maintechnical tool of this paper, and we give a few details of the construction and someof its properties. We begin with some algebra from [HS99], Appendix A.Let m ⊂ E ∼ = W [[ u , · · · , u n − ]] be the maximal ideal. An L -complete E -module M is pro-free if any one of the following equivalent conditions is satisfied:(1) M ∼ = L ( N ) ∼ = N ∧ m for some free E -module N ;(2) the sequence ( p, u , · · · , u n − ) is regular on M ;(3) Tor E ( F p n , M ) = 0;(4) M is projective in the category of L -complete modules.Proposition A.13 of [HS99] implies that a continuous homomorphism M → N ofpro-free modules is an isomorphism if and only if M/ m M → N/ m N is an isomor-phism.If F ⊆ G n is a closed subgroup, then E ∗ E hF is pro-free, by (1.16).By Proposition 8.4 of [HS99], if X is a spectrum with K ( n ) ∗ X concentrated ineven degrees, then E ∗ X is pro-free, concentrated in even degrees. Furthermore, if E ∗ X is pro-free and concentrated in even degrees, then K ( n ) ∗ X ∼ = F p n ⊗ E E ∗ X ∼ = ( E ∗ X ) / m . Remark 1.26.
We now come to the Adams-Novikov Spectral Sequence in K ( n )-local category. As in [Str00], Proposition 15, this spectral sequence is obtained bythe standard cosimplicial cobar complex for E = E n in the K ( n )-local category.Thus we have E s,t ∼ = π s π t ( E • ∧ X ) = ⇒ π t − s L K ( n ) X. The smash products are in the K ( n )-local category. It is a consequence of the proofof [Str00], Proposition 15 that this spectral sequence converges strongly and has ahorizontal vanishing line at E ∞ .We’d now like to rewrite the E -term as group cohomology, at least under somehypotheses. For any group G , let EG be the standard contractible simplicial G -setwith ( EG ) s = G s +1 . This is a bar construction. If G is a topological group, then EG is a simplicial space. Let M be an L -complete Morava module. We define H s ( G n , M ) = π s map G n ( E G n , M )where map G n ( − , − ) denotes the group of continuous G n -maps. Since there is anisomorphism map G n ( G s +1 n , M ) ∼ = map( G sn , M ), we have that H ∗ ( G n , M ) is the s thcohomology group of a cochain complex M (cid:47) (cid:47) map( G n , M ) (cid:47) (cid:47) map( G n , M ) (cid:47) (cid:47) · · · Proposition 1.27.
Suppose X = Y ∧ Z where K ( n ) ∗ Y is concentrated in evendegrees and Z is a finite complex. Then we have an isomorphism π s π t ( E • ∧ X ) ∼ = H s ( G n , E t X ) and the K ( n ) -local Adams-Novikov Spectral Sequence reads H s ( G n , E t X ) = ⇒ π t − s L K ( n ) X. Proof. If Z = S , then E ∗ X is pro-free in even degrees. The result can be foundin Theorem 4.3 of [BH16]. The authors there work with the version of Morava E -theory obtained from the Honda formal group, but they need only the isomorphism E E ∼ = map( G n , E ), which holds in our case. See the remarks after (1.16). Thekey idea is that this last isomorphism can be extended to an isomorphism E ∗ ( E ∧ s ∧ X ) −→ map( G sn , E ∗ X )for any spectrum X with E t X pro-free for t . For more general Z , the isomorphismon E -terms follows from the five lemma. (cid:3) In some crucial cases, it is possible to reduce the E -term to group cohomologyover a finite group. Let F ⊆ G n be a finite subgroup and N an L -complete twisted F -module. Define an L -complete module N ↑ G n F as the set of continuous maps φ : G n → N such that φ ( hx ) = hφ ( x ) for h ∈ F . This becomes a Morava modulewith ( gφ )( x ) = φ ( xg )with g ∈ G n and there is an isomorphismmap G n ( G s +1 n , N ↑ G n F ) ∼ = map F ( G s +1 n , N )of groups of continuous maps. Let H s ( F, N ) = π s map F ( EF, N ). Lemma 1.28 ( Shapiro Lemma).
Let F ⊆ G n be a finite subgroup and sup-pose N is an L -complete twisted F -module with the property that N/ m N is finitedimensional over F p n . Then there is an isomorphism H ∗ ( G n , N ↑ G n F ) ∼ = H ∗ ( F, N ) and, under these hypotheses on N , there is an isomorphism H ∗ ( F, N ) ∼ = lim k H s ( F, N/m k N ) . Proof.
Since N is L -complete and N/ m N is finite, the short exact sequence of (1.15)implies N ∼ = N ∧ m . OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 13 Choose a nested sequence U i +1 ⊆ U i ⊆ G n of finite index subgroups of G n withthe property that ∩ U i = { e } . Then for all s ≥ G n ( G s +1 n , N ↑ G n F ) ∼ = map F ( G s +1 n , N ) ∼ = lim k map F ( G s +1 n , N/ m k N ) ∼ = lim k colim i map F (( G n /U i ) s +1 , N/ m k N ) . The last isomorphism follows from the fact that
N/m k N is discrete and finite. Since F is finite, we have F ∩ U i = { e } for all i greater than some i . Then for i > i itfollows that ( G n /U i ) • +1 is a contractible simplicial free F -set and, thus, we havean isomorphism π s map F (( G n /U i ) • +1 , N/ m k N ) ∼ = H s ( F, N/ m k N ) . Again since F is finite, H s ( F, N/ m k N ) is a finite abelian group. Both isomorphismsof the lemma now follow. (cid:3) Lemma 1.29.
Let Y be a spectrum equipped with an isomorphism of Morava mod-ules E ∗ Y ∼ = map( G n /F, E ∗ ) ∼ = E ∗ E hF where F ⊆ G n is a finite subgroup. Then for all finite spectra Z , there is anisomorphism H ∗ ( G n , E ∗ ( Y ∧ Z )) ∼ = H ∗ ( F, E ∗ Z ) . Proof.
We have an isomorphism of Morava modules E ∗ ( Y ∧ Z ) ∼ = (cid:47) (cid:47) map( G n /F, E ∗ Z )where the action of G n on the target is by conjugation: ( gφ )( x ) = gφ ( g − x ). Thereis an isomorphism of Morava modulesmap( G n /F, E ∗ Z ) ∼ = ( E ∗ Z ) ↑ G n F adjoint to evaluation at eF . The result follows from the Shapiro Lemma 1.28 . (cid:3) Remark 1.30.
Suppose Y = E hF . Then Lemma 1.29 follows from the variantof Theorem 2 of [DH04] appropriate for our formal group; indeed, we need onlyrequire that F be closed. However, the proof relies on the construction of E hF given in that paper and doesn’t a priori apply if we only know E ∗ Y ∼ = E ∗ E hF – aswill be the case in our application. Remark 1.31.
There is a map of Adams-Novikov Spectral SequencesExt sBP ∗ BP (Σ t BP ∗ , BP ∗ ) (cid:43) (cid:51) (cid:15) (cid:15) Z (2) ⊗ π t − s S (cid:15) (cid:15) H s ( G , E t ) (cid:43) (cid:51) π t − s L K (2) S but it takes a little care to define. Let G ( x, y ) ∈ E [[ x, y ]] be the formal group lawof the supersingular curve of (1.9); since G is the formal group of a Weierstrasscurve, it has a preferred coordinate. Let G ∗ ( x, y ) = uG ( u − x, u − y ) ∈ E ∗ [[ x, y ]].Then G ∗ is a formal group over E ∗ with coordinate in cohomological degree 2 and, therefore, is classified by a map Z (2) ⊗ M U ∗ → E ∗ . Since G ∗ is not evidently2-typical, it need not be classified by a map BP ∗ → E ∗ . However, over a Z (2) -algebra, the Cartier idempotent gives an equivalence between the groupoid of allformal group laws and the groupoid of 2-typical formal group laws; hence we havea diagram of spectral sequences as needed:Ext sBP ∗ BP (Σ t BP ∗ , BP ∗ ) (cid:43) (cid:51) Z (2) ⊗ π t − s S Z (2) ⊗ Ext sMU ∗ MU (Σ t M U ∗ , M U ∗ ) (cid:43) (cid:51) ∼ = (cid:79) (cid:79) (cid:15) (cid:15) Z (2) ⊗ π t − s S (cid:79) (cid:79) (cid:15) (cid:15) H s ( G , E t ) (cid:43) (cid:51) π t − s L K (2) S . We will use this below in the section on the cohomology of G .1.6. The action of the Galois group.
We now turn to analyzing E h S as an equi-variant spectrum over the Galois group. As above, we will write Gal = Gal( F p n / F p ),so that G n ∼ = S n (cid:111) Gal.We begin with the following elementary fact: the map Z p → W is Galois withGalois group Gal; thus, it is faithfully flat, ´etale, and the shearing map W ⊗ Z p W → map(Gal , W )sending a ⊗ b to the function g (cid:55)→ ag ( b ) is an isomorphism. In fact, this shearingmap is certainly an isomorphism modulo p ; then the statement for W follows fromNakayama’s Lemma. Faithfully flat descent now implies that the category of Z p -modules is equivalent to the category of twisted W [Gal]-modules under the functor M (cid:55)→ W ⊗ Z p M ; the inverse to this functor sends N to N Gal .This extends to the following result.
Lemma 1.32.
Let K ⊆ G n be a closed subgroup and let K = K ∩ S n . Supposethe canonical map K/K −→ G n / S n ∼ = Gal is an isomorphism. Then for any twisted G n -module M we have isomorphisms H ∗ ( K, M ) ∼ = H ∗ ( K , M ) Gal H ∗ ( K , M ) ∼ = W ⊗ Z p H ∗ ( K, M ) . Proof.
The subgroup S n acts on E through W -algebra homomorphisms; hence itacts on M through W -module homomorphisms. It follows that we can write thefunctor ( − ) K of invariants as a composite functortwisted G n -modules ( − ) K (cid:47) (cid:47) twisted W [Gal]-modules ( − ) Gal (cid:47) (cid:47) Z p -modules . As we just remarked, the second of these two functors is an equivalence of categoriesand in particular it is an exact functor. The first equation follows. The secondequation follows from the first and the fact that the inverse to ( − ) Gal is the functor M (cid:55)→ W ⊗ Z p M . (cid:3) OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 15 We now give a fact seemingly known to everyone, but hard to find in print. DrewHeard was the first to point out an error in our original argument; others followedquickly. We learned the following replacement from Mike Hopkins, Agn`es Beaudry,and the referee. We extend our thanks to everyone.
Lemma 1.33.
For all p and all n ≥ we have isomorphisms H ( G n , E ) ∼ = Z p H ( S n , E ) ∼ = W = W ( F p n ) . Furthermore, H ( G n , E t ) = H ( S n , E t ) = 0 if t (cid:54) = 0 .Proof. By Lemma 1.32 we need only do the case of S n .It is also sufficient to prove this when Γ n is the Honda formal group over F p n .Any other height n formal group becomes isomorphic to the Honda formal groupover the algebraic closure of F p , and the general result could then be deduced fromGalois descent.Write φ for lift of Frobenius to Witt vectors W . For the Honda formal group, S n is the group of units in the endomorphism ringEnd(Γ n ) ∼ = W (cid:104) S (cid:105) / ( S n − p )where W (cid:104) S (cid:105) is non-commutative polynomial ring over W on a variable S with Sa = φ ( a ) S when a ∈ W . Thus we have an inclusion W × ⊂ S n . Every k ∈ Z × p ⊆ S n corresponds to an automorphism Γ n → [ k ] Γ n of Γ n ; in particular Z × p acts triviallyon E and if u ∈ E − is the invertible generator, we have k ∗ u = ku . If follows that H ( Z × p , E t ) = 0 if t (cid:54) = 0.This leaves the case t = 0. Write K = W [ p − ]. Since all elements of S n fix theconstants W ⊆ E there is an inclusion W ∼ = H ( W × , W ) → H ( W × , E ) . Furthermore, since E is torsion-free, it is sufficient to show H ( W × , E [ p − ]) ∼ = K . By [DH95], Lemma 4.3, there are power series w i = w i ( u , u , . . . , u n − ) ∈ K [[ u , · · · , u n − ]] and an inclusion W [[ w , · · · , w n − ]] −→ K [[ u , · · · , u n − ]]onto an G n -equivariant sub-algebra which becomes an isomorphism after inverting p in the source. Thus it is sufficient to show that H ( W × , J ) = 0where J = W [[ w , · · · , w n − ]] / W .By [DH95], Proposition 3.3, the action of W × on W [[ w , · · · , w n − ]] is diagonal:if a ∈ W × , then a ∗ w i = φ i ( a ) a − w i . Let ω ∈ W be a primitive ( p n − a = 1 + pω . Then φ (1 + pω ) = 1 + pω p . If some i j (cid:54) = 0 it follows that(1 + pω ) ∗ w i . . . w i n − n − (cid:54) = w i . . . w i n − n − as needed. (cid:3) Remark 1.34.
Notice that the proof of Lemma 1.33 actually shows we need onlyrelatively small subgroups of G n to get the full invariants. Specifically, we have Z p ∼ = H ( W × (cid:111) Gal , E )and if t (cid:54) = 0, then H ( Z × p , E t ) = 0. We won’t need this stronger result. Remark 1.35.
In the proof of Lemma 1.36 below we will use the following ob-servation. Suppose we have a spectral sequence { E s,tr } that is multiplicative; thatis, E ∗ , ∗ r is a bigraded ring which is commutative up to sign and d r satisfies theLeibniz rule, again up to sign. Further suppose R ⊆ E , r is a commutative subringof d r -cycles and R ⊆ S is an ´etale extension in E , r . Then every element of S is a d r -cycle. To see this, note that d r restricted to S is a derivation over R and anysuch derivation must vanish; indeed, depending on your foundations, the vanishingof such derivations may even be part of your definition of ´etale. Lemma 1.36.
For all p and all n ≥ there is a Gal -equivariant equivalence
Gal + ∧ L K ( n ) S → E h S n . Proof.
We first prove we have injection W → π E h S n . We begin with the isomor-phism W ∼ = H ( S n , E ) of Lemma 1.33. Since Z p ⊆ W is an ´etale extension andthe Adams-Novikov Spectral Sequence for S n is a spectral sequence of rings, all of W survives to E ∞ and the edge homomorphism provides a surjection π E h S n → W of rings. The kernel of this map nilpotent as an ideal because the spectral sequencehas a horizontal vanishing line at E ∞ . See Remark 1.26. We then have a diagram Z p (cid:47) (cid:47) ⊆ (cid:15) (cid:15) π E h S n (cid:15) (cid:15) W = (cid:47) (cid:47) (cid:59) (cid:59) W Again since Z p ⊆ W is ´etale, the dashed arrow can be completed uniquely. Thisyields the injection we need.Now define ω : S → E h S n to be a representative of the homotopy class definedby a primitive ( p n − W . We can extend ω to a Gal-equivariantmap f : Gal + ∧ S → E h S inducing the splitting W → π E h S ; here we use thatthe map Z p [Gal] → W (cid:88) a g g (cid:55)−→ (cid:88) a g g ( ω )is an isomorphism. The map f extends to an isomorphism of twisted G n -modules E ∗ f : E ∗ (Gal + ∧ S ) ∼ = map(Gal , E ∗ ) ∼ = map( G n / S n , E ∗ ) ∼ = E ∗ E h S n thus completing the argument. (cid:3) The following result is a topological analog of Lemma 1.32.
Lemma 1.37.
Let K ⊆ G n be a closed subgroup and let K = K ∩ S n . Supposethe canonical map K/K −→ G n / S n ∼ = Gal is an isomorphism. Then there is a Gal -equivariant equivalence
Gal + ∧ E hK → E hK . OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 17 Proof.
This follows from Lemma 1.36. Define a map E h S n ∧ E hK → E hK by thecomposition(1.38) E h S n ∧ E hK → E hK ∧ E hK → E hK where the first map is given by the inclusion and the last map is multiplication.We have E ∗ ( E h S n ∧ E hK ) ∼ = map( G n / S n , E ∗ ) ⊗ E ∗ map( G n /K, E ∗ ) ∼ = map( G n / S n × G n /K, E ∗ ) . In E ∗ -homology, the map of (1.38) then becomes the mapmap( G n / S n , E ∗ ) ⊗ E ∗ map( G n /K, E ∗ ) ∼ = map( G n / S n × G n /K, E ∗ ) −→ map( G n /K , E ∗ )induced by the maps on cosets G n /K → G n /K × G n /K → G n / S n × G n /K where the first map is the diagonal and the second map is projection. Since K/K ∼ = G n / S n , this map on cosets is an isomorphism; therefore, the map of (1.38) is an E ∗ -isomorphism. By Remark 1.17 this map is a weak equivalence. (cid:3) Remark 1.39.
Combining Lemma 1.32 and Lemma 1.37 yields an isomorphism ofspectral sequences W ⊗ H ∗ ( K, E ∗ ) (cid:43) (cid:51) ∼ = (cid:15) (cid:15) W ⊗ π ∗ E hK ∼ = (cid:15) (cid:15) H ∗ ( K , E ∗ ) (cid:43) (cid:51) π ∗ E hK where the differentials on the top line are the W -linear differentials extended fromthe spectral sequence for K . Remark 1.40. At n = 2 and p = 2, Lemma 1.37 applies to the case of K = G ;then K = G . This implies that any of the spectra X ( i, j ) def = Σ i E hG ∨ Σ j E hG has the property that E ∗ X ( i, j ) ∼ = E ∗ E hG . But X ( i, j ) = Σ i E hG if and onlyif i ≡ j mod 8. This means that in some of the arguments we give to prove ourmain result we will have to produce two homotopy classes rather than one. SeeTheorem 5.8.2. The homotopy groups of homotopy fixed point spectra
Here we collect what we will need about the homotopy groups of E hF , where F runs through the finite subgroups of G of Remark 1.13. We will be working entirelyat n = p = 2 and using the formal group from the supersingular curve of (1.8).Much of what’s needed is in the literature and we’ll do our best to give references.However, much of what is written is for calculations over Hopf algebroids, whichis not quite what we’re doing, and the results need translation. In addition, manyof the results as written include some variant of the phrase “we neglect the bo -patterns”. We make this thought precise with the following ad hoc definition. Definition 2.1.
Let F ⊆ G be any finite subgroup containing C = {± } . Thenwe define the bo -patterns L ( π ∗ E hF ) of π ∗ E hF to be the image of the map inhomotopy π ∗ E hF −→ π ∗ L K (1) E hF . We also define the pure K (2) -classes M ( π ∗ E hF ) to be the kernel of the samemap.Thus we have a short exact sequence0 → M ( π ∗ E hF ) → π ∗ E hF → L ( π ∗ E hF ) → . Notice that the bo -patterns are defined as a quotient. In most cases, this sequenceis not split as modules over the homotopy groups of spheres. Remark 2.2.
The name bo -patterns is something of a misnomer, as KO -patternswould be more accurate. Here KO is 8-periodic 2-complete real K -theory. In allour examples we will have an isomorphism R ( F ) ⊗ Z KO ∗ ∼ = π ∗ L K (1) E hF for some Z -algebra R ( F ) in degree zero. While R ( F ) ⊗ Z KO ∗ is 8-periodic, L ( π ∗ E hF ) will typically have 8 k -periodicity for some k >
1. As a warning, wemention that this isomorphism is simply as rings; we are not claiming L K (1) E hF isa KO -algebra. Remark 2.3.
Here is more detail, to explain our thinking.Let us write S/ n for the Z / n -Moore spectrum. Then there is a weak equiva-lence L K (1) X (cid:39) holim v − ( X ∧ S/ n )and, if X is K (2)-local, a corresponding localized Adams-Novikov Spectral Sequence(2.4) lim v − H ∗ ( G , ( E ∗ X ) / n ) = ⇒ π ∗ L K (1) X. This spectral sequence doesn’t obviously converge.Now suppose C = {± } ⊆ F ⊂ G . Then the spectral sequence (2.4) for X = E hF becomes(2.5) lim v − H ∗ ( F, ( E ∗ ) / n ) = ⇒ π ∗ L K (1) E hF . Using Strickland’s formulas [Str] it is possible to show thatlim v − H ∗ ( F, ( E ∗ ) / n ) ∼ = lim v − H ∗ ( C , ( E ∗ ) / n ) F/C and that v − H ∗ ( C , ( E ∗ ) / n ) ∼ = W (( u ))[ u ± , η ]where W (( u )) = lim( W / n )[[ u ± ]] and η ∈ H ( C , E ) detects the class of thesame name in π S . (See (2.6) below.) From this it follows that the spectralsequence (2.5) is completely determined by the standard differential d ( v ) = (cid:15)η ,where v = u u − and (cid:15) ∈ F (( u )) × is a unit. We can conclude that the spectralsequence converges and π ∗ L K (1) E hF ∼ = W (( u )) F ⊗ Z KO ∗ OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 19 and in particular, that π ∗ L K (1) E hF and L ( π ∗ E hF ) are both concentrated in de-grees congruent to 0, 1, 2, and 4 modulo 8. It then remains to analyze the pure K (2)-local classes.Now, not much of what we just wrote is explicitly in print, and it would takequite a few pages to prove in detail. But we will put together what we can from theexisting literature to cover the main points case-by-case below. See Propositions2.7, 2.11, and 2.172.1. The homotopy groups of E hC and E hC . The standard source here isMahowald-Rezk [MR09], which uses the Hopf algebroid approach, but also uses thesame elliptic curve we have chosen. So the translation is straightforward. Here is asummary.The central C ⊆ G acts trivially on E and by multiplication by − u ;hence(2.6) H ∗ ( C , E ∗ ) ∼ = W [[ u ]][ u ± , α ] / (2 α )where α ∈ H ( C , E ) is the image of the generator of H ( C , Z (cid:104) sgn (cid:105) ) under themap which sends the generator of the sign representation to u − . Since v = u u − ∈ H ( C , E / u α ∈ H ( C , E ) is the image of v under the integral Bockstein,the class η ∈ π S is detected by u α . We will also write η = u α . Proposition 2.7.
The class b def = u u − reduces to v in v − H ∗ ( C , E ∗ / . Thereis an isomorphism W (( u ))[ b ± , η ] / (2 η ) ∼ = lim v − H ∗ ( C , E ∗ / n ) . The standard differential d ( v ) = η (see Lemma 2.21 below) forces a differential d ( u − ) = (cid:15)u α where (cid:15) ∈ F [[ u ]] × . Using the Mahowald-Rezk transfer argument [MR09, Prop.3.5] we have ν ∈ π S is non-zero in π E hC and detected by α ; this in turn forcesa differential d ( u − ) = α = αν . The spectral sequence collapses at E and we have the following result. Proposition 2.8.
The homotopy ring π ∗ E hC is periodic of period 16 with peri-odicity generator e detected by u − . The bo -patterns L ( E hC ) are concentratedin degrees congruent to , , , and modulo 8 and the group of pure K (2) -localclasses M ( E hC ) is generated by the classes α i e k , k ∈ Z , i = 3 , , , . To get the homotopy of E hC , with C = C × F × , we need to know the actionof F × . We can use Strickland’s calculations [Str] or interpret the Mahowald-Rezkresults. Let ω ∈ F × be a primitive cube root of unity. Then, ω ∗ u = ωu and ω ∗ u = ωu ; it follows that ω ∗ α = ω − α . The next result can be deduced fromthese formulas and the fact that π ∗ E hC ∼ = ( π ∗ E hC ) F × . Proposition 2.9.
The homotopy ring π ∗ E hC is periodic of period 48 with peri-odicity generator e detected by u − . The bo -patterns L ( E hC ) are concentratedin degrees congruent to , , , and modulo 8 and the group of pure K (2) -localclasses M ( E hC ) is generated by the classes e k α e k α e k +116 α e k +216 α of degrees k + 3 , k + 6 , k + 20 , and k + 37 respectively. Furthermore, thehomotopy class ν ∈ π S is detected by the class α and the class κ ∈ π S isdetected by e α . The homotopy groups of E hC . The standard reference for this calculationis Behrens-Ormsby [BO16] § C on a differentversion of Morava E -theory. There are two possible solutions. One is to do thecalculations over again, using Strickland’s formulas. The other is to notice that thetwo supersingular curves become isomorphic over the algebraically closed field F and to use descent to make the calculations. Using either method we obtain thefollowing result. Let i ∈ C be a generator and let z = u + i ∗ u ∈ H ( C , E ) . There are further cohomology classes b ∈ H ( C , E ) δ ∈ H ( C , E )and γ ∈ H ( C , E ) ξ ∈ H ( C , E ) . Let η ∈ H ( C , E ) and ν ∈ H ( C , E ) be the images of the like-named classesfrom the BP -based Adams-Novikov Spectral Sequence (see Remark 1.31). Proposition 2.10.
There is an isomorphism H ∗ ( C , E ∗ ) ∼ = W [[ z ]][ b , δ ± , η, ν, γ, ξ ] /R where R is the ideal of relations given by η = 2 γ = 4 ξ = 0 and b ≡ z δ mod 2 and δη = b ξ = γ b γ = zδη and b η = zγ γη = zξ and the final relations involving ν : ν = 2 ξ ν = zν = ην = b ν = γν = 0 Proof.
This can be obtained from Perspective 2 (after Remark 2.1.9) of [BO16] bya three-step process. First set ˜ γ = γ , ˜ j = z −
2, and β = δ − ξ . Second, invert δ .Finally, complete at the maximal ideal of H ( C , E ). (cid:3) OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 21 This result is displayed in Figure 1 below, presented in the standard Adamsformat: the x -axis is t − s ; the y -axis is s . In this chart, the square box (cid:3) representsa copy of W [[ z ]], the circle ◦ a copy of F [[ z ]], and the crossed circle ⊗ a copy of W [[ z ]] / (4 , z ) generated by a class of the form ξ j δ i . A solid dot is a copy of F annihilated by z ; it is generated by a class of the form ξ j ν . The solid lines aremultiplication by η or ν , as needed, and a dashed line indicates that xη = zy ,where x and y are generators in the appropriate bidegree. ◦ ◦ ⊗ ◦ ◦ (cid:3) ◦ ⊗ ◦ ◦ ◦ ⊗ ◦ ◦ (cid:3) ◦ ⊗ ◦ ◦ ◦ ⊗ ◦ ◦ (cid:3) ◦ ⊗ ◦ ◦ ◦ ⊗ (cid:3) ◦ ◦ ◦ ⊗ ◦ ◦ ◦ ⊗ (cid:3) ◦ ◦ ◦ ⊗ ◦ ◦ ◦ ⊗ (cid:3) ◦ ◦ ◦ ⊗ ◦ ◦ ◦ ⊗ (cid:3) ◦ ◦ κδγξ Figure 1.
The cohomology of C Notice that H ∗ ( C , E ∗ ) is 8-periodic with the periodicity class δ and that × ξ : H s ( C , E ∗ ) → H s +2 ( C , E ∗ +8 )is onto for s ≥ s >
0. In fact δ − ξ ∈ H ( C , E ) is,up to multiplication by a unit, the image of the periodicity class for the groupcohomology of C under the inclusion of trivial coefficients: Z / ∼ = H ( C , Z ) ∼ = H ( C , W ) → H s ( C , E ) . Proposition 2.11.
Modulo we have an equivalence b ≡ v and then an isomor-phism W (( z ))[ b ± , η ] / (2 η ) ∼ = lim v − H ∗ ( C , E ∗ / n ) . The differentials and extensions in this spectral sequence go exactly as in Behrensand Ormsby [BO16], Theorem 2.3.12. We end with the following result.
Proposition 2.12.
The homotopy ring π ∗ E hC is periodic of period 32 with pe-riodicity generator e detected by δ . The bo -patterns L ( E hC ) are concentratedin degrees congruent to , , , and modulo 8 and the group of pure K (2) -localclasses M ( E hC ) is generated by the classes e k x where x is from the followingtable Class Degree Order E -namea δ − ξνaη δ − ξ ν ν νaν δ − ξν ν ν (cid:15) δ − ξ ν = η(cid:15) κ δ − ξνκ
14 2 δν b
19 4 δ νκ
20 4 δξ ηκ
21 2 ηξ bν
22 4 δ ν c
27 2 δ γξcη
28 2 δ − ξ The pure K (2) classes in π ∗ E hC are presented in the Figure 2: the horizontalbar is multiplication by ν , the diagonal bar is multiplication by η . Note that κη = 2 bν . The horizontal scale is the degree of the element, but the vertical scalehas no meaning. Many of the additive and multiplicative relations are given byexotic extensions in this spectral sequence and the meaning of the original Adams-Novikov filtration becomes attenuated as a result; see Behrens-Ormsby [BO16],especially figure 9, for details. a ν (cid:15) κ cb κ Figure 2.
The pure K (2) classes in π ∗ E hC The homotopy groups of E hG and E hG . Remarks 1.39 and 1.40 yieldan isomorphism of spectral sequences W ⊗ Z H ∗ ( G , E ∗ ) (cid:43) (cid:51) ∼ = (cid:15) (cid:15) W ⊗ π ∗ E hG ∼ = (cid:15) (cid:15) H ∗ ( G , E ∗ ) (cid:43) (cid:51) π ∗ E hG . Therefore, we focus on the case of G . Here the standard sources are [Bau08],[DFHH14] and [HM98] although it requires some translation in each case to get theresults we want. OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 23 The ring H ( G , E ∗ ) is isomorphic to the ring of modular forms for supersin-gular elliptic curves at the prime 2. Then there are elements c ∈ H ( G , E ) c ∈ H ( G , E ) ∆ ∈ H ( G , E )obtained from the modular forms of the same name for our supersingular curve.Since this curve is smooth, ∆ is invertible and the j -invariant of our curves j = c / ∆ ∈ H ( G , E ) is defined. Then we get an isomorphism Z [[ j ]][ c , c , ∆ ± ] / ( c − c = (12) ∆ , ∆ j = c ) ∼ = H ( G , E ∗ ) . Modulo 2 we get a slightly simpler answer: F [[ j ]][ v , ∆ ± ] / ( j ∆ = v ) ∼ = H ( G , E ∗ / . Modulo 2 we have congruences(2.13) c ≡ v and c ≡ v . To describe the higher cohomology, we make a table of multiplicative generators.For each x , the bidegree of x is ( s, t ) if x ∈ H s ( G , E t ). All but µ detect theelements of the same name in π ∗ S . Furthermore, all elements but κ are in theimage of the map (see Remark 1.31)Ext ∗ , ∗ BP ∗ BP ( BP ∗ , BP ∗ ) → H ∗ ( G , E ∗ ) → H ∗ ( G , E ∗ ) . Hence we also give the name (the “MRW” is for Miller-Ravenel-Wilson) of a preim-age. The Greek letter notation is that of [MRW77].Class Bidegree Order MRW η (1 ,
2) 2 α ν (1 ,
4) 4 α / µ (1 ,
6) 2 α (cid:15) (2 ,
10) 2 β κ (2 ,
16) 2 β κ (4 ,
24) 8 − The class κ ∈ π S is detected by the image of β in H ( G , E ∗ ). The class µ hasa special role which we discuss below in Lemma 2.21, but we would like to noteright away that(2.14) v η ≡ µ modulo 2 . The following result is actually much easier to visualize than to write down. Seethe Figure 3 below.
Theorem 2.15.
There is an isomorphism H ( G , E ∗ )[ η, ν, µ, (cid:15), κ, κ ] /R ∼ = H ∗ ( G , E ∗ ) where R is the ideal defined by (1) the order of the elements of positive cohomological degree: η = 4 ν = 2 µ = 2 (cid:15) = 2 κ = 8 κ = 0;(2) the relations for ν : ην = 2 ν = ν = µν = 0; (3) the relations for (cid:15) : η(cid:15) = ν , ν(cid:15) = (cid:15) = µ(cid:15) = 0;(4) the relations for κ : ν κ = 4 κ, η κ = (cid:15)κ = κ = µκ = 0;(5) the elements annihilated by modular forms c ν = c ν = c (cid:15) = c (cid:15) = c κ = c κ = 0;(6) the relations between κ and modular forms; c κ = ∆ η , c κ = ∆ η µ ;(7) and the relations indicated by the congruences of (2.13) and (2.14): µ = c η c µ = c η c µ = c η . This result is presented graphically in Figure 3 below. We present it as the E page of the Adams-Novikov Spectral Sequence. The cohomology is 24-periodic on∆ , and the spectral sequence fills the entire upper-half plane. In Figure 3, thesquare box (cid:3) represents a copy of Z [[ j ]], the circle ◦ a copy of F [[ j ]], and thecrossed circle ⊗ a copy of Z [[ j ]] / (8 , j ) generated by a class of the form ∆ i κ j . Thesolid bullet represents a class of order 2 annihilated by j and the doubled bullet aclass of order 4 annihilated by j ; these last classes are always of the form ∆ i κ j ν .The solid lines are multiplication by η or ν , as needed, and a dashed line indicatesthat xη = jy , where x and y are generators in the appropriate bidegree. (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) ⊗ (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) µ (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) ⊗ (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) (cid:7) (cid:7) ⊗ κ (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7)(cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7)(cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7) (cid:7)(cid:7) (cid:7) (cid:7) (cid:7) (cid:3) (cid:7) (cid:7) Figure 3.
The cohomology of G Remark 2.16. (1) Many of the later relations can be rephrased as relations formultiplication by j = c / ∆. For example Theorem 2.15 (4) implies jν = j(cid:15) = jκ = 0 OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 25 and (5) implies jκ = c η and (6) implies jµ = c c ∆ − η. These last two equations explain the dashed lines in Figure 4.(2) Multiplication by κ : H s ( G , E t ) → H s +4 ( G , E t +24 ) is surjective and anisomorphism if s >
0. In fact, up to a unit, ∆ − κ ∈ H ( G , E ) is the imageof the periodicity class in group cohomology for Q under the inclusion of trivialcoefficients: Z / ∼ = H ( Q , W ) G /Q ∼ = H ( G , W ) → H ( G , E ) . The congruence (2.13) and the relations of Theorem 2.15 now give the followingresult. Note that the class c becomes invertible in lim v − H ∗ ( G , E ∗ / n ) and wemay define b = c /c . ( Warning:
This class b is related to, but not quite thesame, as the class b of Proposition 2.10. Both uses of b appear in the literature.) Proposition 2.17.
The class b reduces to v in v − H ∗ ( G , E ∗ / . There areisomorphisms Z (( j ))[ b ± , η ] / (2 η ) ∼ = lim v − H ∗ ( G , E ∗ / n ) . and F (( j ))[ v ± , η ] ∼ = v − H ∗ ( G , E ∗ / . Under the reduction map H ∗ ( G , E ∗ ) → H ∗ ( G , E ∗ / we have c (cid:55)→ v c (cid:55)→ v c κ (cid:55)→ v κ = ∆ η µ (cid:55)→ v η . Under the localization map H ∗ ( G , E ∗ ) → v − H ∗ ( G , E ∗ / we have ∆ (cid:55)→ v /jκ (cid:55)→ v η /j and that ν , (cid:15) , and κ map to zero. We have the following; see [HM98], [Bau08], or [DFHH14].
Proposition 2.18.
The homotopy ring π ∗ E hG is periodic of period 192 withperiodicity generator detected by ∆ . The bo -patterns L ( E hG ) are concentratedin degrees congruent to , , , and modulo 8. Remark 2.19.
We will not try to enumerate the pure K (2)-classes of M ( E hG );this information is known (by the same references as for Proposition 2.18), but wewon’t need that information in its entirety and it is rather complicated to writedown. What we will need can be read off of Figure 4, which is adapted from thecharts created by Tilman Bauer [Bau08], Section 8. This chart shows a section of the E ∞ -page of the Adams-Novikov Spectral Se-quence H s ( G , E t ) = ⇒ π t − s E hG . It is in the standard Adams bigrading ( t − s, s ).
40 44 48 52 56 60 64 6804812 ∆ κη ∆ κ η (cid:0) (cid:12) ◦ ◦ ◦ (cid:2) ◦ ◦ (cid:2) ◦ ◦ (cid:2) (cid:2) Figure 4.
The homotopy groups π i E hG for 40 ≤ i ≤ j κ i , have been left in gray for orientation, eventhough they do not last to the E ∞ -page. Bullets with circles are elements of order4; bullets with two circles are elements of order 8. Vertical lines are extensionsby multiplication by 2, lines raising homotopy degree by 1 are η -extensions, linesraising homotopy degree by 3 are ν -extensions.The lines 0 ≤ s ≤ bo -patterns; the adorned boxes and circles allrepresent ideals of either Z [[ j ]] or F [[ j ]]: (cid:3) ∼ = Z [[ j ]] (cid:2) ∼ = (2) ⊆ Z [[ j ]] (cid:0) ∼ = (4 , j ) ⊆ Z [[ j ]] ◦ ∼ = F [[ j ]] (cid:12) ∼ = ( j ) ⊆ F [[ j ]] . Elements not falling into one of these patterns are annihilated by j . The η -extensionfrom ( t − s, s ) = (65 ,
3) entry is ambiguous. We mark it as non-zero because wemay choose, as Bauer does, the two generators of the group of pure K (2)-classes in π E hG to be e [45 , κ and e [51 , κ where e [45 , ∈ π E hG and e [51 , ∈ π E hG are generators detected by ∆ κη and ∆ ν respectively. The class e [51 , κ is detected on the s = 3 line by ∆ κν and e [51 , κη (cid:54) = 0.We now record, from Figure 4, some data about our crucial homotopy classes. OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 27 Lemma 2.20.
There is an isomorphism Z / ∼ = π E hG . The generator is detected by the class ∆ κη ∈ H ( G , E ) . The class ∆ κ η ∈ H ( G , E ) is a non-zero permanent cycle detecting a gen-erator of the subgroup π E hG of the pure K (2) -classes of that degree. We close with some remarks on the role of µ in the d differentials. Lemma 2.21.
Let µ ∈ H ( G , E ) be the image of the class α ∈ Ext BP ∗ BP (Σ BP ∗ , BP ∗ ) . Then in any of the Adams-Novikov Spectral Sequences H ∗ ( F, E ∗ ) = ⇒ π t − s E hF and for any x ∈ H ∗ ( F, E ∗ ) we have d ( xµ ) = d ( x ) µ + xη . In the spectral sequences (2.22) H ∗ ( F, E ∗ /
2) = ⇒ π t − s ( E hF ∧ S/ we have d ( v x ) = d ( x ) v + xη + y where yη = 0 . Finally, in the spectral sequence (2.22) we have d ( v x ) = v d ( x ) . Proof.
In the Adams-Novikov spectral sequenceExt sBP ∗ BP (Σ t BP ∗ , BP ∗ ) = ⇒ Z (2) ⊗ π t − s S . we have d ( α ) = η . (In fact, by [MRW77], Corollary 4.23, η (cid:54) = 0 at E and E , ∼ = Z / α . The differential is then forced.) Since the fixed pointspectral sequence is a module over this standard Adams-Novikov Spectral Sequence,the first formula follows. The second formula follows because v η = α inExt sBP ∗ BP (Σ t BP ∗ , BP ∗ / . The third formula follows from the fact that S/ v -self map. (cid:3) Algebraic and topological resolutions
In this section we review the centralizer resolution constructed by Hans-WernerHenn [Hen07] § S is a virtual Poincar´e duality group of dimension 3. Thecentralizer resolution on the other hand, is much closer to being an Adams-Novikov tower as there is an underlying relative homological algebra in the spirit of Miller[Mil81]. See Remark 3.24 below.3.1. The centralizer resolution.
Henn’s centralizer resolutions grew out of hispaper [Hen98] which used the centralizers of elementary abelian subgroups of S n to detect elements in the cohomology of S n . At the prime 2, this approach needsa slight modification, as the maximal finite 2-group in S is Q , which is not ele-mentary abelian. Remark 3.1.
In (1.10) we defined G ⊆ S as the image of a group of auto-morphisms of a supersingular elliptic curve. The group S fits into a short exactsequence 1 −→ S (cid:47) (cid:47) S N (cid:47) (cid:47) Z −→ N is the reduced determinant map of (1.7). Let π = 1 + 2 ω be an element of S , where ω ∈ W × is a cube root of unity. Notice that π is not anelement of S because N ( π ) = 3 . Then we define G (cid:48) := πG π − ⊆ S . This is asubgroup isomorphic to G , but not conjugate to G in S .Note that multiplication by π defines an equivalence E hG (cid:39) E hG (cid:48) . For com-plete details on this and more, see [Bea15].We now have the following result from § algebraiccentralizer resolution . Theorem 3.2.
There is an exact sequence of continuous S -modules → Z [[ S /C ]] → Z [[ S /C ]] → Z [[ S /C ]] ⊕ Z [[ S /C ]] → Z [[ S /G ]] ⊕ Z [[ S /G (cid:48) ]] (cid:15) −→ Z → . (3.3) The map (cid:15) is the sum of the augmentation maps.
We will call this a resolution, even though the terms are not projective as Z [[ S ]]-modules. It is an F -projective resolution, an idea we explore below in Remark 3.24. Remark 3.4.
Suppose we write P = Z [[ S /G ]] × Z [[ S /G (cid:48) ]] P = Z [[ S /C ]] × Z [[ S /C ]] P = Z [[ S /C ]] P = Z [[ S /C ]] . Then for any profinite S -module M , we get a spectral sequence E p,q ∼ = Ext q Z [[ S ]] ( P p , M ) = ⇒ H p + q ( S , M ) . The E -terms can all be written as group cohomology groups; for example E ,q ∼ = H q ( G , M ) × H q ( G (cid:48) , M ) . OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 29 We will call this the algebraic centralizer resolution spectral sequence . In manyapplications, the distinction between the groups G and G (cid:48) disappears. For ex-ample, if M is a G -module (such as E n X for some spectrum X ) then multiplicationby π induces an isomorphism H q ( G , M ) ∼ = H q ( G (cid:48) , M ). Remark 3.5.
We can induce the resolution (3.3) of S -modules up to a resolutionof G -modules and obtain an exact sequence0 → Z [[ G /C ]] → Z [[ G /C ]] → Z [[ G /C ]] ⊕ Z [[ G /C ]] → Z [[ G /G ]] ⊕ Z [[ G /G ]] → Z [[ G / S ]] → . Since G (cid:48) is conjugate to G in G , we have Z [[ G /G ]] ∼ = Z [[ G /G (cid:48) ]] as G -modules and we have made that substitution. If F is any closed subgroup of G ,then the equivalence of (1.19) gives us an isomorphism of twisted G -modulesHom Z [[ G ]] ( Z [[ G /F ]] , E ∗ ) ∼ = E ∗ E hF . Combining these observations, we a get an exact sequence of twisted G -modules(3.6) 0 → E ∗ E h S → E ∗ E hG × E ∗ E hG → E ∗ E hC × E ∗ E hC → E ∗ E hC → E ∗ E hC → . We then have the following result; this is the topological centralizer resolution ofTheorem 12 of [Hen07].
Theorem 3.7.
The algebraic resolution of 3.6 can be realized by a sequence ofspectra E h S p −→ E hG × E hG → E hC × E hC → E hC → E hC All compositions and all Toda brackets are zero modulo indeterminacy.
Remark 3.8.
The vanishing of the Toda brackets in this result has several impli-cations. To explain these and for future reference we write, echoing the notation ofRemark 3.4: F = E hG × E hG F = E hC × E hC (3.9) F = E hC F = E hC . Then the resolution of Theorem 3.7 can be refined to a tower of fibrations under E h S :(3.10) E h S (cid:47) (cid:47) Y (cid:47) (cid:47) Y (cid:47) (cid:47) E hG × E hG = F Σ − F (cid:79) (cid:79) Σ − F (cid:79) (cid:79) Σ − F (cid:79) (cid:79) Alternatively we could refine the resolution into a tower over E h S . Let us write X (cid:31) (cid:31) Z (cid:111) (cid:111) Y (cid:64) (cid:64) for a cofiber sequence (i.e., a triangle) X → Y → Z → Σ X . Then we have adiagram of cofiber sequences(3.11) E h S p (cid:33) (cid:33) C (cid:111) (cid:111) (cid:30) (cid:30) C (cid:111) (cid:111) (cid:30) (cid:30) C (cid:111) (cid:111) (cid:39) (cid:30) (cid:30) F (cid:64) (cid:64) F (cid:64) (cid:64) F (cid:64) (cid:64) F where each of the compositions F i − → C i → F i is the map F i − → F i in theresolution.The towers (3.10) and (3.11) determine each other. This is because there is adiagram with rows and columns cofibration sequences(3.12) Σ − ( s +1) C s +1 (cid:47) (cid:47) (cid:15) (cid:15) E h S (cid:47) (cid:47) (cid:15) (cid:15) Y s (cid:15) (cid:15) Σ − s C s (cid:47) (cid:47) (cid:15) (cid:15) E h S (cid:47) (cid:47) (cid:15) (cid:15) Y s − (cid:15) (cid:15) Σ − s F s (cid:47) (cid:47) ∗ (cid:47) (cid:47) Σ − s +1 F s Using the tower over E h S of (3.11) we get a number of spectral sequences; forexample, if Y is any spectrum, we get a spectral sequence for the function spectrum F ( Y, E h S )(3.13) E s,t = π t F ( Y, F s ) = ⇒ π t − s F ( Y, E h S ) . Up to isomorphism, this spectral sequence can be obtained from the tower of (3.10);this follows from (3.12).
Remark 3.14.
It is direct to calculate E ∗ C s and E ∗ Y s for the layers of the twotowers. If we define K s ⊆ E ∗ F s to be the image of E ∗ F s − → E ∗ F s , then E ∗ C s ∼ = K s and, more, if we apply E ∗ to (3.11) we get a collection of short exact sequences: E ∗ E h S (cid:36) (cid:36) (cid:36) (cid:36) E ∗ C (cid:111) (cid:111) (cid:34) (cid:34) (cid:34) (cid:34) E ∗ C (cid:111) (cid:111) (cid:34) (cid:34) (cid:34) (cid:34) E ∗ C (cid:111) (cid:111) ∼ = (cid:35) (cid:35) E ∗ F (cid:60) (cid:60) (cid:60) (cid:60) E ∗ F (cid:60) (cid:60) (cid:60) (cid:60) E ∗ F (cid:60) (cid:60) (cid:60) (cid:60) E ∗ F . Notice that this implies that each of the dotted arrows of (3.11) has Adams-Novikovfiltration one. Finally, the cofibration sequence E h S → Y s → Σ − s C s +1 induces ashort exact sequence 0 → E ∗ E h S → E ∗ Y s → Σ − s K s +1 → . OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 31 The duality resolution, first steps.
We have the algebraic duality resolu-tion from [Bea15]. The groups G and G (cid:48) are defined in Remark 3.1. Theorem 3.15.
There is an exact sequence of continuous S -modules → Z [[ S /G (cid:48) ]] → Z [[ S /C ]] → Z [[ S /C ]] → Z [[ S /G ]] (cid:15) → Z → where (cid:15) is the augmentation. Remark 3.17.
As in Remark 3.4 we get a spectral sequence. Suppose we write Q = Z [[ S /G ]] Q = Q = Z [[ S /C ]] Q = Z [[ S /G (cid:48) ]]Then for any profinite S -module M, such as E ∗ X = ( E ) ∗ X for some finite spec-trum X , we get a spectral sequence E p,q ∼ = Ext q Z [[ S ]] ( Q p , M ) = ⇒ H p + q ( S , M ) , which we will call the algebraic duality resolution spectral sequence .As in Remark 3.5 and (3.6) we immediately have the following consequence. Corollary 3.18.
There is an exact sequence of twisted G -modules → E ∗ E h S → E ∗ E hG → E ∗ E hC → E ∗ E hC → E ∗ E hG → . The first of these maps is induced by the map on homotopy fixed point spectra E h S → E hG induced by the subgroup inclusion G ⊆ S . We’d now like to prove the following result, paralleling Theorem 3.7; it alsoappears in [Hen07]. The main work of the next two sections and, indeed, the maintheorem of this paper is to identify X . Proposition 3.19.
The algebraic resolution of 3.18 can be realized by a sequenceof spectra E h S q −→ E hG → E hC → E hC → X with E ∗ X ∼ = E ∗ E hG as a twisted G -module. All compositions and all Todabrackets are zero modulo indeterminacy. Remark 3.20.
A consequence of the last sentence of this result is that this reso-lution can be refined to a tower of fibrations under E h S (3.21) E h S (cid:47) (cid:47) Z (cid:47) (cid:47) Z (cid:47) (cid:47) E hG Σ − X (cid:79) (cid:79) Σ − E hC (cid:79) (cid:79) Σ − E hC (cid:79) (cid:79) or to a tower over E h S (3.22) E h S q (cid:36) (cid:36) D (cid:111) (cid:111) (cid:34) (cid:34) D (cid:111) (cid:111) (cid:34) (cid:34) D (cid:111) (cid:111) (cid:39) (cid:32) (cid:32) E hG (cid:60) (cid:60) E hC (cid:60) (cid:60) E hC (cid:60) (cid:60) X .
As in Remark 3.14 the dotted arrows have Adams-Novikov filtration 1. Examiningthis last diagram, we see that X can be defined as the cofiber of D → E hC andit will follow, as in Remark 3.14, that E ∗ X ∼ = E ∗ E hG . Thus Proposition 3.19 isequivalent to the following result. See also [Hen07] or [Bob14]. Lemma 3.23.
The truncated resolution → E ∗ E h S → E ∗ E hG → E ∗ E hC → E ∗ E hC can be realized by a sequence of spectra E h S −→ E hG → E hC → E hC such that all compositions are zero and the one Toda bracket is zero modulo inde-terminacy.Proof. The map E h S q −→ E hG is induced by the inclusion map on homotopy fixedpoint spectra induced by the subgroup inclusion G ⊆ S . To realize the othermaps and to show the compositions are zero, we prove that Hurewicz map π F ( E hF , E hC ) −→ Hom
Mor ( E E hF , E E hC )to the category of Morava modules is an isomorphism for F = S or F = G . Tosee this, first note there is an isomorphism π F ( E hF , E hC ) = π E [[ G /F ]] hC ∼ = H ( C , E [[ G /F ]]) . This follows from (1.25) and the fact that π E hK = H ( K, E ) whenever K ⊆ C ;see § H ( C , E [[ G /F ]]) ∼ = Hom E [[ G ]] ( E [[ G /C ]] , E [[ G /F ]]) ∼ = Hom Mor ( E E hF , E E hC ) . This leaves the Toda bracket. To see that it is zero modulo indeterminacy weshow that the indeterminacy is the entire group. To be specific, we show that theinclusion E h S → E hG induces a surjection π ∗ F ( E hG , E hC ) −→ π ∗ F ( E h S , E hC ) . Using (1.21), we can rewrite this map as π ∗ E [[ G /G ]] hC −→ π ∗ E [[ G / S ]] hC . Again using (1.21), the inclusion E h S → E hG induces a map of C -spectra E [[ G /G ]] (cid:39) F ( E hG , E ) → F ( E h S , E ) (cid:39) E [[ G / S ]] . It is thus sufficient to show that the quotient map on cosets G /G −→ G / S has a C -splitting. Since G ∼ = S (cid:111) Gal( F / F ), every coset in G / S has a rep-resentative of form π i φ (cid:15) S where π is as in Remark 3.1, φ ∈ Gal( F / F ) is theFrobenius, i ∈ Z , and (cid:15) = 0 or 1. The splitting is then given by π i φ (cid:15) S (cid:55)−→ π i φ (cid:15) G . (cid:3) OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 33 Comparing the two resolutions.
There is a map from the centralizer towerto the duality tower; we will not prove that here. In the end we will only need asmall part of the data given by such a map, and what we need is in Remark 3.27.
Remark 3.24.
The underlying algebra for the centralizer resolutions fits well therelative homological algebra usually deployed in building an Adams-Novikov tower;this goes back to Miller in [Mil81], among other sources.Here is more detail. Let F be the set of conjugacy classes of finite subgroups of S . A continuous S -module P is F -projective if the natural map (cid:77) F ∈F Z [[ S ]] ⊗ Z [[ F ]] P −→ P is split surjective, where F runs over representatives for the classes in F . The classof F -projectives is the smallest class of continuous S -modules closed under finitesums, retracts, and containing all induced modules Z [[ S ]] ⊗ Z [[ F ]] M , where M isa continuous F -module.The class of F -projectives defines a class of F -exact morphisms, there are enough F -projectives, there are F -projective resolutions, and so on. All of this and moreis discussed in § Proposition 3.25.
The centralizer resolution (3.3) is an F -projective resolutionof the trivial S -module Z . Thus if we write P • → Z for the centralizer resolution (3.3) and Q • → Z forthe duality resolution (3.16), then standard homological algebra gives us a map ofresolutions, unique up to chain homotopy Q • (cid:47) (cid:47) g • (cid:15) (cid:15) Z (cid:15) (cid:15) P • (cid:47) (cid:47) Z . The map g : Q → P can be chosen to be the inclusion onto the first factor(3.26) Q = Z [[ S /G ]] i (cid:47) (cid:47) Z [[ S /G ]] ⊕ Z [[ S /G (cid:48) ]] = P . Remark 3.27.
This immediately gives a map from the centralizer resolution spec-tral sequence of Remark 3.4 to the duality resolution spectral sequence of Remark3.17. This map is independent of the choice of g • at the E -page. This can belifted to a map from the centralizer tower to the duality tower, although we don’tneed that here and won’t prove it. We note that (3.26) implies there is a commu-tative diagram where the horizontal maps are the edge homomorphisms of the two spectral sequences H ∗ ( S , E ∗ ) p ∗ (cid:47) (cid:47) = (cid:15) (cid:15) H ∗ ( G , E ∗ ) × H ∗ ( G , E ∗ ) ( g ) ∗ (cid:15) (cid:15) H ∗ ( S , E ∗ ) q ∗ (cid:47) (cid:47) H ∗ ( G , E ∗ )and ( g ) ∗ is projection onto the first factor. This can be realized by a diagram ofspectra, where the map g is again projection onto the first factor E h S p (cid:47) (cid:47) = (cid:15) (cid:15) E hG × E hG g (cid:15) (cid:15) E h S q (cid:47) (cid:47) E hG . Constructing elements in π k +48 X We now turn to the analysis of the homotopy groups of X , where Σ − X is thetop fiber in the duality tower; see Proposition 3.19. We have an isomorphism ofMorava modules E ∗ X ∼ = E ∗ E hG and hence, by Proposition 1.27 and Lemma 1.29,a spectral sequence H ∗ ( G , E ∗ ) = ⇒ π ∗ X. See also Remark 1.30. The cohomology if G is discussed in Theorem 2.15. In thissection we show, roughly, that ∆ k +2 ∈ H ( G , E k +48 ) is a permanent cycle—which would certainly be necessary if our main result is true. The exact result isbelow in Corollary 4.7. In the next section, we will use this and a mapping spaceargument to finish the identification of the homotopy type X .The statements and the arguments in this section have a rather fussy naturebecause the spectrum X has no a priori ring or module structure and, in particular,the W -algebra structure on H ∗ ( G , E ∗ ) does not immediately extend to a W -module structure on π ∗ X .The results of this section were among the main results in the first author’s thesis[Bob14] and the key ideas for the entire project can be found there.We begin by combining Remark 1.39 and Lemma 2.20 to obtain the followingresult. Note that 45 ≡ bo -patternsin that degree. In all degrees the bo -patterns lie in Adams-Novikov filtration at most2. The following is an immediate consequence of Lemma 1.37 and Lemma 2.20. Lemma 4.1.
There is an isomorphism F ∼ = π E hG . We can chose an F generator detected by the class ∆ κη ∈ H ( G , E ) . OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 35 The class ∆ κ η ∈ H ( G , E ) is a non-zero permanent cycle detecting an F generator of the subgroup of π E hG consisting of the elements of Adams-Novikovfiltration greater than . Let p : E h S → E hG × E hG be the augmentation in the topological centralizerresolution of Theorem 3.7. This is the same map as from the top to the bottom ofthe centralizer resolution tower (3.10). Lemma 4.2.
Let k ∈ Z . The map p ∗ : π k +45 E h S −→ π k +45 ( E hG × E hG ) is surjective. If x ∈ π k +45 E h S has the property that p ∗ ( x ) (cid:54) = 0 , then x hasAdams-Novikov filtration at most , xκη (cid:54) = 0 , and xκη is detected by a class ofAdams-Novikov filtration at most 10.Proof. For the first statement we examine the homotopy spectral sequence of thecentralizer tower (3.13). In this case this spectral sequence reads π t F s = ⇒ π t − s E h S . The fibers F s are spelled out in (3.9). We are asking that the edge homomorphism p ∗ : π k +45 E h S −→ π k +45 F be surjective. The crucial input is that π k E hC ⊆ π k E hC = 0for k = 45, 46, and 47 and that π E hC = 0. See Proposition 2.9 and Figure 2.The final statement follows from Lemma 4.1. (cid:3) Now let q : E h S → E hG be the augmentation in the topological duality reso-lution of Proposition 3.19. This is also the projection from the top to the bottomof the duality resolution tower (3.21). Let i : Σ − X → E h S be the map from thetop fiber of the duality tower. Consider the commutative diagram(4.3) π k +45 Σ − X i ∗ (cid:47) (cid:47) r (cid:15) (cid:15) π k +45 E h S = (cid:47) (cid:47) p ∗ (cid:15) (cid:15) π k +45 E h S q ∗ (cid:15) (cid:15) F (cid:47) (cid:47) π k +45 ( E hG × E hG ) ( g ) ∗ (cid:47) (cid:47) π k +45 E hG . The bottom row is short exact and induced by the map between the resolutions.See Remark 3.27. The map r is defined by this diagram and the fact that thecomposition π ∗ Σ − X −→ π ∗ E h S q ∗ −→ π ∗ E hG is zero. Proposition 4.4.
The map r : π k +45 Σ − X → F
46 IRINA BOBKOVA AND PAUL G. GOERSS is surjective. If y ∈ π k +45 Σ − X is any class so that r ( y ) (cid:54) = 0 , then y is detectedby a class f def = f ( j )∆ k +2 ∈ H ( G , E k +48 ) ∼ = W [[ j ]]∆ k +2 . Furthermore f κη ∈ H ( G , E k +73 ) is a non-zero permanent cycle in the spectral sequence for π ∗ X .Proof. In the diagram (4.3), the map p ∗ is onto, and any element x in the kernelof ( g ) ∗ must have filtration at least 1 in the homotopy spectral sequence of theduality tower. Since π k E hC = 0 for k = 46 and k = 47, by Proposition 2.12, anysuch element must be the image of class y ∈ π ∗ Σ − X . This shows r is surjective.The map Σ − X → E h S raises Adams-Novikov filtration by 3; see the diagramof (3.22) and the remarks thereafter. If r ( y ) = p ∗ i ∗ ( y ) (cid:54) = 0, then by Lemma 4.2, y must have Adams-Novikov filtration at most 2; however, by Theorem 2.15 and thechart of Figure 3 we have that H ( G , E k +49 ) = 0 = H ( G , E k +50 ) . Thus y must have filtration 0. Similarly 0 (cid:54) = yκη must have filtration at leastfive and at most 7. Again we examine the chart of Figure 3 to find it must havefiltration 5 and be detected in group cohomology, as claimed. (cid:3) Recall that we are writing S/ Proposition 4.5.
Let y ∈ π k +48 X be detected by f = f ( j )∆ k +2 ∈ H ( G , E ∗ ) . If r ( y ) (cid:54) = 0 , then f and f κη are non-zero permanent cycles in the spectral sequence H ∗ ( G , E ∗ /
2) = ⇒ π ∗ ( X ∧ S/ . Proof.
This follows from Proposition 4.4 and the fact that H ( G , E k +73 ) → H ( G , E k +73 / → E ∗ → E ∗ → E ∗ / → H ( G , E k +73 ) = 0 . See Figure 3. (cid:3)
The crucial theorem then becomes:
Theorem 4.6.
Let y ∈ π k +48 X and let f = f ( j )∆ k +2 ∈ H ( G , E ∗ ) be the image of y under the edge homomorphism π ∗ X −→ H ( G , E ∗ ) . If f ( j ) ≡ modulo (2 , j ) , then r ( y ) = 0 . OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 37 Proof.
We will show that if f (0) ≡ f κ = 0 in E ∗ , ∗ ( X ∧ S/ f (0) ≡ f = jg ( j )∆ k +2 ∈ H ( G , E ∗ / . We will show that in the Adams-Novikov Spectral Sequence for L K (2) ( X ∧ S/
2) wehave d ( v g ( j )∆ k +2 µ ) = f κ . The result will follow.We appeal to Theorem 2.15, Remark 2.16, and the chart of Figure 3. We have jκ = c η = v η and, hence, that f κ = jg ( j )∆ k +2 κ = v g ( j )∆ k +2 η . Since f κ is a d -cycle, d is η -linear, and η : E k +48 , / ∼ = H ( G , ( E/ k +48 ) → H ( G , E k +52 ) ∼ = E k +48+4 , is injective we have that d ( v g ( j )∆ k +2 ) = 0 . It now follows from Lemma 2.21 that d ( v g ( j )∆ k +2 µ ) = v g ( j )∆ k +2 η = f κ. This is what we promised. (cid:3)
The next result has a slightly complicated statement because we don’t know yetthat π ∗ X is a W -module. Corollary 4.7.
There is a commutative diagram π k +48 X (cid:47) (cid:47) r (cid:15) (cid:15) H ( G , E k +48 ) (cid:15) (cid:15) (cid:15) F ∼ = (cid:47) (cid:47) F where the bottom map is some possibly non-trivial isomorphism of groups and (cid:15) ( f ( j )∆ k +2 ) = f (0) mod (2 , j ) . There are homotopy classes x k,i ∈ π k +48 X , i = 1 , detected by classes f i ( j )∆ k +2 ∈ H ( G , E k +48 ) so that f (0) and f (0) span F as an F vector space.Proof. This is an immediate consequence of Theorem 4.6. (cid:3)
Remark 4.8.
Lemma 4.4 produces classes f ∈ π Σ − X for which f κη (cid:54) = 0. Theimage of any such f in π E h S is non-zero and has Adams-Novikov filtration atleast 3. It is natural to ask what we know about these classes. In particular, doesone of these classes come from the homotopy groups of sphere itself under the unitmap π ∗ S → π ∗ E h S ?There are classes x ∈ π S detected in the Adams Spectral Sequence by h = h h . Note any such class has filtration 3. There is a choice for x that seem tosupport non-zero multiplications by many of the basic elements in π ∗ S , including η and κ . It would be easy to guess that this class is mapped to the class we haveconstructed, but we’ve not yet settled this one way or another.For more about this class in π S , see the chart “The E ∞ -page of the classicalAdams Spectral Sequence” in [Isa14]. The fact that some class detected by h h can have a non-zero κ multiplication can be found in Lemma 4.114 of [Isa]. Seealso Table 33 of that paper. Note that Isaksen is careful to label this lemma astentative, as this is in the range where the homotopy groups of spheres still needexhaustive study. In the table A.3.3 of [Rav86], a related class is posited to bedetected in the Adams-Novikov Spectral Sequence by the class γ , also of filtration3, but note the question mark there.5. The mapping space argument
We would like to extend the results of the Section 4 in the following way. Let ι : S → E hG be the unit and r the composition π k +48 X ∼ = π k +45 (Σ − X ) → π k +45 E hG ∼ = F defined in (4.3). Theorem 5.1.
The composite π k +48 F ( E hG , X ) ι ∗ (cid:47) (cid:47) π k +48 X r (cid:47) (cid:47) F is surjective. We can use this result to build maps out of E hG as follows. Recall from (1.21)that if F ⊆ G is a closed subgroup, then there is an isomorphism E ∗ [[ G /F ]] ∼ = E ∗ E hF of E ∗ [[ G ]]-modules. Also, Proposition 3.19 and the Universal CoefficientTheorem give an isomorphism E ∗ X ∼ = E ∗ [[ G /G ]]. OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 39 Consider the following diagram. Note we are using that E − = E .(5.2) π ( E hG , X ) i ∗ (cid:47) (cid:47) H (cid:15) (cid:15) π X H (cid:15) (cid:15) Hom G ( E [[ G /G ]] , E − [[ G /G ]]) ι ∗ (cid:47) (cid:47) Hom G ( E [[ G /G ]] , E − ) ∼ = (cid:15) (cid:15) W [[ j ]]∆ ∼ = ( E ) G (cid:15) (cid:15) (cid:15) F where the maps labelled H are the Hurewicz maps for E ∗ ( − ) and the map (cid:15) reducesmod (2 , j ). By Corollary 4.7, the vertical composition on the right is r up to someautomorphism of F . Proposition 4.4 and Corollary 4.7 then yield the followingcorollary to Theorem 5.1, using the case when k = 0. Corollary 5.3.
Let f ( j )∆ ∈ H ( G , E ) . Then there is a map φ : Σ E hG → X so that ι ∗ ( φ ) ≡ f (0) modulo . We will use this result to show that there is an equivalence Σ E hG → X . SeeTheorem 5.8 below.We now begin the proof of Theorem 5.1. Let p : E h S → E hG × E hG be theprojection from the top to the bottom of the centralizer resolution tower. Lemma 5.4.
Let k ∈ Z . The map p ∗ : π k +45 F ( E hG , E h S ) → π k +45 ( F ( E hG , E hG ) × F ( E hG , E hG )) is surjective.Proof. We apply F ( E hG , − ) to the centralizer tower and examine the resultingspectral sequence in homotopy. See (3.13). The spectral sequence reads π t F ( E hG , F s ) = ⇒ π t − s F ( E hG , E h S )and the fibers F s are described in (3.9). Thus we need to know0 = π k +45 F ( E hG , E hC ∨ E hC )= π k +46 F ( E hG , E hC )= π k +47 F ( E hG , E hC ) . We can use (1.24). Note that C is central, so all of the subgroups F x contain C .Therefore, the crucial input is as before: π k E hC ⊆ π k E hC = 0for k = 45, 46, and 47 and that π E hC = 0. See Propositions 2.8, 2.9, and 2.12.See also Figure 2. (cid:3) Let q : E h S → E hG be the projection from the top to the bottom of the dualityresolution tower. Using Remark 3.27 we now can produce a commutative diagram,where we have abbreviated F ( E hG , Y ) as F ( Y ) and we’re writing n = 192 k + 45.(5.5) π n F (Σ − X ) (cid:47) (cid:47) r (cid:15) (cid:15) π n F ( E h S ) = (cid:47) (cid:47) p ∗ (cid:15) (cid:15) π n F ( E h S ) q ∗ (cid:15) (cid:15) π n F ( E hG ) (cid:47) (cid:47) ι ∗ (cid:15) (cid:15) π n ( F ( E hG ) × F ( E hG )) ( g ) ∗ (cid:47) (cid:47) ι ∗ (cid:15) (cid:15) π n F ( E hG ) ι ∗ (cid:15) (cid:15) π n E hG (cid:47) (cid:47) π n ( E hG × E hG ) ( g ) ∗ (cid:47) (cid:47) π n E hG . The maps labelled ( g ) ∗ are induced from the map g : E hG × E hG → E hG discussed in Remark 3.27. It is projection onto the first factor. The lower two rowsare split short exact, the maps labelled ι ∗ are all onto, and the maps p ∗ and q ∗ areonto. The map r is then defined by the requirement that the upper right squarecommute. It is the analog of the map r of (4.3), and in fact we have a commutativediagram π k +45 F ( E hG , Σ − X ) ι ∗ (cid:47) (cid:47) r (cid:15) (cid:15) π k +45 Σ − X r (cid:15) (cid:15) π k +45 F ( E hG , E hG ) ι ∗ (cid:47) (cid:47) π k +45 E hG ∼ = F . Since E hG is an E hG -module spectrum, the evaluation map ι : F ( E hG , E hG ) → E hG is split. Hence, Theorem 5.1 follows from the next result. Lemma 5.6.
The map r : π k +45 F ( E hG , Σ − X ) → π k +45 F ( E hG , E hG ) is onto.Proof. The proof is an exact copy of the first part of the argument for Proposition4.4, generalized to mappings spaces; that is, we examine the homotopy spectralsequence built from the duality tower for F ( E hG , E h S ). In the diagram (5.5), themap q ∗ is onto, and any element x in the kernel of ( g ) ∗ must have filtration at least 1in the homotopy spectral sequence of the duality tower. Since π k F ( E hG , E hC ) =0 for k = 46 and k = 47, using (1.24) and Propositions 2.8 and 2.9, any suchelement must be the image of class from π ∗ F ( E hG , Σ − X ). (cid:3) Remark 5.7.
Note that all of these arguments would work with replacing E hG with E hG .The next result is our main theorem. Theorem 5.8.
There is an equivalence Σ E hG (cid:39) −→ X OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 41 realizing the given isomorphism of Morava modules E ∗ E hG ∼ = −→ E ∗ X. Proof.
We actually use the given isomorphism of Morava modules to produce a(non-equivariant) equivalence C +2 ∧ Σ E hG (cid:39) Σ ( E hG ∨ E hG ) → X. Here C = Gal = Gal( F / F ). Then we will apply Lemma 1.37.We begin with some algebra. Recall from Remark 1.14 that the 2-Sylow subgroup S ⊆ S can be decomposed as K (cid:111) Q . Since G = Q (cid:111) F × and G = G (cid:111) Galwe have that E ∗ [[ G /G ]] and E ∗ [[ G /G ]] are free E ∗ [[ K ]] modules of rank 2 andrank 1 respectively. Since K is a finitely generated pro-2-group, the ring E [[ K ]]is a complete local ring with maximal ideal m K given by the kernel of the reducedaugmentation E [[ K ]] −→ E −→ F . We will produce a map f : Σ ( E hG ∨ E hG ) → X so that the map of E [[ K ]]-modules E ∗ f : E X −→ E (Σ ( E hG ∨ E hG ))is an isomorphism modulo m K . Then, by the appropriate variant of Nakayama’sLemma (see Lemma 4.3 of [GHMR05]), it will be an isomorphism of E [[ K ]]-modules and, hence, of E [[ G ]]-modules, as required.Since G ∼ = Q (cid:111) F × and G ∼ = (( K (cid:111) Q ) (cid:111) F × ) (cid:111) Gal , we have an isomorphismof K -sets G /G ∼ = K (cid:116) Kφ where φ is the Frobenius in the Galois group. Then wehave the following commutative diagram. It is an expansion of the diagram (5.2).Hom G ( E [[ G /G ]] , E − [[ G /G ]]) (cid:47) (cid:47) ι ∗ (cid:15) (cid:15) Hom K ( E [[ K ]] ⊕ E [[ Kφ ]] , E − [[ K ]]) ι ∗ (cid:15) (cid:15) Hom G ( E [[ G /G ]] , E − ) (cid:47) (cid:47) ∼ = (cid:15) (cid:15) Hom K ( E [[ K ]] ⊕ E [[ Kφ ]] , E − ) ⊆ (cid:15) (cid:15) ( E ) G ∼ = W [[ j ]] (cid:47) (cid:47) (cid:15) (cid:15) E ∼ = ( W [[ u ]]) (cid:15) (cid:15) F (cid:47) (cid:47) F . The top two horizontal maps are forgetful maps, remembering only the K action.The third horizontal map is the composition W [[ j ]] ⊆ (cid:47) (cid:47) W [[ u ]] × φ (cid:47) (cid:47) ( W [[ u ]]) . and the bottom map is x (cid:55)→ ( x, φ ( x )). The top vertical maps are induced by themap E ∗ ( E hG ) → E ∗ S given by the unit, the middle vertical map evaluates a homomorphism at 1 ∈ E [[ G /G ]]; note we are again using that E − ∼ = E .The final map is reduction modulo the maximal ideals in both cases.By Corollary 5.3 we can produce two maps f i : Σ E hG → X, i = 1 , f i ) ∗ ( ι ) ≡ ω i ∆ modulo (2 , j ) , where ω ∈ F is the primitive cube root ofunity. We now examine the fate of E ( f i ) : E [[ G /G ]] −→ E − [[ G /G ]]as we work from the upper left to the bottom right of this diagram. Using theformulas of the previous paragraph we have E ( f ) (cid:55)→ ( ω, ω ) and E ( f ) (cid:55)→ ( ω , ω ) . Finally, let f = f ∨ f : Σ ( E hG ∨ E hG ) → X. If we apply E ∗ ( − ) to this map we get a map E ∗ f : E [[ G /G ]] → E − [[ G /G ]] × E − [[ G /G ]]which yields a map of K -modules E ∗ f : E [[ K ]] −→ E − [[ K ]] which, modulo the maximal ideal m K , gives the map F −→ F given by the matrix (cid:18) ω ω ω ω (cid:19) with determinant ω + ω = 1. Thus E ∗ f is an isomorphism, as needed. (cid:3) We can now complete the proof of Theorem 0.1 and construct the topologicalduality resolution. Proposition 3.19 and Theorem 5.8 imply the following. Recallfrom Theorem 2.9 that E hC is 48-periodic. Corollary 5.9.
There exists a resolution of E h S in the K (2) -local category at theprime 2 E h S → E hG → E hC → Σ E hC → Σ E hG . Remark 5.10.
In [GHMR05], working at the prime 3, the authors were able toproduce a topological resolution for L K (2) S = E h G itself by the same methodsthat produced the resolution for E h G . The resolution for the sphere was essentiallya double of that for E h G . Not only do the methods of [GHMR05] not apply to thecase p = 2, it’s very unlikely that there is a topological resolution of the sphere,or for E h S , which could be obtained by doubling the resolution of E h S . Thereare any number of difficulties, but the first obstacle is that we have only a semi-direct decomposition S (cid:111) Z ∼ = S at p = 2, rather than the product decomposition G × Z ∼ = G at the prime 3. This makes the algebra much harder, and it only getsworse from there. Other short topological resolutions are possible, of course, andcould be very instructive. This is the subject of current research by Agn`es Beaudryand Hans-Werner Henn. OPOLOGICAL RESOLUTIONS IN K (2)-LOCAL HOMOTOPY THEORY 43 References [Bau08] T. Bauer. Computation of the homotopy of the spectrum tmf . In
Groups, homotopyand configuration spaces , volume 13 of
Geom. Topol. Monogr. , pages 11–40. Geom.Topol. Publ., Coventry, 2008.[Bea15] Agn`es Beaudry. The algebraic duality resolution at p = 2. Algebr. Geom. Topol. ,15(6):3653–3705, 2015.[Bea17] Agn`es Beaudry. The chromatic splitting conjecture at n = p = 2. Geom. Topol. ,21(6):3213–3230, 2017.[Beh06] Mark Behrens. A modular description of the K (2)-local sphere at the prime 3. Topol-ogy , 45(2):343–402, 2006.[Beh12] Mark Behrens. The homotopy groups of S E (2) at p ≥ Adv. Math. ,230(2):458–492, 2012.[BH16] Tobias Barthel and Drew Heard. The E -term of the K ( n )-local E n -Adams spectralsequence. Topology Appl. , 206:190–214, 2016.[BO16] Mark Behrens and Kyle Ormsby. On the homotopy of Q (3) and Q (5) at the prime 2. Algebr. Geom. Topol. , 16(5):2459–2534, 2016.[Bob14] Irina Bobkova.
Resolutions in the K(2)-local Category at the Prime 2 . ProQuest LLC,Ann Arbor, MI, 2014. Thesis (Ph.D.)–Northwestern University.[Bou79] A. K. Bousfield. The localization of spectra with respect to homology.
Topology ,18(4):257–281, 1979.[DFHH14] C.L. Douglas, J. Francis, A.G. Henriques, and M.A. Hill.
Topological Modular Forms ,volume 210 of
Mathematical Surveys and Monographs . Amer. Math. Soc., 2014.[DH95] Ethan S. Devinatz and Michael J. Hopkins. The action of the Morava stabilizer groupon the Lubin-Tate moduli space of lifts.
Amer. J. Math. , 117(3):669–710, 1995.[DH04] Ethan S. Devinatz and Michael J. Hopkins. Homotopy fixed point spectra for closedsubgroups of the Morava stabilizer groups.
Topology , 43(1):1–47, 2004.[GH04] P. G. Goerss and M. J. Hopkins. Moduli spaces of commutative ring spectra. In
Structured ring spectra , volume 315 of
London Math. Soc. Lecture Note Ser. , pages151–200. Cambridge Univ. Press, Cambridge, 2004.[GH16] Paul G. Goerss and Hans-Werner Henn. The Brown-Comenetz dual of the K (2)-localsphere at the prime 3. Adv. Math. , 288:648–678, 2016.[GHM04] Paul Goerss, Hans-Werner Henn, and Mark Mahowald. The homotopy of L V (1) forthe prime 3. In Categorical decomposition techniques in algebraic topology (Isle ofSkye, 2001) , volume 215 of
Progr. Math. , pages 125–151. Birkh¨auser, Basel, 2004.[GHM14] P. Goerss, H.-W. Henn, and M. Mahowald. The rational homotopy of the K (2)-localsphere and the chromatic splitting conjecture at the prime 3 and level 2. Doc. Math. ,19:1271–1290, 2014.[GHMR05] P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk. A resolution of the K (2)-localsphere at the prime 3. Ann. of Math. (2) , 162(2):777–822, 2005.[GHMR15] P. Goerss, H.-W. Henn, M. Mahowald, and C. Rezk. On Hopkins Picard groups forthe prime 3 and chromatic level 2.
J. Topol. , 8(1):267–294, 2015.[Hen98] Hans-Werner Henn. Centralizers of elementary abelian p -subgroups and mod- p coho-mology of profinite groups. Duke Math. J. , 91(3):561–585, 1998.[Hen07] H.-W. Henn. On finite resolutions of K ( n )-local spheres. In Elliptic cohomology , vol-ume 342 of
London Math. Soc. Lecture Note Ser. , pages 122–169. Cambridge Univ.Press, Cambridge, 2007.[HG94] M. J. Hopkins and B. H. Gross. The rigid analytic period mapping, Lubin-Tate space,and stable homotopy theory.
Bull. Amer. Math. Soc. (N.S.) , 30(1):76–86, 1994.[HKM13] H.-W. Henn, N. Karamanov, and M. Mahowald. The homotopy of the K (2)-localMoore spectrum at the prime 3 revisited. Math. Z. , 275(3-4):953–1004, 2013.[HM98] M. Hopkins and M. Mahowald. From elliptic curves to homotopy theory. 1998.Unpublished. Available from http://hopf.math.purdue.edu/Hopkins-Mahowald/eo2homotopy.pdf .[HS99] M. Hovey and N. P. Strickland. Morava K -theories and localisation. Mem. Amer.Math. Soc. , 139(666):viii+100, 1999.[Isa] Daniel C. Isaksen. Stable stems.
Mem. Amer. Math. Soc. , To Appear. [Isa14] Daniel C. Isaksen. Classical and motivic Adams charts.
ArXiv1401. 4983[ mathAT] ,January 2014.[Lad13] O. Lader. Une r´esolution projective pour le second groupe de morava pour p ≥ https://tel.archives-ouvertes.fr/tel-00875761/document , 2013.[Mil81] Haynes R. Miller. On relations between Adams spectral sequences, with an applicationto the stable homotopy of a Moore space. J. Pure Appl. Algebra , 20(3):287–312, 1981.[MR09] M. Mahowald and C. Rezk. Topological modular forms of level 3.
Pure Appl. Math.Q. , 5(2, Special Issue: In honor of Friedrich Hirzebruch. Part 1):853–872, 2009.[MRW77] Haynes R. Miller, Douglas C. Ravenel, and W. Stephen Wilson. Periodic phenomenain the Adams-Novikov spectral sequence.
Ann. of Math. (2) , 106(3):469–516, 1977.[Rav84] D. C. Ravenel. Localization with respect to certain periodic homology theories.
Amer.J. Math. , 106(2):351–414, 1984.[Rav86] Douglas C. Ravenel.
Complex cobordism and stable homotopy groups of spheres , vol-ume 121 of
Pure and Applied Mathematics . Academic Press, Inc., Orlando, FL, 1986.[Sil09] Joseph H. Silverman.
The arithmetic of elliptic curves , volume 106 of
Graduate Textsin Mathematics . Springer, Dordrecht, second edition, 2009.[Str] N. P. Strickland. Level three structures. Unpublished.[Str00] N. P. Strickland. Gross-Hopkins duality.
Topology , 39(5):1021–1033, 2000.[SW02] Katsumi Shimomura and Xiangjun Wang. The Adams-Novikov E -term for π ∗ ( L S )at the prime 2. Math. Z. , 241(2):271–311, 2002.[SY95] Katsumi Shimomura and Atsuko Yabe. The homotopy groups π ∗ ( L S ). Topology ,34(2):261–289, 1995.
School of Mathematics, Institute for Advanced Study, Princeton, NJ 08540, U.S.A.
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