Odd primary analogs of Real orientations
aa r X i v : . [ m a t h . A T ] S e p ODD PRIMARY ANALOGS OF REAL ORIENTATIONS
JEREMY HAHN, ANDREW SENGER, AND DYLAN WILSON
Abstract.
We define, in C p -equivariant homotopy theory for p ą
2, a notion of µ p -orientation analogous to a C -equivariant Real orientation. The definition hinges on a C p -space CP µ p , which we prove to be homologically even in a sense generalizing recent C -equivariant work on conjugation spaces.We prove that the height p ´ E -theory is µ p -oriented and that tmf p q is µ -oriented. We explain how a single equivariant map v µ p : S ρ Ñ Σ CP µ p completelygenerates the homotopy of E p ´ and tmf p q , expressing a height-shifting phenomenonpervasive in equivariant chromatic homotopy theory. Contents
1. Introduction 11.1. Homological and homotopical evenness 41.2. A view to the future 52. Orientation theory 73. Evenness 103.1. Homological Evenness 103.2. Homotopical Evenness 124. The homological evenness of CP µ p p ´ E -theory 175.2. The spectrum tmf p q as a form of BP x y µ v µ p and a formula for its span 206.1. The non-equivariant v as a p th power 206.2. The equivariant v µ p as a norm 226.3. A formula for v µ p in terms of v v µ p in height p ´ Introduction
The complex conjugation action on CP gives rise to a C -equivariant space, CP R , withfixed points RP . The subspace CP R is invariant and equivalent as a C -space to S ρ , theone-point compactification of the real regular representation of C . A C -equivariant ringspectrum R is Real oriented if it is equipped with a mapΣ CP R Ñ Σ ρ R such that the restriction S ρ “ Σ CP R Ñ Σ CP R Ñ Σ ρ R is the Σ ρ -suspension of the unit map S Ñ R . Such a Real orientation induces a homotopyring map MU R Ñ R, with domain the spectrum of Real bordism [AM78, HK01]. These orientations have provedinvaluable to the study of 2-local chromatic homotopy theory, leading to an explosion ofprogress surrounding the Hill–Hopkins–Ravenel solution of the Kervaire invariant one Prob-lem [HHR16, GM17, HM17, KLW17, HLS18, HSWX19, BBHS19, LLQ20, LSWX19, HS20,BHSZ20, MSZ20].The above papers solve problems, at the prime p “
2, that admit clear but often unap-proachable analogs for odd primes. To give two examples, the 3 primary Kervaire problemremains unresolved [HHR11], and substantially less precise information is known about oddprimary Hopkins–Miller EO -theories [BC20, Conjecture 1.12].To rectify affairs at p ą
2, the starting point must be to find a C p -equivariant spaceplaying the role of CP R . This paper began as an attempt of the first two authors tounderstand a space proposed by the third. Construction 1.1 (Wilson) . For any prime p , let CP µ p denote the fiber of the C p -equivariantmultiplication map p CP q ˆ p Ñ CP , where the codomain has trivial C p -action. In other words, a map of spaces X Ñ CP µ p consists of the data of: ‚ A p -tuple of complex line bundles p L , L , ¨ ¨ ¨ , L p q on X . ‚ A trivialization of the tensor product L b L b ¨ ¨ ¨ b L p .The action on CP µ p is given by p L , L , ¨ ¨ ¨ , L p q ÞÑ p L p , L , ¨ ¨ ¨ , L p ´ q . Remark 1.2.
There is an equivalence of C -spaces CP µ » CP R . In general, the non-equivariant space underlying CP µ p is equivalent to p CP q ˆ p ´ . The fixed points ´ CP µ p ¯ C p are equivalent to the classifying space BC p , as can be seen by applying the fixed pointsfunctor p´q C p to the defining fiber sequence for CP µ p . The key point here is that the C p -fixed points of p CP q ˆ p consist of the diagonal copy of CP , and BC p is the fiber of the p th tensor power map CP Ñ CP .To formulate the notion of Real orientation, it is essential to understand the inclusion ofthe bottom cell S ρ “ CP R Ñ CP R . At an arbitrary prime, the analog of this bottom cell is described as follows:
Notation 1.3.
We let S (cid:7) denote the cofiber of the unique non-trivial map of pointed C p -spaces from p C p q ` to S . This is the spoke sphere , and it is a wedge of p p ´ q copies of S with action on reduced homology given by the augmentation ideal in the group ring Z r C p s .We denote the suspension Σ S (cid:7) of the spoke sphere by either S ` (cid:7) or CP µ p , and Remark 1.6provides a natural inclusion S ` (cid:7) “ CP µ p Ñ CP µ p . We will often also use S ` (cid:7) to denote Σ S ` (cid:7) .With this bottom cell in hand, we propose the following generalization of Real orientationtheory: DD PRIMARY ANALOGS OF REAL ORIENTATIONS 3
Definition 1.4. A µ p -orientation of a C p -equivariant ring R is a map of spectraΣ CP µ p Ñ Σ ` (cid:7) R such that the composite S ` (cid:7) “ Σ CP µ p Ñ Σ CP µ p Ñ Σ ` (cid:7) R is the S ` (cid:7) -suspension of the unit map S Ñ R . Remark 1.5.
Applying the geometric fixed point functor Φ C p to a µ p -orientation we learnthat the non-equivariant spectrum Φ C p R has p “ Remark 1.6.
Let Z – H Z denote the C p -equivariant Eilenberg–MacLane spectrum as-sociated to the constant Mackey functor. Then there is an equivalence of C p -equivariantspaces Ω Σ ` (cid:7) Z » CP µ p . Indeed, suspending and rotating the defining cofiber sequence p C p q ` Ñ S Ñ S (cid:7) gives riseto a cofiber sequence S ` (cid:7) Ñ p C p q ` b S Ñ S . Tensoring with Z and applying Ω yieldsthe defining fiber sequence for CP µ p .Under this identification, the natural inclusion CP µ p Ñ CP µ p is simply adjoint to theΣ ` (cid:7) -suspension of the unit map S Ñ Z . In particular, the identification CP µ p » Ω p Σ ` (cid:7) Z q gives a canonical µ p -orientation of Z . In contrast, Bredon cohomology with coefficients inthe Burnside Mackey functor cannot be µ p -oriented, since p is nonzero in the geometricfixed points.In this paper we explore the interaction between µ p -orientations and chromatic homotopytheory in the simplest possible case: chromatic height p ´
1. Specifically, we study thefollowing height p ´ E -ring spectra: Notation 1.7.
We let E p ´ denote the height p p ´ q Lubin–Tate theory associated to theHonda formal group law over F p p ´ , with C p -action given by a choice of order p element inthe Morava stabilizer group. At p “
3, we let tmf p q denote the 3-localized connective ringof topological modular forms with full level 2 structure [Sto12]. The ring tmf p q naturallyadmits an action by Σ – SL p F q , and we restrict along an inclusion C Ă Σ to viewtmf p q as a C -equivariant ring spectrum.The underlying homotopy groups of these spectra are given respectively by π e ˚ p E p ´ q – W p F p p ´ q J u , u , ¨ ¨ ¨ , u p ´ K r u ˘ s , | u i | “ , | u | “ ´ , and π e ˚ p tmf p qq – Z p q r λ , λ s , | λ i | “ . We will review the C p -actions on the homotopy groups in Section 5. Theorem 1.8.
For all primes p , there exists a µ p -orientation of the C p -equivariant Morava E -theory E p ´ . Theorem 1.9.
The ( -localized) C -equivariant ring tmf p q of topological modular formswith full level structure admits a µ -orientation. Our second main result concerns the fact that, while π ˚ E p ´ – W p F p p ´ q J u , u , . . . , u p ´ K r u ˘ s has p p ´ q distinct named generators, the conglomeration of them is generated under the µ p -orientation by a single equivariant map v µ p . JEREMY HAHN, ANDREW SENGER, AND DYLAN WILSON
Construction 1.10.
In Section 6, we will construct a map of C p -equivariant spectra v µ p : S ρ Ñ Σ CP µ p . This map should be viewed as canonical only up to some indeterminacy, just as the classicalclass v is only well-defined modulo p . As was pointed out to the authors by Mike Hill, onechoice of this map is given by norming a non-equivariant class in π e CP µ p . Construction 1.11.
Suppose a C p -equivariant ring R is µ p -oriented via a mapΣ CP µ p Ñ Σ ` (cid:7) R, so that we may consider the composite S ρ Σ CP µ p Σ ` (cid:7) R. v µp Using the dualizability of S ` (cid:7) , this composite is equivalent to the data of a map S ρ ´ ´ (cid:7) Ñ R. The non-equivariant spectrum underlying S ρ ´ ´ (cid:7) is (non-canonically) equivalent to a directsum of p ´ S p ´ . In particular, by applying π e p ´ to the map S ρ ´ ´ (cid:7) Ñ R, one obtains a map from a rank p ´ Z p p q -module to π e p ´ R . Definition 1.12.
Given a C p -equivariant ring R with a µ p -orientation, the span of v µ p willrefer to the subset of π e p ´ R consisting of the image of the rank p ´ Z p p q -moduleconstructed above. Theorem 1.13.
For any µ -orientation of tmf p q , the span of v µ in π e tmf p q is all of π e tmf p q . Theorem 1.14.
For any µ p -orientation of the height p ´ Morava E -theory E p ´ , the spanof v µ p inside π e p ´ E p ´ maps surjectively onto π e p ´ E p ´ {p p, m q . Homological and homotopical evenness
Non-equivariantly, complex orientation theory is intimately tied to the notion of evenness.A fundamental observation is that, since CP has a cell decomposition with only even-dimensional cells, any ring R with π ˚´ R – C -equivariant homotopy theory, a ring R is called even if π C ˚ ρ ´ R – π e ˚´ R –
0, andit is a basic fact that any even ring is Real orientable [HM17, § C p -equivariant homotopy theory, we propose the appropriate notion of evenness to becaptured by the following definition, which we discuss in more detail in Section 3: Definition 1.15.
We say that a C p -equivariant spectrum E is homotopically even if thefollowing conditions hold for all n P Z :(1) π e n ´ E “ π C p nρ ´ E “ π C p nρ ´ ´ (cid:7) E “ Remark 1.16. A C -spectrum E is homotopically even, according to our definition above,if and only if it is even in the sense of [HM17, § Theorem 1.17.
If a p -local C p -ring spectrum R is homotopically even, then it is also µ p -orientable. DD PRIMARY ANALOGS OF REAL ORIENTATIONS 5
The key point here, as we explain in Section 4, is that CP µ p admits a slice cell decomposi-tion with even slice cells. An even more fundamental fact, which turns out to be equivalentto the slice cell decomposition, is a splitting of the homology of CP µ p : Definition 1.18.
We say that a C p -spectrum X is homologically even if there is a directsum splitting X b Z p p q » à k A k b Z p p q , where each A k is equivalent, for some n P Z , to one of p C p q ` b S n , S nρ , S nρ ` ` (cid:7) . Theorem 1.19.
The space CP µ p is homologically even. Remark 1.20.
The notion of homological evenness we propose in this paper restricts, when p “
2, to the notion studied by Hill in [Hil19, Definition 3.2]. Notably, our definition differsfrom Hill’s when p ą C , work of Pitsch, Ricka, and Scherer relates a ver-sion of homological evenness to the study of conjugation spaces [PRS19]. An interest-ing example of a conjugation space, generalized in [HH18] and its in-progress sequel, isBU R “ Ω Σ ρ BP x y R . It would be very interesting to develop a C p -equivariant version ofconjugation space theory. Since tmf p q is a form of BP x y µ (cf. Question 7), we wonderwhether there is an interesting slice cell decomposition of Ω Σ ` (cid:7) tmf p q .1.2. A view to the future
The most natural next question, after those tackled in this paper, is the following:
Question 1.
Let n ě
1, and fix a formal group Γ of height n p p ´ q over a perfect field k of characteristic p . When is the associated Lubin–Tate theory E k, Γ µ p -orientable?We have not fully answered this question even for n “
1, since we focus attention on theHonda formal group.It seems likely that further progress on Question 1, at least for n ě
2, must wait forwork in progress of Hill–Hopkins–Ravenel, who have a program by which to understand the C p -action on Lubin–Tate theories. As the authors understand that work in progress, it isto be expected that the height n p p ´ q Morava E -theory has homotopy generated by n copies of the reduced regular representation, v µ p , v µ p , ¨ ¨ ¨ , v µ p n . One expects to be able toconstruct µ p Morava K -theories, generated by a single v µ p i , and we expect at least theseMorava K -theories to be homotopically even in the sense of this paper. Question 2.
Can one construct homotopically even µ p Morava K -theories?In light of the orientation theory of Section 2, it seems useful to know if µ p Morava K -theories admit norms . Indeed, at p “ K -theories all admit thestructure of E σ -algebras. Since the first µ Morava K -theory should be TMF p q{
3, orperhaps L K p q TMF p q{
3, it seems pertinent to answer the following question first:
Question 3.
At the prime p “
3, what structure is carried by the C -equivariant spectrum L K p q TMF p q{
3? Is there an analog of the E σ structure carried by KU R { Question 4.
Is there an analog of the notion of µ p -orientation related to the Q -actions onLubin–Tate theories at the prime 2? JEREMY HAHN, ANDREW SENGER, AND DYLAN WILSON
One may also go beyond finite groups and ask for notions capturing other parts of theMorava stabilizer group, such as the central Z ˆ p that acts on CP » B Z p after p -completion.To make full use of all these ideas, one would like not only an analog of CP R , but also ananalog of at least one of MU R or BP R . Attempts to construct such analogs have consumedthe authors for many years; we consider it one of the most intriguing problems in stablehomotopy theory today. Question 5. (Hill–Hopkins–Ravenel [HHR11]) Does there exist a natural C p -ring spectrum,BP µ p , with ‚ Underlying, non-equivariant spectrum the smash product of p p ´ q copies of BP. ‚ Geometric fixed points Φ C p BP µ p » H F p .At p “
2, it should be the case that BP µ “ BP R .To the above we may add: Question 6.
Does such a natural BP µ p orient all µ p -orientable C p -ring spectra, or at leastall those that admit norms in the sense of Section 2?Most of our attempts to build BP µ p have proceeded via obstruction theory, while MU R is naturally produced via geometry. It would be extremely interesting to see a geometricdefinition of an object MU µ p . Alternatively, it would be very clarifying if one could provethat a reasonable BP µ p does not exist. As some evidence in that direction, the authorsdoubt any variant of BP µ p can be homotopically even.Even if BP µ p cannot be built, or cannot be built easily, it would be excellent to knowwhether it is possible to build C p -ring spectra BP x y µ p . Question 7.
Does there exist, for each prime p , a C p -ring BP x y µ p satisfying the followingproperties: ‚ BP x y µ is the 2-localization of ku R , and BP x y µ is the 3-localization of tmf p q . ‚ The homotopy groups are given by π e ˚ BP x y µ p – Z p p q r λ , λ , ¨ ¨ ¨ , λ p ´ s , with | λ i | “ p ´
2. The C p action on these generators should make π e p ´ BP x y µ p into a copy of the reduced regular representation. ‚ There is a C p -ring map BP x y µ p Ñ E p ´ . ‚ BP x y µ p is homotopically even, and in particular µ p -orientable. ‚ The underlying spectrum ` BP x y µ p ˘ e additively splits into a wedge of suspensionsof BP x p ´ y . ‚ We have Φ C p BP x y µ p » F p r y s for a generator y of degree 2 p .It is plausible that BP x y µ p should come in many forms, in the sense of Morava’s formsof K -theory [Mor89]. A natural E form might be obtained by studying compactificationsof the Gorbounov–Hopkins–Mahowald stack [GM00, Hil06] of curves of the form y p ´ “ x p x ´ qp x ´ a q ¨ ¨ ¨ p x ´ a p ´ q . Studying the uncompactified stack, it is possible to construct a C p -equivariant E ring E p q µ p which is a µ p analog of uncompleted Johnson-Wilson theory. Remark 1.21.
The C p -action on CP µ p is naturally the restriction of an action by Σ p . Infact, most objects in this paper admit actions of Σ p , or at least of C p ´ ˙ C p , but these areconsistently ignored. The reader is encouraged to view this as an indication that the theoryremains in flux, and welcomes further refinement. DD PRIMARY ANALOGS OF REAL ORIENTATIONS 7
Remark 1.22.
Since work of Quillen [Qui69], the notion of a complex orientation has beenintimately tied to the notion of a formal group law. There are hints throughout this paper,particularly in Section 2 and Section 6, that the norm and diagonal maps on CP µ p lead toequivariant refinements of the p -series of a formal group. It may be interesting to develop thepurely algebraic theory underlying these constructions, particularly if algebraically defined v µ p i turn out to be of relevance to higher height Morava E -theories.1.3. Notation and Conventions ‚ If X is a C p -space, we use X e to denote the underlying non-equivariant space, andwe use X C p to denote the fixed point space. If X is a C p -spectrum, we will useeither Φ e X or X e to denote the underlying spectrum, and we use Φ C p X to denotethe geometric fixed points. ‚ We fix a prime number p , and throughout the paper all spectra and all (nilpotent)spaces are implicitly p -localized. In the C p -equivariant setting, this means that weimplicitly p -localize both underlying and fixed point spaces and spectra. ‚ If X is a C p -space or spectrum, we use π e ˚ X to denote the homotopy groups of X e ,considered as a graded abelian group with C p -action . If V is a C p -representation,we use π C p V X to denote the set of homotopy classes of equivariant maps from S V to X . ‚ We let S (cid:7) denote the cofiber of the C p -equivariant map p C p q ` Ñ S , and we alsouse S (cid:7) to refer to the suspension C p -spectrum of this C p -space. We let S ´ (cid:7) denotethe Spanier-Whitehead dual of the C p -spectrum S (cid:7) . Given a C p -representation V and a C p -spectrum X , we will use π C p V ` (cid:7) X and π C p V ´ (cid:7) to denote the set of homotopyclasses of equivariant maps from S V ` (cid:7) : “ S V b S (cid:7) and S V ´ (cid:7) : “ S V b S ´ (cid:7) to X . ‚ We let CP µ p denote the fiber of the C p -equivariant multiplication map p CP q ˆ p Ñ CP . ‚ If R is a classical commutative ring, we use ¯ ρ R to denote the R r C p s -module givenby the augmentation ideal ker p R r C p s Ñ R q . This is a rank p ´ R -module withgenerators permuted by the reduced regular representation of C p . We similarly use R to denote the R r C p s -module that is isomorphic to R with trivial action. Wesometimes use ρ R to denote R r C p s itself, and write free to denote a sum of copiesof ρ R . 1.4. Acknowledgments
The authors thank Mike Hill, Mike Hopkins, and Doug Ravenel for inspiring numerousideas in this document, as well as their consistent encouragement and interest in the work.We would especially like to thank Mike Hill for suggesting that our earlier definition of v µ p might be more conceptually viewed as a norm. We additionally thank Robert Burklund,Hood Chatham and Danny Shi for several useful conversations.The first author was supported by NSF grant DMS-1803273, and thanks Yuzhi Chen andWenyun Liu for their hospitality during the writing of this paper. The second author wassupported by an NSF GRFP fellowship under Grant No. 1122374. The third author wassupported by NSF grant DMS-1902669.2. Orientation theory
Non-equivariantly, one may study complex orientations of any unital spectrum R . How-ever, if R is further equipped with the a homotopy commutative multiplication, then the JEREMY HAHN, ANDREW SENGER, AND DYLAN WILSON theory takes on extra significance: in this case, a complex orientation of R provides anisomorphism R ˚ p CP q – R ˚ rr x ss .In this section, we work out the analogous theory for µ p -orientations. In particular, wefind that the theory of µ p -orientations takes on special significance for C p homotopy ringspectra R that are equipped with a norm N C p e R Ñ R refining the underlying multiplication.Recall the following definition from the introduction: Definition 2.1. A µ p -orientation of a unital C p -spectrum R is a mapΣ CP µ p ÝÑ Σ ` (cid:7) R such that the composite S ` (cid:7) ÝÑ Σ CP µ p ÝÑ Σ ` (cid:7) R is equivalent to Σ ` (cid:7) of the unit.For any C p representation sphere S V , it is traditional to denote by S r S V s the free E -ringspectrum S r S V s “ S ‘ S V ‘ S V ‘ S V ‘ ¨ ¨ ¨ Below, we extend this construction to take input not only representation spheres S V , butspoke spheres as well. Definition 2.2.
For integers n , let S r S nρ ´ ´ (cid:7) s : “ N C p e p S r S np ´ sq b S r S nρ ´ s S , where we consider N C p e S r S np ´ s as a S r S nρ ´ s -bimodule via the E -map induced by thecomposite S nρ ´ Ñ p C p q ` b S np ´ Ñ N C p e S r S np ´ s . In this composite, the first map is adjoint to the identity on S np ´ and the second map isthe canonical inclusion. Note that S r S nρ ´ ´ (cid:7) s is a unital left module over N C p e S r S np ´ s .Furthermore, given a C p -equivariant spectrum R , we set R r S nρ ´ ´ (cid:7) s – R b S r S nρ ´ ´ (cid:7) s . Construction 2.3.
Suppose that R is a homotopy ring in C p -spectra, further equippedwith a genuine norm map N C p e R Ñ R which is unital and restricts on underlying spectra to the composite p Φ e R q b p id b γ b¨¨¨b γ p ´ ÝÝÝÝÝÝÝÝÝÝÑ p Φ e R q b p m ÝÑ Φ e R, where γ P C p is the generator and m is the p -fold multiplication map.If R is µ p -oriented by a map S ´ ´ (cid:7) Ñ R CP µp ` then we may produce a map R r S ´ ´ (cid:7) s Ñ R C P µp ` as follows. First, the composite S ´ e Ñ Φ e p C p ` ^ S ´ q Ñ Φ e p S ´ ´ (cid:7) q Ñ Φ e p R CP µp ` q , where the map e is the inclusion of the factor of S ´ corresponding to the identity in C p ,extends to a map S r S ´ s Ñ Φ e p R CP µp ` q DD PRIMARY ANALOGS OF REAL ORIENTATIONS 9 since the target is a homotopy ring. Norming up, and combining the norm on R with thediagonal map CP µ p Ñ Map p C p , CP µ p q , we get a mapN C p e p S r S ´ sq Ñ N C p e p R CP µp ` q Ñ R CP µp ` . Finally, the extension of C p ` ^ S ´ Ñ R CP µp ` over S ´ ´ (cid:7) provides a nullhomotopy of thecomposite S ´ Ñ p C p q ` b S ´ Ñ R CP µp ` , producing a map S r S ´ ´ (cid:7) s Ñ R CP µp ` . We finish by extending scalars to R . Construction 2.4. If R is µ p -oriented then so too is the Postnikov truncation R ď n . Theconstruction above is natural, and so we may form a map R rr S ´ ´ (cid:7) ss : “ lim ÐÝ R ď n r S ´ ´ (cid:7) s Ñ lim ÐÝp R ď n q CP µp ` » R CP µp ` . Theorem 2.5.
Suppose R is a µ p -oriented homotopy C p ring, further equipped with a unitalhomotopy N C p e R -module structure such that the unit N C p e R Ñ R respects the underlying multiplication in the sense of Construction 2.3. Then, with notationas above, the map R rr S ´ ´ (cid:7) ss ÝÑ R CP µp ` is an equivalence.Proof. By construction, it suffices to prove that the map R ď n r S ´ ´ (cid:7) s ÝÑ p R ď n q CP µp ` is an equivalence for each n ě
0. This is clear on underlying spectra. On geometric fixedpoints we can factor this map as p Φ C p R ď n qr S ´ s Ñ p Φ C p R ď n q B C p ` Ñ Φ C p ´ p R ď n q CP µp ` ¯ , being careful to interpret the source as a module (this is not a map of rings). Specifically,the above composite is one of unital Φ C p N C p e S r S ´ s » S r S ´ s -modules and, separately,one of Φ C p R ď n -modules.The second map is an equivalence by Lemma 2.6 below, so we need only prove the firstmap is an equivalence. Since p “ C p R , the Atiyah-Hirzebruch spectral sequencecomputing π ˚ p Φ C p R ď n q BC p ` has E -page given by π ˚ p Φ C p R ď n q b F p Λ F p p x q b F p F p r y s The class x is realized by applying geometric fixed point to the µ p -orientation. The powersof y are obtained from the unit of the unital S r S ´ s -module structure. Using the Φ C p R ď n -module structure, this implies that the spectral sequence degenerates and moreover that thefirst map is an equivalence. (cid:3) Lemma 2.6. If R is bounded above, and X is a C p -space of finite type, then the map p Φ C p R q X Cp ` Ñ Φ C p p R X ` q is an equivalence. Proof.
Write X “ colim X n where the X n are skeleta for a C p -CW-structure on X witheach X n finite. Then the fiber ofΦ C p p R X ` q Ñ Φ C p p R X n ` q becomes increasingly coconnective, and hence the mapΦ C p p R X ` q Ñ lim ÐÝ Φ C p p R X n ` q is an equivalence. We are thus reduced to the case X “ X n finite, where the result followssince Φ C p p´q is exact. (cid:3) Since Z is µ p -oriented by Remark 1.6 and truncated, we have the following corollary ofTheorem 2.5: Corollary 2.7.
There is a natural equivalence Z r S ´ ´ (cid:7) s » Z CP µp ` . Evenness
In this section, we will introduce a notion of evenness in C p -equivariant homotopy the-ory. This is a generalization of the notion of evenness in non-equivariant homotopy theory.Evenness comes in two forms: homological evenness and homotopical evenness. Homolog-ical evenness is a C p -equivariant version of the condition that a spectrum have homologyconcentrated in even degrees, and homotopical evenness corresponds to the condition thata spectrum have homotopy concentrated in even degrees.The main results in this section are Proposition 3.9, which shows that, under certainconditions, a bounded below homologically even spectrum admits a cell decomposition into even slice spheres (defined below), and Proposition 3.16, which shows that there are noobstructions to mapping in a bounded below homologically even spectrum to a homotopicallyeven spectrum. 3.1. Homological Evenness
We begin our discussion of evenness with the definition of an even slice sphere.
Definition 3.1.
We say that a C p -equivariant spectrum is an even slice sphere if it isequivalent to one of the following for some n P Z : p C p q ` b S n , S nρ , S nρ ` ` (cid:7) . A dual even slice sphere is the dual of an even slice sphere. The dimension of a (dual) evenslice sphere is the dimension of its underlying spectrum. Remark 3.2.
The phrase slice sphere is taken from [Wil17b, Definition 2.3], where a G -equivariant slice sphere is defined to be a compact G -equivariant spectrum, each of whosegeometric fixed point spectra is a finite direct sum of spheres of a given dimension.It is easy to check that the (dual) even slice spheres of Definition 3.1 are slice spheres inthis sense. Remark 3.3.
In the case p “
2, the even slice spheres are precisely those of the form p C q ` b S n or S nρ for some n P Z . DD PRIMARY ANALOGS OF REAL ORIENTATIONS 11
Definition 3.4.
We say that a C p -equivariant spectrum X is homologically even if there isan equivalence of Z p p q -modules X b Z p p q » à n S n b Z p p q , where S n is a direct sum of even slice spheres of dimension 2 n . Remark 3.5.
When p “
2, this recovers the notion of homological purity given in [Hil19,Definition 3.2]. However, when p is odd, our definition of homological evenness differsfrom Hill’s definition of homological purity. The most important difference is that we allowthe spoke spheres S nρ ` ` (cid:7) to appear in our definition. This is necessary for CP µ p to behomologically even.As in the non-equivariant case, homological evenness for a bounded below spectrum isequivalent to the existence of an even cell structure. To prove this, we need to recall thefollowing definition: Definition 3.6.
We say that a C p -equivariant spectrum X is regular slice n -connective if:(1) X e is n -connective, and(2) Φ C p X is r np s -connective.Furthermore, we say that X is bounded below if it is regular slice n -connective for someinteger n . Lemma 3.7.
Let X be a bounded below C p -spectrum with the property that Φ C p X is offinite type. Then X is regular slice n -connective if and only if X b Z p p q is regular slice n -connective.Proof. For the underlying spectrum, the follows from the fact that Z p p q detects connectivityof bounded below p -local spectra. For the geometric fixed points, we use the fact thatΦ C p Z p p q “ F p r y s , | y | “
2, detects connectivity of bounded below p -local spectra which areof finite type, since a finitely generated Z p p q -module is trivial if and only if it is trivial aftertensoring with F p . (cid:3) Lemma 3.8.
Let W denote an even slice sphere of dimension n , and suppose that X isregular slice n -connective. Then we have r W, Σ X s “ .Proof. If W is of dimension n , then its underlying spectrum W e is a direct sum of n -spheresand Φ C p W is a r np s -sphere. It therefore follows that W is a regular slice n -sphere in thesense of [Wil17b, § (cid:3) Proposition 3.9.
Suppose that X is a bounded below, homologically even C p -equivariantspectrum with the property that Φ C p X is of finite type, so that there exists a splitting X b Z p p q » à k ě n S k b Z p p q , where S k is a direct sum of k -dimensional even slice spheres. Then X admits a filtration t X k u k ě n such that X k { X k ´ » S k for each k ě n .Proof. By assumption, we are given a splitting X b Z p p q » à k ě n S k b Z p p q , where S k is a direct sum of 2 k -dimensional even slice spheres. By induction on n , it willsuffice to show that the dashed lifting exists in the diagram XS n À k ě n S k b Z p p q » X b Z p p q , since the cofiber of any such lift is a bounded below homologically even C p -spectrum withΦ C p X of finite type and whose Z p p q -homology is À k ě n ` S k b Z p p q .Note that Lemma 3.7 implies that X is regular slice 2 n -connected. Let F be the fiber ofthe Hurewicz map S Ñ Z p p q . Then F is easily seen to be regular slice 0-connective, so that F b X is regular slice 2 n -connective. This implies that r S n , Σ F b X s “ X Ñ Z p p q b X Ñ Σ F b X. (cid:3) Remark 3.10.
It will follow from Example 3.15 and Proposition 3.17 that the followingconverse of Proposition 3.9 holds: if X is bounded below and admits an even slice cellstructure, then X is homologically even.3.2. Homotopical Evenness
We now introduce the homotopical version of evenness.
Definition 3.11.
We say that a C p -equivariant spectrum E is homotopically even if thefollowing conditions hold for all n P Z :(1) π e n ´ E “ . (2) π C p nρ ´ E “ . (3) π C p nρ ´ ´ (cid:7) E “ . Remark 3.12.
All of the examples of homotopically even C p -spectra that we will encouterwill satisfy the following condition for all n P Z :(4) π C p nρ ` (cid:7) E “ . We will say that a homotopically even C p -spectrum satisfies condition (4) if this holds.In fact, the examples which we study satisfy even stronger evenness properties. We havechosen the weakest possible set of properties for which our theorems hold. Remark 3.13.
If we assume condition (1), then we may rewrite conditions (3) and (4) asfollows:(3 ) the transfer maps π e np ´ E Ñ π C p nρ ´ E are surjective for all n P Z .(4 ) the restrction maps π C p nρ E Ñ π e np E are injective for all n P Z .This follows directly from the cofiber sequences defining S ´ (cid:7) and S (cid:7) : S ´ (cid:7) Ñ S ÝÑ p C p q ` b S p C p q ` b S ÝÝÑ S Ñ S (cid:7) . Remark 3.14. If p “
2, Definition 3.11 reduces to the requirement that, for all n P Z :(1) π e n ´ E “ . (2) π C nρ ´ E “ . A C -equivariant spectrum is therefore homotopically even if and only if it is even in thesense of [HM17, Definition 3.1]. Moreover, condition (4) is redundant in the C -equivariantsetting. DD PRIMARY ANALOGS OF REAL ORIENTATIONS 13
Example 3.15.
The Eilenberg–Maclane spectra F p and Z p p q are examples of homotopicallyeven C p -spectra which satisfy condition (4). To verify this, we refer to the reader to theappendix of third author’s thesis [Wil17a, § A], where one may find a computation of theslice graded homotopy groups of F p and Z p p q .At the prime p “
2, there are many examples of homotopically even C -spectra in theliterature, such as MU R , BP R , BP x n y R , E p n q R , K p n q R and E n , where E n is equipped withthe Goerss-Hopkins C -action [HM17, HS20].The main result of Section 5 is that the C p -spectra E p ´ and the C -spectrum tmf p q are homotopically even and satisfy condition (4).When trying to map a bounded below homologically even C p -spectrum into a homotopi-cally even C p -spectrum, there are no obstructions: Proposition 3.16.
Let E be a homotopically even C p -spectrum, and suppose that X is a C p -spectrum equipped with a bounded below filtration t X k u k ě n such that each S k : “ X k { X k ´ is a direct sum of k -dimensional even slice spheres.Then, for any k ě n , every C p -equivariant map X k Ñ E extends to an equivariant map X Ñ E .Proof. It suffices to prove by induction that any map X k Ñ E extends to a map X k ` Ñ E .Using the cofiber sequence Σ ´ S k ` Ñ X k Ñ X k ` , we just need to know that any map from the desuspension of an even slice sphere into E isnullhomotopic. This follows precisely from the definition of homotopical evenness. (cid:3) If E further satisfies condition (4), we have the stronger result: Proposition 3.17.
Let E be a homotopically even C p -ring spectrum which satisfies con-dition (4), and suppose that X is a C p -spectrum equipped with a bounded below filtration t X k u k ě n such that each S k : “ X k { X k ´ is a direct sum of k -dimensional even slice spheres.Then there is a splitting of the induced filtration on X b E by E -modules: X b E » à k ě n S k b E. Proof.
We need to show that the filtration t X k u k ě n splits upon smashing with E . Workingby induction, we see that it suffices to show that all maps S k Ñ Σ S m b E, where k ą m , are automatically null. Enumerating through all of the possible even slicespheres that can appear in S k and S m , and making use of the (non-canonical) equivalence S (cid:7) b S ´ (cid:7) » S ‘ à p ´ ` p C p q ` b S ˘ , we find that this follows precisely from the hypothesis that E is homotopically even andsatisfies condition (4). (cid:3) The homological evenness of CP µ p The main goal of this section is to prove the following theorem:
Theorem 4.1.
The C p -spectrum Σ CP µ p is homologically even. Noting that Φ C p Σ CP µ p “ Σ BC p is of finite type, we may apply Proposition 3.9 andso deduce the following corollary: Corollary 4.2.
There is a filtration t Σ CP nµ p u n ě of Σ CP µ p with subquotients as follows Σ CP nµ p { Σ CP n ´ µ p » $’&’% S mρ ‘ À pp C p q ` b S n q , if n “ mpS mρ ` ` (cid:7) ‘ À pp C p q ` b S n q , if n “ mp ` À pp C p q ` b S n q , otherwise . Warning 4.3.
We believe that there is a filtration t CP nµ p u n ě of the space CP µ p thatrecovers t Σ CP nµ p u n ě upon applying Σ , but we do not prove this here. As such, ourname Σ CP nµ p must be regarded as an abuse of notation: we do not prove that Σ CP nµ p isΣ of a C p -space CP nµ p . In light of the Dold-Thom theorem, it seems likely that the space CP nµ p could be defined as the n th symmetric power of S ` (cid:7) . Remark 4.4.
The identification of the particular even slice spheres appearing in this decom-position is determined by the cohomology of CP µ p as a C p -representation, and in particularfrom the combination of Corollary 2.7, Lemma 4.9 and Proposition 4.10.As an application, we obtain the following analog of the fact that any ring spectrum withhomotopy groups concentrated in even degrees admits a complex orientation: Corollary 4.5.
Let E be a homotopically even C p -ring spectrum. Then E is µ p -orientable.Proof. We wish to show that that the p ` (cid:7) q -suspension of the unit map factors as S ` (cid:7) Ñ Σ CP µ p Ñ Σ ` (cid:7) E. This is an immediate consequence of Corollary 4.2 and Proposition 3.16. (cid:3)
We devote the remainder of the section to the proof of Theorem 4.1. By Corollary 2.7,there is an equivalence Z r S ´ ´ (cid:7) s » Z CP µp ` . This is of finite type, so to prove Theorem 4.1 it will suffice to prove the following theoremand dualize:
Theorem 4.6.
As a C p -equivariant spectrum, S r S nρ ´ ´ (cid:7) s is a direct sum of dual evenslice spheres for all n P Z . To prove this, we will construct a map in from a wedge of dual even slice spheres whichis an equivalence on underlying spectra and geometric fixed points.
Construction 4.7.
The composition S nρ ´ Ñ p C p q ` b S np ´ Ñ N C p e S r S np ´ s Ñ S r S nρ ´ ´ (cid:7) s is canonically null, and hence induces a map r x : S nρ ´ ´ (cid:7) Ñ S r S nρ ´ ´ (cid:7) s . On the other hand, letting x : S np ´ Ñ S r S np ´ s denote the canonical inclusion, there is the norm map Nm p x q : S p np ´ q ρ Ñ N C p e S r S np ´ s Ñ S r S nρ ´ ´ (cid:7) s . Since S r S nρ ´ ´ (cid:7) s is a module over N C p e S r S np ´ s , this implies the existence of maps Nm p x q k ¨ r x ε : S k p np ´ q ρ ` ε p nρ ´ ´ (cid:7) q Ñ S r S nρ ´ ´ (cid:7) s for k P N and ε P t , u . DD PRIMARY ANALOGS OF REAL ORIENTATIONS 15
We first show that the sum of these maps induces an equivalence on geometric fixedpoints:
Proposition 4.8.
Let
Ψ : à k ě ε Pt , u S k p np ´ q ρ ` ε p nρ ´ ´ (cid:7) q Ñ S r S nρ ´ ´ (cid:7) s denote the direct sum of the maps Nm p x q k ¨ r x ε . Then Φ C p p Ψ q is an equivalence.Proof. We have an identificationΦ C p S r S nρ ´ ´ (cid:7) s » S r S np ´ s b S r S n ´ s S » S r S np ´ s b p S b S r S n ´ s S q . Under this identification, the mapΦ C p p Nm p x qq : S np ´ Ñ Φ C p S r S nρ ´ ´ (cid:7) s corresponds to the inclusion of S np ´ into the left factor.Moreover, there is an isomorphismH e ˚ p S b S r S n ´ s S ; Z q – Λ Z p x n ´ q , and the map Φ C p p r x q : S n ´ Ñ Φ C p S r S nρ ´ ´ (cid:7) s sends the fundamental class of S n ´ to x n ´ .It follows that Φ C p p Ψ q induces an isomorphism on homology, so is an equivalence. (cid:3) Our next task is to extend Ψ to a map that also induces an equivalence on underlyingspectra. We will see that this can be accomplished by taking the direct sum with maps frominduced even spheres, which are easy to produce. The main input is a computation of thehomology of the underlying spectrum of S r S nρ ´ ´ (cid:7) s as a C p -representation. Lemma 4.9.
There is a C p -equivariant isomorphism H e ˚ p S r S nρ ´ ´ (cid:7) s ; Z q – Sym ˚ Z p ρ q , where ρ lies in degree .Proof. There are equivariant isomorphismsH e ˚ p S r S nρ ´ s ; Z q – Sym ˚ Z p x q and H e ˚ p N C p e S r S np ´ s ; Z q – Sym ˚ Z p ρ q , where x and ρ both lie in degree 2. Since S r S nρ ´ ´ (cid:7) s is a unital N C p e S r S np ´ s -module,we obtain a map Sym ˚ Z p ρ q Ñ H e ˚ p S r S nρ ´ ´ (cid:7) s ; Z q of Sym ˚ Z p ρ q -modules. Since x goes to zero in H e ˚ p S r S nρ ´ ´ (cid:7) s ; Z q , it follows that this factorsthrough a map Sym ˚ Z p ρ q – Sym ˚ Z p ρ q b Sym ˚ Z p x q Z Ñ H e ˚ p S r S nρ ´ ´ (cid:7) s ; Z q . Examining the K¨unneth spectral sequence, we see that this map must be an isomorphism. (cid:3)
The following theorem in pure algebra determines the structure of the mod p reductionSym ˚ F p p ρ q as a C p -representation: Proposition 4.10 ([AF78, Propositions III.3.4-III.3.6]) . Let ρ denote the reduced regularrepresentation of C p over F p , and let e , . . . e p P ρ denote generators which are cyclicallypermuted by C p and satisfy e ` ¨ ¨ ¨ ` e p “ . We set Nm “ e ¨ ¨ ¨ e p P Sym p F p p ρ q . Then thesymmetric powers of ρ decompose as follows: Sym k F p p ρ q – $’&’% t Nm ℓ u ‘ free if k “ ℓ ¨ pρ t Nm ℓ e , . . . , Nm ℓ e p u ‘ free if k “ ℓ ¨ p ` otherwise.Proof of Theorem 4.6. Let Ψ be as in Proposition 4.8. It follows from Lemma 4.9 andProposition 4.10 that the mod p homology of Φ e p S r S nρ ´ ´ (cid:7) sq splits as im p H e ˚ p Ψ qq ‘ free.Moreover, Ψ is an equivalence on geometric fixed points by Proposition 4.8.It therefore suffices to show that, given any summand of H e k p S r S nρ ´ ´ (cid:7) s ; F p q isomor-phic to ρ , there is a map p C p q ` b S k Ñ S r S nρ ´ ´ (cid:7) s whose image is that summand.Taking the direct sum of Ψ with an appropriate collection of such maps, we obtain an F p -homology equivalence. Since both sides have finitely-generated free Z -homology, this mustin fact be a p -local equivalence, as desired.To prove the remaining claim, it suffices to show that the mod p Hurewicz map π e ˚ p S r S nρ ´ ´ (cid:7) sq Ñ H e ˚ p S r S nρ ´ ´ (cid:7) s ; F p q is surjective in every degree. This follows from the following square π e ˚ p N C p e S r S np ´ sq H e ˚ p N C p e S r S np ´ s ; F p q π e ˚ p S r S nρ ´ ´ (cid:7) sq H e ˚ p S r S nρ ´ ´ (cid:7) s ; F p q , where the top horizontal arrow is a surjection because N C p e S r S np ´ s is a non-equivariantdirect sum of spheres, and the right vertical arrow is a surjection by the proof of Lemma 4.9. (cid:3) Examples of homotopical evenness
In this section, we introduce our principal examples of homotopically even C p -ring spec-tra. By Corollary 4.5, they are also µ p -orientable.Our first examples are the the Morava E -theories E p ´ associated to the height p ´ E p ´ admits an essentially unique C p -action by E -automorphisms. We use this action to view E p ´ as a Borel C p -equivariant E -ring.Our second example is the connective E -ring tmf p q of topological modular forms withfull level 2 structure. The group GL p Z { Z q – Σ acts on tmf p q via modification of thelevel 2 structure, and we view tmf p q as a C -equivariant E -ring via the inclusion C Ă Σ .We will discuss this example in Section 5.2.The main result of this section is the homotopical evenness of the above C p -ring spectra: Theorem 5.1.
The Borel C p -equivariant height p ´ Morava E -theories E p ´ associatedto the Honda formal group over F p p ´ are homotopically even and satisfy condition (4). Theorem 5.2.
The C -ring spectrum tmf p q of connective topological modular forms withfull level structure is homotopically even and satisfies condition (4). DD PRIMARY ANALOGS OF REAL ORIENTATIONS 17
Applying Corollary 4.5, we obtain the following corollary:
Corollary 5.3.
The C p -ring spectra E p ´ and tmf p q are µ p -orientable. Height p ´ Morava E -theory Given a pair p k, G q , where k is a perfect field of characterstic p ą G is a formalgroup G over k of finite height h , we may functorially associate an E -ring E p k, G q , theLubin-Tate spectrum or Morava E -theory spectrum of p k, G q [GH04, Lur18]. There is anon-canonical isomorphism π ˚ E p k, G q – W p k q J u , . . . , u h ´ K r u ˘ s , where | u i | “ | u | “ ´ p and finite height h , a formal group particularly well-studied in homotopytheory is the Honda formal group. The Honda formal group G Honda h is defined over F p , sothe Frobenius isogney may be viewed as a endomorphism F : G Honda h Ñ G Honda h . The Honda formal group is uniquely determined by the condition that F h “ p in End p G Honda h q .The endomorphism ring of the base change of G Honda h to F p h is the maximal order O h inthe division algebra D h of Hasse invariant 1 { h and center Q p . By the functoriality of theLubin-Tate theory construction, the automorphism group S h “ O ˆ h of G Honda h over F p h actson E p F p h , G Honda h q . To keep our notation from becoming too burdensome, we set E p ´ : “ E p F p p ´ , G Honda p ´ q . There is a subgroup C p Ă S p ´ , which is unique up to conjugation. Indeed, such sub-groups correspond to embeddings Q p p ζ p q Ă D p ´ . Since Q p p ζ p q is of degree p ´ Q p ,it follows from a general fact about division algebras over local fields that such a subfieldexists and is unique up to conjugation (cf. [Ser67, Application on pg. 138]). Using any such C p , we may view E p ´ as a Borel C p -equivariant E -ring spectrum.Homotopical evenness of E p ´ will follow from the computation of the homotopy fixedpoint spectral sequence for E hC p p ´ , which was first carried out by Hopkins and Miller and hasbeen written down in [Nav10] and again reviewed in [HMS17]. We recall this computationbelow. The homotopy fixed point spectral sequence takes the formH s p C p , π t E p ´ q ñ π t ´ s E hC p p ´ , so the first step is to compute the action of C p on π ˚ E p ´ .This action may be determined as follows. Abusing notation, let v P π p ´ E p ´ denotea lift of the canonically defined element v P π p ´ E p ´ { p . The element v is fixed modulo p by the S p ´ and in particular the C p -action on E p ´ , so if we fix a generator γ P C p we findthat the element v ´ γv is divisible by p . Set v “ v ´ γv p . Then the two key properties of v are that:(1) v ` γv ` ¨ ¨ ¨ ` γ p ´ v “ v is a unit in π ˚ E p ´ . As a consequence, Nm p v q “ v ¨ γv ¨ ¨ ¨ γ p ´ v is a unit in π ˚ E p ´ which is fixed by the C p -action [Nav10, pg. 498].The existence of an element v satisfying the above two conditions completely determinesthe action of C p on π ˚ E p ´ , as follows. First, let r w P π ´ E p ´ denote any unit, and set w “ v ¨ Nm p r w q P π ´ E p ´ . Then w continues to satisfy (1) and (2) above and determines amap of C p -representations ρ W p F pp ´ q Ñ π ´ E p ´ . This determines a C p -equivariant mapSym ˚ W p F pp ´ q p ρ qr Nm p w q ´ s Ñ π ˚ E p ´ , which identifies π ˚ E p ´ with the graded completion of Sym ˚ W p F pp ´ q p ρ qr Nm p w q ´ s at thegraded ideal generated by the kernel of the essentially unique nonzero map of W p F p p ´ qr C p s -modules ρ W p F pp ´ q Ñ F pp ´ . Remark 5.4.
In Section 7, we will see that the element v is intimately related to the µ p -orientability of E p ´ . For later use, we note that it follows from the above analysis that themap ρ F pp ´ Ñ π p ´ E p ´ {p p, m q induced by v is an isomorphism.Using the above determination of the C p -action on π ˚ E p ´ , as well as Proposition 4.10,one may obtain with some work the following description of H s p C p , π t E p ´ q : Proposition 5.5 (Hopkins–Mahowald, cf. [HMS17, Proposition 2.6]) . There is an exactsequence π ˚ E p ´ tr ÝÑ H ˚ p C p , π ˚ E p ´ q Ñ F p p ´ r α, β, δ ˘ s{p α q Ñ , (1) where | α | “ p , p ´ q , | β | “ p , p ´ p q , and | δ | “ p , p q . Finally, we must recall the differentials in the homotopy fixed point spectral sequence.We let . “ denote equality up to multiplication by an element of W p F p p ´ q ˆ . Then, asexplained in [HMS17, § d p p ´ q` p δ q . “ αβ p ´ δ ´p p ´ q and d p p ´ q ` p δ p p ´ q α q . “ β p p ´ q ` , along with the fact that all differentials vanish on the image of the transfer map.In particular, on the E -page of the homotopy fixed point spectral sequence there are noelements in positive filtration in total degrees 0, ´ ´
2. Indeed, there are no elementsat all in the p´ q -stem.We now have enough information to establish the homotopical evenness of E p ´ . Proof of Theorem 5.1.
Let u P π e E p ´ denote the periodicity element. Then Nm p u q P π C p ρ E p ´ is also invertible, so the RO p C p q -graded equivariant homotopy of E p ´ is 2 ρ -periodic.Therefore, using Remark 3.13, we see that it suffices to show that:(1) π e ´ E p ´ “ π ´ E hC p p ´ “ π e ´ E p ´ Ñ π ´ E hC p p ´ is a surjection.(4) The restriction map π E hC p p ´ Ñ π e E p ´ is an injection.Condition (1) is immediate from the fact that E p ´ is even periodic. Condition (2) is adirect consequence of the above computation of the homotopy fixed point spectral sequence.Condition (3) follows from the following two facts: ‚ The short exact sequence (1) implies that H p C p , π ´ E p ´ q is spanned by the imageof the transfer. ‚ On the E -page of the homotopy fixed point spectral sequence, there are no positivefiltration elements in stem ´ -page of the homotopy fixed point spectralsequence, there are no positive filtration elements in the zero stem. (cid:3) DD PRIMARY ANALOGS OF REAL ORIENTATIONS 19
The spectrum tmf p q as a form of BP x y µ Recall from [Sto12] or [HL16] the spectrum tmf p q of connective topological modularforms with full level 2 structure. In this section we will consider tmf p q as implictly 3-localized. It is a genuine Σ -equivariant E -ring spectrum with Σ -fixed points tmf p q Σ “ tmf, the (3-localized) spectrum of connective topological modular forms. We view tmf p q as a C -spectrum via restriction along an inclusion C Ă Σ .This spectrum has been well-studied by Stojanoska [Sto12]. In particular, Stojanoskacomputes π e ˚ tmf p q “ Z p q r λ , λ s , where | λ i | “ γ of C acts by λ ÞÑ λ ´ λ and λ ÞÑ ´ λ . It follows that λ and λ span a copy of ρ , so that π ˚ tmf p q – Sym ˚ Z p q p ρ q . The corresponding family of elliptic curves is cut out by the explicit equation y “ x p x ´ λ qp x ´ λ q . For later use, we note down some facts about the associated formal group law.
Proposition 5.6.
The -series of the formal group law associated to tmf p q is given by thefollowing formula: r sp x q “ x ` p λ ` λ q x ` p λ ´ λ λ ` λ q x ` p λ ´ λ λ ´ λ λ ` λ q x ` p λ ´ λ λ ` λ λ ´ λ λ ` λ q x ` O p x q It follows that we have the following formulas for v and v : v ” ´ λ ´ λ mod 3 and v ” λ ” λ mod p , v q . Proof.
This is an elementary computation using the method of [Sil09, § IV.1]. (cid:3)
Remark 5.7.
Let v “ ´ λ ´ λ , so that v ” v mod p . Then we have γv ´ v “ pp λ ´ λ q ` λ q ` λ ` λ “ λ , so that γv ´ v “ λ . Note that this element generates π ˚ tmf p q as a Z p q -algebra with C -action. In Section 7,we will relate this element to the µ -orientation of tmf p q .In his thesis, the third author has computed the slices of tmf p q (cf. [HHR16, § Proposition 5.8 ([Wil17a, Corollary 3.2.1.10]) . Given a C p -equivariant spectrum X , let P nn X denote the n th slice of X . The slices of tmf p q are of the form: à n P nn tmf p q » Z p q r S ρ ´ ´ (cid:7) s . We now turn to the proof of Theorem 5.2. Given the computation of the slices of tmf p q in Proposition 5.8, this will follow from Theorem 4.6 and the following proposition: Proposition 5.9.
Let X be a C p -spectrum whose slices are of the form P nn X » S n b Z p p q ,where S n is a direct sum of dual even slice n -spheres. Then X is homotopically even andsatisfies condition (4). The spectrum tmf p q is obtained from the spectrum Tmf p q discussed in the references by taking theΣ -equivariant connective cover. Using the slice spectral sequence, the proof of Proposition 5.9 reduces to the followinglemma:
Lemma 5.10.
Let S denote a dual even slice sphere. Then S b Z p p q is homotopically evenand satisfies condition (4).Proof. If S » S n b p C p q ` , then this follows from the fact that π n ´ Z p p q “ n P Z .If S » S nρ , then this follows from the fact that Z p p q is homotopically even, since thedefinition of homotopically even is invariant under 2 ρ -suspension.If S » S nρ ´ ´ (cid:7) , then condition (1) of Definition 3.11 is clearly satisfied, and conditions(2)-(4) follow from the following statements for all n P Z , which may be read off from[Wil17a, § A.2]: ‚ π C p nρ ` (cid:7) Z p p q “ ‚ π C p nρ ´ Z p p q “ ‚ π C p nρ ` ` λ Z p p q “ , where in the proofs of (3) and (4) we have implicitly used the existence of equivalences S (cid:7) b S ´ (cid:7) » S ‘ à p ´ p C p q ` b S and S (cid:7) b S (cid:7) » S λ ‘ à p ´ p C p q ` b S . (cid:3) v µ p and a formula for its span In this section, given a µ p -oriented C p -ring spectrum R , we will define a class v µ p P π C p ρ p Σ ` (cid:7) R q – π C p ρ ´ ´ (cid:7) R. When p “
2, our construction agrees with the class v R P π C ρ R in the homotopy of a Realoriented C -ring spectrum. Just as v is well-defined modulo p , we will see that v µ p iswell-defined modulo the transfer. We will also give a formula for the image of v µ p in thethe underlying homotopy of R in terms of the classical element v and the C p -action.To define v µ p , we first construct a class v µ p P π C p ρ Σ CP µ p , and then we take its imagealong the µ p -orientation Σ CP µ p Ñ Σ ` (cid:7) R . To begin, we recall an analogous constructionof the classical element v .6.1. The non-equivariant v as a p th power We recall some classical, non-equivariant theory that we will generalize to the equivariantsetting in the next section.
Notation 6.1.
We let β : S » Σ CP Ñ Σ CP denote a generator of the stablehomotopy group π p Σ CP q .Since CP » Ω Σ Z is an infinite loop space, its suspension spectrum Σ CP is anon-unital ring spectrum. This allows us to make sense of the following definition. Definition 6.2.
We define the class v P π p Σ CP to be β p , the p th power of the degree2 generator.There are at least two justifications for naming this class v , which might more commonlybe defined as the coefficient of x p in the p -series of a complex-oriented ring. The relationshipis expressed in the following proposition: DD PRIMARY ANALOGS OF REAL ORIENTATIONS 21
Proposition 6.3.
Let R denote a (non-equivariant) homotopy ring spectrum, equipped witha complex orientation Σ ´ Σ CP Ñ R, which can be viewed as a class x P R p CP q . Then the composite S p ´ v Ñ Σ ´ Σ CP Ñ R records, up to addition of a multiple of p , the coefficient of x p in the p -series r p s F p x q .Proof. Consider the p -fold multiplication map of infinite loop spaces p CP q ˆ p m Ñ CP Applying R ˚ to the above, we obtain a map R ˚ J x K Ñ R ˚ J x , x , ¨ ¨ ¨ , x p K . By the definition of the formal group law ´ ` F ´ associated to the complex orientation, theclass x P R p CP q is sent to the formal sum f p x , x , ¨ ¨ ¨ , x p q “ x ` F x ` F ¨ ¨ ¨ ` F x p . The commutativity of the formal group law ensures that this power series is invariant undercyclic permutation of the x i . The composite in π p ´ R that we must compute is the coeffi-cient of the product x x ¨ ¨ ¨ x p in f p x , x , ¨ ¨ ¨ , x p q . We can consider the power series in asingle variable r p sp x q “ f p x, x, ¨ ¨ ¨ , x q . Since the only degree p monomial in x , ..., x p thatis invariant under cyclic permutation of the x i is the product x ¨ ¨ ¨ x p , the coefficient of x p in r p sp x q will be equal to the coefficient of x x ¨ ¨ ¨ x p in f p x , x , . . . , x p q up to addition ofa multiple of p . (cid:3) Remark 6.4.
The integral homology H ˚ p CP ; Z p p q q is a divided power ring on the Hurewiczimage of β . In particular, the Hurewicz image of v “ β p is a multiple of p times a generatorof H p p CP ; Z p p q q .Consider the ring spectrum M U together with its canonical complex orientationΣ ´ Σ CP Ñ M U.
The integral homology H ˚ p M U ; Z q is the symmetric algebra on the image, under this map,of r H ˚ p CP ; Z q . In particular, the Hurewicz image of v in H p p Σ ´ Σ CP ; Z p p q q is sent to p times an indecomposable generator of H p ´ p M U ; Z p p q q . By [Mil60], this provides anotherjustification for the name v . Remark 6.5.
One might ask whether higher v i , with i ą
1, can be defined in π ˚ p Σ CP q .A classical argument with topological K -theory [Mos68] shows that the Hurewicz image of π ˚ p Σ CP q inside of H ˚ p Σ CP ; Z p p q q is generated as a Z p p q -module by powers of β . For i larger than 1, β p i is not simply p times a generator of H p i p CP ; Z p p q q , so it is impossible tolift the corresponding indecomposable generators of π ˚ p M U q to π ˚ p Σ ´ Σ CP q . However,it may be possible to lift multiples of such generators.Finally, we record the following proposition for later use: Proposition 6.6.
Let A denote a (non-equivariant) homotopy ring spectrum, equipped witha map f : Σ CP Ñ Σ A that induces the zero homomorphism on π (in particular, f is not a complex orientation).Then the image of v in π p ´ A is a multiple of p . Proof.
Let Cα denote the cofiber of α : S p ´ Ñ S .We recall first that, p -locally, the spectrumΣ CP p admits a splitting as Σ Cα ‘ À p ´ k “ S k . Indeed, since α is the lowest positive degreeelement in the p -local stable stems, most of the attaching maps in the standard cell structurefor CP p are automatically p -locally trivial. The only possibly non-trivial attaching map isbetween the p p q th cell and the bottom cell, and this attaching map is detected by the P action on H ˚ p CP ; F p q .By cellular approximation, v : S p Ñ Σ CP must factor through Σ CP p , and againthe lack of elements in the p -local stable stems ensures a further factorization of v throughΣ Cα . Thus, to determine the image of v in π p p Σ A q , it suffices to consider the composite˜ f : Σ Cα Ñ Σ CP p Ñ Σ CP Ñ Σ A. There is by definition a cofiber sequence S Ñ Σ Cα Ñ S p . By the assumption that f is trivial on π , ˜ f must factor as a compositeΣ Cα Ñ S p Ñ Σ A. We now finish by noting that the composite v : S p Ñ Σ Cα Ñ S p must be a multipleof p , because otherwise Cα would split as S p ‘ S . (cid:3) The equivariant v µ p as a norm As we defined the non-equivariant v P π p Σ CP to be the p th power of a degree 2class, we similarly define an equivariant v µ p P π C p ρ Σ CP µ p to be the norm of a degree 2class. We thank Mike Hill for suggesting this conceptual way of constructing v µ p . To seethat Σ CP µ p is equipped with norms, we will make use of the following proposition: Proposition 6.7.
There is an equivalence of C p -equivariant spaces Ω Σ ` (cid:7) Z » CP µ p , where Z denotes the C p -equivariant Eilenberg–Maclane spectrum associated to the constantMackey functor.Proof. This is Remark 1.6. (cid:3)
Construction 6.8.
The above proposition equips the space CP µ p with a natural norm ,meaning a map N C p e pp CP µ p q e q Ñ CP µ p . Indeed, any C p -equivariant infinite loop space Ω Y , like Ω S ` (cid:7) Z , is equipped with a norm N C p e p Ω Y q e Ñ Ω Y. This norm is Ω applied to the C p -spectrum map p C p q ` b Y Ñ Y that is induced from the identity on Y e . Convention 6.9.
For the remainder of this section we fix a (non-canonical) equivalence p CP µ p q e » p CP q ˆ p ´ . The natural map of C p -spaces S ` (cid:7) “ CP µ p Ñ CP µ p DD PRIMARY ANALOGS OF REAL ORIENTATIONS 23 then induces an (again, non-canonical) equivalence p S ` (cid:7) q e » ł p ´ S , giving p ´ classes β , β , . . . , β p ´ P π e p CP µ p q . Choosing our non-canonical equivalence appropriately, we may suppose that the C p -actionon π e p CP µ p ; Z p p q q is given by the rules(1) γ p β i q “ β i ` , if ď i ď p ´ (2) γ p β p ´ q “ ´ β ´ β ´ ¨ ¨ ¨ ´ β p ´ . Definition 6.10.
We let v µ p : S ρ Ñ Σ CP µ p . denote the norm of β . Explicitly, norming the non-equivariant β map yields a map S ρ » N C p e S Ñ N C p e p Φ e p Σ CP µ p qq , and we may compose this with the norm map of Construction 6.8 to make the class v µ p P π C p ρ p Σ CP µ p q . Remark 6.11.
Of course, the choice of the class β above is not canonical. We view thisas a mild indeterminancy in the definition of v µ p , related to the fact that the classical v should only be well-defined modulo p . As we will see later, many formulas we write for v µ p will similarly be well-defined only modulo transfers.6.3. A formula for v µ p in terms of v Our next aim will be to give an explicit formula for the image of v µ p in the underlyinghomotopy of a µ p -oriented cohomology theory. Our formula is stated as Theorem 6.20. Tobegin its derivation, our first order of business is to give a different formula for v µ p modulotransfers: Proposition 6.12. In π e p p Σ CP µ p q , the class pv µ p and the class Tr p β p q differ by p times atransferred class. In particular, Tr p β p q is divisible by p , and the class Tr p β p q p is the restrictionof a class in π C p ρ Σ CP µ p .Proof. Identifying π e p Σ CP µ p q with ρ Z p p q and using the nonunital E -ring structure onΣ CP µ p , we obtain a map Sym p Z p p q p ρ Z p p q q Ñ π e p p Σ CP µ p q under which the norm class Nm maps to the image of v µ p . The conclusion of the propositionthen follows from Lemma 6.13 below. (cid:3) Lemma 6.13.
Let ρ Z p p q denote the reduced regular representation of C p over Z p p q , andlet e , . . . , e p P ρ Z p p q denote generators which are cyclically permuted by C p and satisfy e ` ¨ ¨ ¨ ` e p “ . We set Nm “ e ¨ ¨ ¨ e p P Sym p Z p p q p ρ Z p p q q .Then Tr p e p q is divisible by p , and Nm and Tr p e p q p differ by a transferred class in Sym p Z p p q p ρ Z p p q q . Proof.
To see that Tr p e p q is divisible by p , we expand it out in terms of the basis e , . . . , e p ´ of ρ Z p p q : Tr p e p q “ e p ` ¨ ¨ ¨ ` e pp ´ ` p´ e ´ e ´ ¨ ¨ ¨ ´ e p ´ q p . It is clear from linearity of the Frobenius modulo p that Tr p e p q is divisible by p . Our nextgoal is to show that Nm ´ Tr p e p q p is a transferred class. It is clearly fixed by the C p -action,so we wish to show that its image in ´ Sym p Z p p q p ρ Z p p q q ¯ C p Tr ´ Sym p Z p p q p ρ Z p p q q ¯ is zero. Since p times any fixed point of C p is the transfer of an element, there is anisomorphism ´ Sym p Z p p q p ρ Z p p q q ¯ C p Tr ´ Sym p Z p p q p ρ Z p p q q ¯ – ´ Sym p F p p ρ F p q ¯ C p Tr ´ Sym p F p p ρ F p q ¯ . By Proposition 4.10, there is an isomorphism of C p -representationsSym p F p p ρ F p q – F p t Nm u ‘ free , so that any choice of C p -equivariant map Sym p F p p ρ F p q Ñ F p which is nonzero on Nm restrictsto an isomorphism ´ Sym p F p p ρ F p q ¯ C p Tr ´ Sym p F p p ρ F p q ¯ – F p . A choice of such a map may be made as follows. First, let f : ρ F p Ñ F p denote theequivariant map sending each e i to 1. This induces a map Sym p F p p f q : Sym p F p p ρ F p q Ñ Sym p F p p F p q – F p which sends Nm to 1. We now need to show that the image of Tr p e p q p under Sym p F p p f q is also equal to 1. WritingTr p e p q p “ e p ` ¨ ¨ ¨ ` e pp ´ ` p´ e ´ e ´ ¨ ¨ ¨ ´ e p ´ q p p , we find that its image of Sym p F p p f q is equal to p ´ ´ p p ´ q p p “ p ´ ´ p´ ` O p p qq p ” p, as desired. (cid:3) Proposition 6.12 can be read as the statement that Tr p β p q p is a formula for v µ p P π C p ρ Σ CP µ p ,if one is only interested in v µ p modulo transfers. We often find this formula for v µ p to bemore useful in computational contexts. Convention 6.14.
For the remainder of this section, we fix a C p -ring R together with a µ p -orientation Σ CP µ p Ñ Σ ` (cid:7) R. DD PRIMARY ANALOGS OF REAL ORIENTATIONS 25
Definition 6.15.
The µ p -orientation of R gives rise to a map p Σ CP µ p q e Ñ p Σ ` (cid:7) R q e , which under our fixed identification of p CP µ p q e is given by a mapΣ p CP q ˆ p ´ Ñ à p ´ Σ R. By mapping in the first of the p p ´ q copies of CP , and then projecting to the first of the p p ´ q copies of R , we obtain the underlying complex orientation of R . Warning 6.16.
While it is convenient to give formulas in terms of the underlying complexorientation of Definition 6.15, we stress once again that this is non-canonical, dependingon Convention 6.9. There is no canonical classical complex orientation associated to a µ p -oriented C p -ring. Notation 6.17.
Using Definition 6.2, the underlying complex orientation of R gives rise toa class v “ β p P π e p ´ R . Notation 6.18.
Recall our fixed non-canonical identification p S ` (cid:7) q e » À p ´ S . Let y i P π e S ` (cid:7) correspond to the i th copy of S , so that we have(1) γ p y i q “ y i ` if 1 ď i ď p ´
2, and(2) γ p y p ´ q “ ´ y ´ ¨ ¨ ¨ ´ y p ´ .Then a generic class r P π e p p Σ ` (cid:7) R q – π e S ` (cid:7) b π e p ´ R may be written as r “ y b r ` y b r ` ¨ ¨ ¨ ` y p ´ b r p ´ , where r i P π e p ´ R .The key relationship between the equivariant v µ p and non-equivariant v is expressed inthe following lemma: Lemma 6.19.
The class v “ β p P π e p Σ CP µ p maps to y b v plus a multiple of p in π e p p Σ ` (cid:7) R q . Proof.
The class β p maps to y b r ` y b r ` ¨ ¨ ¨ ` y p ´ b r p ´ for some collection ofelements r , r , ..., r p ´ P π e p ´ R .By Definition 6.2, r “ v , so it suffices to show that each of r , ..., r p ´ is divisible by p .These statements in turn each follow by application of Proposition 6.6. (cid:3) At last, we are ready to state the main result of this section:
Theorem 6.20.
Suppose that the underlying homotopy groups π e ˚ R are torsion-free. Thenthe class v µ p P π e p p Σ ` (cid:7) R q is given, modulo transfers, by the class y b v ´ γ p ´ v p ` y b γv ´ v p ` ¨ ¨ ¨ ` y p ´ b γ p ´ v ´ γ p ´ v p . Proof.
By Proposition 6.12, it is equivalent to show the above formula determines Tr p β p q{ p P π e p p Σ ` (cid:7) R q modulo transfers. But this may be computed directly from Lemma 6.19. (cid:3) Remark 6.21.
Consider the class y b v ´ γ p ´ v p ` y b γv ´ v p ` ¨ ¨ ¨ ` y p ´ b γ p ´ v ´ γ p ´ v p . of Theorem 6.20. If in this formula we replace v by v “ v ` px , for an arbitrary class x P π e p ´ R , the resulting expression differs from the original by y b p x ´ γ p ´ x q ` y b p γx ´ x q ` ¨ ¨ ¨ ` y p ´ b p γ p ´ x ´ γ p ´ x q . This is exactly the transfer, in π e p p Σ ` (cid:7) R q , of y b x . Thus, altering v by a multiple of p does not change the class v µ p modulo transfers.7. The span of v µ p in height p ´ theories In this section, we use the formula of Theorem 6.20 to compute the span of v µ p in theheight p ´ E p ´ and tmf p q , which we verified were µ p -orientable in Section 5.Our main result, stated in Theorems 7.3 and 7.4, proves that the span of v µ p generates thehomotopy of these theories in a suitable sense. This demonstrates a height-shifting phe-nomenon in equivariant homotopy theory: though these theories are height p ´ v µ p indicates that they should be regarded asheight 1 objects in C p -equivariant homotopy theory. Notation 7.1.
Let R denote a C p -ring spectrum, equipped with a µ p -orientationΣ CP µ p Ñ Σ ` (cid:7) R. Precomposition with v µ p then yields a map S ρ Ñ Σ ` (cid:7) R, which by the dualizability of S ` (cid:7) is equivalent to a map of C p -spectra S ρ ´ ´ (cid:7) Ñ R. Engaging in a slight abuse of notation, we will throughout this section denote this map by v µ p : S ρ ´ ´ (cid:7) Ñ R. Definition 7.2.
Given a µ p -oriented C p -ring R , applying π e p ´ gives a homomorphism of Z p p q r C p s -modules π e p ´ v µ p : π e p ´ S ρ ´ ´ (cid:7) Ñ π e p ´ R. The main theorems of this section are as follows:
Theorem 7.3.
Suppose that Σ CP µ p Ñ Σ ` (cid:7) tmf p q is any µ -orientation of tmf p q . Then the map π e v µ p : π e S ρ ´ ´ (cid:7) Ñ π e tmf p q is anisomorphism of Z p q -modules, and thus also of Z p q r C s -modules. Theorem 7.4.
Suppose that Σ CP µ p Ñ Σ ` (cid:7) E p ´ is any µ p -orientation of E p ´ . Then the image of π e p ´ v µ p in π e p ´ E p ´ maps surjectivelyonto the degree p ´ component of π ˚ p E p ´ q{p p, m q . DD PRIMARY ANALOGS OF REAL ORIENTATIONS 27
Remark 7.5.
Note that the map π e S ρ ´ ´ (cid:7) Ñ π e tmf p q of Theorem 7.3 is a map of rank2 free Z p q -modules. Thus, it is an isomorphism if and only if its mod 3 reduction is, whichis a map of rank 2 vector spaces over F .Similarly, the degree 2 p ´ π ˚ p E p ´ q{p p, m q is a rank p ´ F p , generated by u p ´ , u u p ´ , u u p ´ , ¨ ¨ ¨ , u p ´ u p ´ . The map π e p ´ S ρ ´ ´ (cid:7) Ñ π p ´ p E p ´ {p p, m qq of Theorem 7.4 factors through the mod p reduction of its domain,after which it becomes a map of rank p ´ F p .Both Theorems 7.3 and 7.4 thus reduce to a question of whether maps of rank p ´ F p are isomorphisms. These maps are furthermore equivariant, or mapsof F p r C p s -modules, with the actions of C p given by reduced regular representations. We willtherefore find Lemma 7.7 below particularly useful. First, we recall some basic facts fromrepresentation theory. Recollection 7.6.
Given two F p r C p s -modules V and W , the space Hom F p p V, W q inheritsthe structure of a C p -module via conjugation, where γ P C p sends F : V Ñ W to γ ˝ F ˝ γ ´ .Then there is an identificationHom F p p V, W q C p “ Hom F p r C p s p V, W q , so that the transfer determines a linear mapTr : Hom F p p V, W q Ñ
Hom F p r C p s p V, W q . Lemma 7.7.
Let ¯ ρ denote the F p r C p s -module corresponding to the reduced regular repre-sentation of C p . Then a homomorphism φ P Hom F p r C p s p ¯ ρ, ¯ ρ q is an isomorphism if and only if φ ` Tr p ψ q is for any transferred homomorphism Tr p ψ q .More precisely, Hom F p r C p s p ¯ ρ, ¯ ρ q is a local F p r C p s -algebra, with maximal ideal the ideal oftransferred homomorphisms.Proof. Note that ρ is a uniserial F p r C p s -module, i.e. its submodules are totally ordered byinclusion. Since the endomorphism ring of a uniserial module over a Noetherian ring is local[Lam01, Proposition 20.20], the ring Hom F p r C p s p ρ, ρ q is local.There is an identification ρ C p “ , so we obtain a ring homomorphismHom F p r C p s p ρ, ρ q Ñ Hom F p r C p s p ρ C p , ρ C p q “ Hom F p r C p s p , q “ F p . Since this homomorphism is clearly surjective, we learn that its kernel must be equal to themaximal ideal of Hom F p r C p s p ρ, ρ q .On the other hand, for any x P ρ C p and ψ P Hom F p p ρ, ρ q , we haveTr p ψ qp x q “ p ´ ÿ i “ γ i ψ p γ ´ i x q “ p ´ ÿ i “ γ i ψ p x q “ Tr p ψ p x qq “ , where the last equality follows from the fact that the transfer is zero on ρ . It follows thatTr p ψ q lies in the maximal ideal of Hom F p r C p s p ρ, ρ q .Finally, the equivalence Hom F p p ρ, ρ q – t id ρ u ‘ free , shows that the maximal ideal is equal to the image of Tr for dimension reasons. (cid:3) Proof of Theorem 7.3.
Recall that π e tmf p q is a free Z p q -module with basis λ and λ . Inlight of Remark 7.5, it suffices to analyze the image of v µ in its mod 3 reduction, which isa free F -module generated by the reductions of λ and λ . By combining Lemma 7.7 withTheorem 6.20, it suffices to show that a basis for this rank 2 F -module is given by the mod3 reduction of classes v ´ γ v , γv ´ v P π e tmf p q . Here, v P π e tmf p q refers to the class of Notation 6.17, which depends on the chosen µ -orientation. By combining Remark 6.21 and Proposition 5.6, we may as well set v to be ´ λ ´ λ . Using the formulas of [Sto12, Lemma 7.3] (cf. Remark 5.7), we calculate v ´ γ v ” ´ λ mod 3 , and γv ´ v ” λ mod 3 . These clearly generate all of π e tmf p q modulo 3, as desired. (cid:3) Proof of Theorem 7.4.
By arguments analogous to those in the previous proof, it suffices tocheck that v ´ γ p ´ v p , γv ´ v p , ¨ ¨ ¨ , γ p ´ v ´ γ p ´ v p P π e p ´ E p ´ reduce to generators of the degree 2 p ´ π ˚ p E p ´ q{p p, m q . By Remark 6.21,we may assume that γv ´ v p in π e p ´ E p ´ is the element v defined in Section 5.1. Underthis assumption, the p ´ v and its translates under the C p actionon π e p ´ E p ´ . As noted in Remark 5.4, these span π e p ´ E p ´ {p p, m q . (cid:3) References [AF78] Gert Almkvist and Robert Fossum. Decomposition of exterior and symmetric powers of indecom-posable Z { p Z -modules in characteristic p and relations to invariants. In S´eminaire d’Alg`ebre PaulDubreil, 30`eme ann´ee (Paris, 1976–1977) , volume 641 of
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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
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