CC -EQUIVARIANT TOPOLOGICAL MODULAR FORMS DEXTER CHUA
Abstract.
We compute
TMF B C , the C fixed points of equivariant topolog-ical modular forms. Contents
1. Introduction 12. Equivariant elliptic cohomology 53. The E page of the DSS 94. Differentials in the DSS 135. Identification of the last factor 256. Further questions 29Appendix A. Sage scripts 30References 321. Introduction
Topological K -theory is one of the first examples of generalized cohomologytheories. It admits a natural equivariant analogue — for a G -space X , the group KO G ( X ) is the Grothendieck group of G -equivariant vector bundles over X . Inparticular, KO G ( ∗ ) = Rep( G ) is the representation ring of G .As in the case of non-equivariant K -theory, this extends to a G -equivariantcohomology theory KO G , and is represented by a genuine G -spectrum. We shallcall this G -spectrum KO , omitting the subscript, as we prefer to think of this as aglobal equivariant spectrum — one defined for all compact Lie groups. The G -fixedpoints of this, written KO B G , is a spectrum analogue of the representation ring,with π KO B G = KO G ( ∗ ) = Rep( G ) (more generally, π n KO B G = KO − nG ( ∗ ) ).These fixed point spectra are readily computable as KO -modules. For example, KO B C = KO ∨ KO , KO B C = KO ∨ KU . This corresponds to the fact that C has two real characters, while C has a realcharacter plus a complex conjugate pair. If one insists, one can write KU = KO ∧ Cη ,providing an arguably more explicit description of KO B C as a KO -module. Ingeneral, KO B G decomposes as a direct sum of copies of KO , KU and KSp , with thefactors determined by the representation theory of G [Seg68, p.133–134].From the chromatic point of view, the natural object to study after K -theory iselliptic cohomology, or its universal version, topological modular forms. Equivariantelliptic cohomology, in various incarnations, has been of interest to many people,include geometric representation theorist and quantum field theorists. Most recently,in [GM20], Gepner and Meier constructed integral equivariant elliptic cohomologyand topological modular forms for compact abelian Lie groups, following the outlinein [Lur09] and the groundwork in [Lur18a; Lur18b; Lur19]. The introduction in[GM20] provides a nice overview of the relevant history, whose efforts we shall notattempt to reproduce. a r X i v : . [ m a t h . A T ] J a n DEXTER CHUA
The spectra
TMF B C n can be constructed as follows: in [Lur18b], Lurie constructedthe universal (derived) oriented elliptic curve, which we shall denote p : E → M .Equivariant
TMF is then constructed with the property that
TMF B C n = Γ( E [ n ]; O E [ n ] ) , TMF B S = Γ( E ; O E ) , where E [ n ] is the n -torsion points of the elliptic curve. This is to be compared tothe homotopy fixed points (with trivial group action), where E is replaced by theformal group ˆ E .We are interested in explicit descriptions of these spectra as TMF -modules.Much work was done by Gepner–Meier themselves: in [GM20, Theorem 1.1], theycomputed
TMF B S = TMF ⊕ ΣTMF . This corresponds to the fact that the coherent cohomology of a (classical) ellipticcurve is concentrated in degrees and by Serre duality.As for finite groups, [GM20, Example 9.4] argues that if (cid:96) (cid:45) | G | or (cid:96) > , then TMF B G(cid:96) splits as sums of shifts of
TMF (3) , TMF (2) and TMF . Further,
TMF (3) and TMF (2) can themselves be described as the smash product of TMF with an 8-and 3-cell complex respectively (see [Mat16, Section 4] for details). Thus, we havean explicit description of
TMF B G(cid:96) as a
TMF (cid:96) -module.This leaves us with the case where (cid:96) = 2 , and (cid:96) | | G | . In this paper, we compute TMF B C at the prime . Theorem 1.1.
There is a (non-canonical) isomorphism of -completed TMF -modules
TMF B C ∼ = TMF ⊕ TMF ⊕ TMF ∧ DL, where DL is the spectrum S − ∪ ν S − ∪ η S − ∪ S − , as depicted in Figure 1. This space DL is so named because its dual L is a split summand of the spectrum L defined in [BR19, Definition 2.3]; in fact, L = L ⊕ S . − − − − νη Figure 1.
Cell diagram of DL Remark.
Despite being a -cell complex, the TMF -module
TMF ∧ DL is reallya rank- -module. After base change to the flat cover TMF (3) = BP (cid:104) (cid:105) , allof , η, ν vanish, and the ( − -cell is attached to the ( − -cell via v . Since v isinvertible in TMF (3) , these two cells kill each other off, and we are left with twofree cells in degrees − and − . “oriented” refers to complex orientation of the associated cohomology theory -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 3 Remark.
While the theorem is stated for
TMF , the same result holds for all ellipticcohomology theories. Indeed, by [MM15, Theorem 7.2], taking global sections givesan equivalence of ∞ -categories Γ : QCoh( M ) ∼ → Mod
TMF . Thus, as quasi-coherent sheaves, we have p ∗ O E [2] ∼ = O M ⊕ O M ⊕ O M ⊗ DL.
By the universality of M , the same must hold for all other elliptic cohomologytheories.1.1. Outline of proof.
To prove the theorem, we begin by computing the homotopygroups of
TMF B C . As in the case of TMF , there is a descent spectral sequencecomputing π ∗ TMF B C whose E page is the coherent cohomology of the -torsionpoints of the (classical) universal elliptic curve.Upon computing the E page for TMF B C , one immediately observes that thereare two copies of TMF ’s E page as direct summands (as one would expect fromthe answer). We can identify these copies as follows:(1) Applying Γ to the map p : E [2] → M induces M ; O M ) → Γ( E [2]; O E [2] ) .This is split by the identity section.(2) Since TMF is a genuine C -equivariant cohomology theory, we get a normmap TMF hC = TMF ∧ RP ∞ + → TMF B C . Restricting to the bottom cellof RP ∞ + gives us a map tr : TMF → TMF B C , which we call the transfer.We will explore these further in Section 2.2.To simplify the calculation, we can quotient out these factors, and rephrase ouroriginal theorem as Theorem 1.2.
There is an isomorphism
TMF B C ≡ TMF B C / (1 , tr) (cid:39) TMF ∧ DL.
This is proven by computing the homotopy groups of
TMF B C via its descentspectral sequence, which is now reasonably sparse, followed by an obstruction theoryargument. This implies the original theorem via the observation Lemma 1.3.
Any cofiber sequence of
TMF -modules
TMF ⊕ TMF → ? → TMF ∧ DL splits.Proof. We have to show that [TMF ∧ DL,
ΣTMF ⊕ ΣTMF]
TMF = 0 . This is equivalent to showing that π − TMF ∧ L = 0 . This follows immediately byrunning the long exact sequences building TMF ∧ L from its cells, since π − TMF = π − TMF = π − TMF = π − TMF = 0 . (cid:3) Remark.
At first I only computed the homotopy groups of
TMF B C . The aboveidentification was discovered when I, for somewhat independent reasons, looked intothe homotopy groups of TMF ∧ L , and observed that they looked almost the sameas that of TMF B C . It is, however, TMF ∧ DL that shows up above; there is acofiber sequence TMF ∧ L → TMF ∧ DL → KO , which induces a short exact sequence in homotopy groups. Thus, the homotopygroups of TMF ∧ DL and TMF ∧ L differ by a single copy of π ∗ KO , which is hardto notice after inverting ∆ . On the other hand, the Adams and Adams–Novikovfiltrations of the classes differ, which makes them easy to distinguish in practice. DEXTER CHUA
Remark.
As part of the proof, we compute the homotopy groups π ∗ TMF B C .To describe the group explicitly, under the decomposition, it remains to specify π ∗ TMF ∧ DL = π ∗ TMF B C . This group is given by the direct sum of the ko -likeparts, namely (cid:77) k ∈ Z π ∗ Σ k ko ⊕ π ∗ Σ k ko , and what is depicted in Figures 11 and 13. In these figures, each dot is a copy of Z / , and the greyed out classes are ones that do not survive the spectral sequence(that is, the homotopy groups are given by the black dots). This part is -periodicvia ∆ -multiplication.1.2. Overview.
In Section 2 we provide relevant background on equivariant ellipticcohomology. Building upon the results in [GM20], we construct C n -equivariantelliptic cohomology as a functor Sp op C n → QCoh( E [ n ]) , which gives us the transfermap tr . We then provide an explicit description of the descent spectral sequence forquasi-coherent sheaves over M .In Section 3, we compute the Hopf algebroid presenting E [2] and subsequentlythe E page of the descent spectral sequence for TMF B C using the -Bocksteinspectral sequence. Unfortunately, the coaction involves division in a fairly complexring, and cocycle manipulations throughout the paper are performed with the aidof sage . The relevant sage code is included in Appendix A.In Section 4, we compute the differentials in the descent spectral sequence. Thekey input here is the fact that there is a norm map TMF hC → TMF B C whosecomposite all the way down to TMF hC is well-understood in terms of stuntedprojective spaces. This provides us with a few permanent classes, which combinedwith the TMF -module structure lets us compute all the differentials. Our calculationswill make heavy use of synthetic spectra [Pst18], whose relation to the Adams spectralsequence is laid out in [BHS19, Theorem 9.19].Finally, in Section 5, we use obstruction theory to construct a map
TMF ∧ DL → TMF B C and show that it is an isomorphism.1.3. Conventions. – All categories are ∞ -categories.– Unless otherwise specified, we work in the category of TMF -modules, andall maps are
TMF -module maps. Further, we implicitly complete at theprime .– Our charts follow the same conventions as, say, [Bau08]. In each bidegree, asolid round dot denotes a copy of Z / . More generally, n concentric circlesdenotes a copy of Z / n . A white square denotes Z . A line of slope denotes h multiplication and a line of slope denotes h multiplication. An arrowwith a negative slope denotes a differential. Dashed lines denote hiddenextensions. In particular, a dashed vertical line is a hidden -extension. Weuse Adams grading, so that the horizontal axis is t − s and vertical axis is s .– All synthetic spectra will be based on BP . We choose our grading con-ventions so that π t − s,s ( νX/τ ) = Ext s,tE ∗ E ( E ∗ , E ∗ X ) , i.e. π x,y shows up atcoordinates ( t − s, s ) = ( x, y ) in an Adams chart. This is not the gradingconvention used by [Pst18] and [BHS19].1.4. Acknowledgements.
I would like to thank Robert Burklund for helpful discus-sions on various homotopy-theoretic calculations, especially regarding the applicationof synthetic spectra in Section 4. Further, I benefited from many helpful discussionswith Sanath Devalapurkar, Jeremy Hahn, and Lennart Meier regarding equivariant -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 5 TMF and equivariant homotopy theory in general. Robert and Lennart also pro-vided helpful comments on an earlier draft. Finally, the paper would not have beenpossible without the support of my advisor, Michael Hopkins, who suggested theproblem and provided useful guidance and suggestions throughout.2.
Equivariant elliptic cohomology
Elliptic cohomology.
Let p : E → X be any (derived) oriented elliptic curve,and C n be the cyclic group of order n . Lemma 2.1.
There is an elliptic cohomology functor E ll EC n : Sp op C n → QCoh( E [ n ]) such that for any m | n , we have a natural identification E ll EC n (( C n /C m ) + ) = O E [ m ] . Moreover, if f : X (cid:48) → X is a morphism almost of finite presentation, then f ∗ E ll EC n ( X ) = E ll f ∗ EC n ( X ) ∈ QCoh( f ∗ E [ n ]) . If there is no risk of confusion, we omit the superscript E . Proof. [GM20, Construction 5.4, Proposition 8.2] constructed the S -equivariantversion of this function — there is a symmetric monoidal functor E ll S : Sp op S → QCoh( E ) , E ll S (( S /C m ) + ) = O E [ m ] . The C n case follows from this formally. Let Ind S C n : Sp C n → Sp S be the inductionmap, left adjoint to the restriction map. Then Ind S C n (( C n /C m ) + ) = ( S /C m ) + .Since the restriction map is symmetric monoidal under the smash product, Ind S C n is oplax monoidal. Thus, the composite E ll ∗ C n : Sp op C n Sp op S QCoh( E ) Ind S Cn E ll S , is lax monoidal. Since S is a coalgebra in Sp C n and every object in Sp C n is naturallyan S -comodule, it follows that this functor canonically factors through the categoryof E ll ∗ C n ( S ) = O E [ n ] -modules in QCoh( E ) , which is equivalent to QCoh( E [ n ]) . Functoriality in X follows from functoriality in the S case as in [GM20, Propo-sition 5.6]. (cid:3) Remark.
Unlike the case of S , the map E ll EC n : Sp op C n → QCoh( E [ n ]) is in generalnot symmetric monoidal. Corollary 2.2.
There is a C n -spectrum R such that for any C n -spectrum X , wehave ( R X ) C n = Γ( E [ n ] , E ll C n ( X )) . We call this R the C n -spectrum associated to the elliptic curve E → X . Forexample, when E → X is the universal elliptic curve, then R = TMF .This follows the argument of [GM20, Construction 8.3]. Proof.
By the spectral Yoneda’s lemma [Lur12, Proposition 4.8.2.18], the Yonedaembedding Sp C n → Fun R (Sp op C n , Sp) is an equivalence. Thus, we have to show thatthe functor X (cid:55)→ Γ( E [ n ] , E ll C n ( X )) preserves limits as a functor Sp op C n → Sp .– By construction E ll S : Sp op S → QCoh( E ) preserves limits. One has to check that the ring structure on O E [ n ] = E ll ∗ C n ( S ) that arises this way is thestandard ring structure, which follows from the construction of E ll S . DEXTER CHUA – Since
Ind S C n : Sp C n → Sp S is a left adjoint, it preserves colimits, hence itsop preserves limits. So E ll ∗ C n : Sp op C n → QCoh( E ) preserves limits.– Since QCoh( E [ n ]) is the category of O E [ n ] -modules in QCoh( E ) , the for-getful functor QCoh( E [ n ]) → QCoh( E ) creates limits. So E ll C n : Sp op C n → QCoh( E [ n ]) preserves limits.– Finally, Γ : QCoh( E [ n ]) → Sp is a right adjoint and preserves limits. (cid:3) We are interested in these global sections, which we can write as Γ( E [ n ]; E ll C n ( X )) = Γ( X ; p ∗ E ll C n ( X )) . By computing p ∗ E ll C n ( S ) , this lets us understand the global sections in terms ofquasi-coherent sheaves on X itself. This pushforward is fairly nice by virtue of Lemma 2.3.
The map [ n ] : E → E is flat, hence so is p : E [ n ] → X .Proof. To check that [ n ] : E → E is flat, observe that the map on underlying(classical) stacks is flat. The condition that [ n ] ∗ π t O E = π t O E as sheaves on theunderlying stack is automatic, since π t O E = p ∗ π t O X and p [ n ] = p .For the second part, we have a pullback square E [ n ] EX E p [ n ] where the bottom map is the identity section, and flat morphisms are closed underpullbacks. (cid:3) Corollary 2.4 ([GM20, Lemma 8.1]) . The underlying stack of E [ n ] is the n -torsionpoints of the underlying stack of E .Proof. More generally, given a pullback of a flat morphism between non-connectivespectral Deligne–Mumford stacks, it is also a pullback on the underlying classicalstacks. To see this, since being flat and a pullback is local, we may assume that thestacks are in fact affine, in which case the result is clear. (cid:3)
The unit and transfer maps.
There are two natural maps , tr : O X → p ∗ O E [ n ] . The map is adjoint to the identity map p ∗ O X = O E [ n ] → O E [ n ] , and is a mapof O X -algebras. In particular, it is an O X -module homomorphism that sends to .If X were affine, then this comes from taking the global sections of p : E [ n ] → X .This map is split by the identity section X → E [ n ] .The trace map tr comes from stable equivariant homotopy theory itself. To avoiddouble subscripts, set G = C n . In the category of G -spectra, there are maps G + → S → G + whose composition is (cid:80) g ∈ G g . The first map comes from applying Σ ∞ + to the mapof unbased G -spaces G → ∗ , whereas the second map is the Spanier–Whitehead dualof the first map, using the self-duality of G + . Intuitively, it sends (cid:55)→ (cid:80) g ∈ G G .Now E ll G ( G + ) = E ll { e } ( S ) = O X , so we get maps of O E [ n ] -modules O X tr → O E [ n ] → O X whose composite is n (since G acts trivially on O X ). Applying p ∗ , we get a map tr : O X → p ∗ O E [ n ] . -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 7 It will be useful to relate this to the norm map of C n -spectra. Let R be the C n -spectrum associated to E . Then unwrapping the definitions, we see that tr isthe C n -fixed points of the map R ⊗ G + −→ R ⊗ S , obtained by tensoring up the unique map G → ∗ . Similarly, the norm map isinduced by R ⊗ EG + −→ R ⊗ S , using the Adams isomorphism ( R ⊗ EG + ) G = R hG . Since G includes into EG , thetrace map factors as tr : R −→ R hG Nm −→ R B G , where the left-hand map is the usual inclusion. Since G acts trivially on theunderlying spectrum R , we have R hG = R ∧ BG + , and the left-hand map is theinclusion of the bottom cell of BG + .We now define p ∗ O E [ n ] by the following cofiber sequence in QCoh( X ) : O X ⊕ O X ⊕ tr −→ p ∗ O E [ n ] −→ p ∗ O E [ n ] . We then write R B C n = Γ( X ; p ∗ O E [ n ] ) ,R B C n = Γ( X ; p ∗ O E [ n ] ) . In particular, when X = M and R = TMF , we have a cofiber sequence TMF ⊕ TMF ⊕ tr −→ TMF B C n −→ TMF B C n . In this paper, we are only interested in the case n = 2 .2.3. The descent spectral sequence.
Our main computational tool is the descentspectral sequence, which we recall in this section.Let X be any non-connective spectral Deligne–Mumford stack and F a quasi-coherent sheaf on X . Let U → X be an étale cover of X . Then the sheaf conditiontells us Γ( X ; F ) = Tot(Γ( U × X · · · × X U ; π ∗ F )) , where U × X · · · × X U is the Cěch nerve of the cover, and π : U × X · · · × X U → X is the projection map. The descent spectral sequence is the Bousfield–Kan spectralsequence for the totalization, and the E page is given by the Cěch cohomology E s,t = ˇ H s ( X cl ; π t F ) of the underlying classical stack X cl with respect to the cover U .For us, we have X = M , and U = M (3) , the moduli stack of elliptic curveswith a Γ (3) -structure (i.e. a choice of -torsion point). By [MM15, Theorem 7.2],the map Γ : QCoh( M ) → Mod
TMF is an equivalence of symmetric monoidal categories. Since i : M (3) → M is affine,we know QCoh( M (3)) = Mod i ∗ O M (QCoh( M ))= Mod TMF (3) (Mod TMF ) = Mod
TMF (3) . Thus, we have Γ( M (3) × M × · · · × M M (3) , π ∗ F ) = TMF (3) ∧ TMF · · · ∧
TMF Γ( M ; F ) . So the descent spectral sequence is also the
TMF (3) -based Adams spectral sequencein Mod
TMF .There is a well-known identification
DEXTER CHUA
Lemma 2.5.
The
TMF (3) -based Adams spectral sequence in Mod
TMF is the sameas the BP -based Adams–Novikov spectral sequence in spectra. We only use this result to apply the machinery of synthetic spectra to the descentspectral sequence; the morally correct approach would be to reproduce the theoryof synthetic spectra inside
Mod
TMF , but we’d rather not take that up.
Proof.
Following [Mil81, Section 1], it suffices to show that any
TMF (3) -resolutionof a TMF -module in
Mod
TMF is also an BP -resolution in Sp . To do so, we have toshow that every TMF (3) -injective module in Mod
TMF is BP -injective in Sp , andevery TMF (3) -exact sequence in Mod
TMF is BP -exact in Sp .(1) We have to show that TMF (3) ⊗ TMF X is BP -injective in Sp for any X ∈ Mod
TMF . Since
TMF (3) is complex orientable, there is a homotopyring map M U → TMF (3) . Thus, TMF (3) ⊗ TMF X is a homotopy M U -module, hence
M U -injective, hence BP -injective.(2) Since F (TMF , − ) is right-adjoint to the forgetful functor Mod
TMF → Sp ,by definition of exactness, it suffices to show that if X is BP -injective, then F (TMF , X ) is TMF (3) -injective. Again we may assume X = BP ∧ Y .Recall that TMF (3) = TMF ∧ Z for some even spectrum Z . By evenness, BP is a retract of BP ∧ DZ . Thus, F (TMF , BP ∧ Y ) is a retract of F (TMF , BP ∧ DZ ∧ Y ) = F (TMF ∧ Z, BP ∧ Y ) = F (TMF (3) , BP ∧ Y ) ,which is a TMF (3) -module, hence TMF (3) -injective. (cid:3) For convenience, set A = π ∗ TMF (3) , Γ = π ∗ TMF (3) ⊗ TMF
TMF (3) . Then (Γ , A ) is a Hopf algebroid, and for any TMF -module N = Γ( M ; F ) , we have Ext s Γ ( A, π t (TMF (3) ⊗ TMF N )) = Ext s Γ ( A, π t ( i ∗ F )) ⇒ π t − s N. To perform calculations, it is of course necessary to identify (Γ , A ) explicitly.From [MR08], we have A ≡ π ∗ TMF (3) = Z [ a , a , ∆ − ] , ∆ = a ( a − a ) , | a i | = 2 i, with associated elliptic curve E (cid:48) : y z + a xyz + a yz = x . Spec Γ is the classifying scheme of two curves of the form E (cid:48) that are abstractlyisomorphic, i.e. related by a coordinate transform. Consider the change of coordinates x (cid:55)→ x + ry (cid:55)→ y + sx + t In order to preserve the form of the equation, we need r − s − a s s − st + a s − a t − a s s − t + 3 a s − a s t + 3 a s − a st + a s − a t So we have
Γ = A [ s, t ] /I , where I is the ideal generated by the relations above(we have eliminated r entirely). One checks that Γ is the free A -module on { , s, s , s , t, st, s t, s t } , and these generators exhibits TMF (3) ⊗ TMF
TMF (3) as the sum of suspended copies of TMF (3) . -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 9 We can read off the structure maps of the Hopf algebroid to be η R ( a ) = a + 2 sη R ( a ) = a + a r + 2 t ∆( s ) = s ⊗ ⊗ s ∆( r ) = r ⊗ ⊗ r ∆( t ) = t ⊗ ⊗ t + s ⊗ r. This Hopf algebroid (or rather, the connective version without inverting ∆ ) wasstudied in detail in [Bau08], whose computations and names we will use significantly.3. The E page of the DSS Computing the comodule.
Let q : E (cid:48) → M (3) be the canonical ellipticcurve over M (3) , so that we have a pullback diagram E (cid:48) EM (3) M . q j pi Then we have
TMF (3) ⊗ TMF
TMF B C = Γ( i ∗ p ∗ O E [2] ) = Γ( q ∗ O E (cid:48) [2] ) = TMF (3) B C , and similarly with the bar version.In this section, we compute π ∗ TMF (3) B C as a Γ -comodule, and then quotientout the image of and tr . By Corollary 2.4, π ∗ TMF (3) B C is given by (the globalsections of) the classical scheme of -torsion points of E (cid:48) .The naïve way to compute E (cid:48) [2] is to write down the duplication formula for E (cid:48) and compute its kernel. However, the duplication formula is unwieldy. Instead,we write down the inversion map i : E (cid:48) → E (cid:48) and compute the equalizer with theidentity map. The inversion map is induced by the map of projective spaces P → P [ x : y : z ] (cid:55)→ [ x : − y − a x − a z : z ] The equalizer of i with the identity is then cut out by the equations x (2 y + a x + a z ) = 0 z (2 y + a x + a z ) = 0 Now observe that the -torsion points are contained in the affine chart y = 1 .Indeed, if y = 0 , then the equation defining E tells us x = 0 . So the unique point onthe curve when y = 0 is [0 : 0 : 1] . But this doesn’t satisfy the last equation abovesince a is invertible.Therefore, we work in the y = 1 chart. Following standard conventions, weredefine z = − xy , w = − zy . Then the -torsion points are cut out by the equations z − w + a zw + a w = 02 z − a z − a zw = 02 w − a zw − a w = 0 . Adding the first and last equation gives z + w = 0 . Eliminating w , we find that E (cid:48) [2] = Spec A [ z ] / (2 z − a z + a z ) . In other words, π ∗ TMF (3) B C = A [ z ] / (2 z − a z + a z ) , | z | = − . Since a is invertible, this is a free A -module of rank .The Γ -coaction on π ∗ TMF (3) B C comes directly from the construction of Γ itself;it is given by z (cid:55)→ z − rz − sz + tz . Theorem 3.1.
The map tr : TMF (3) → TMF (3) B C sends to − a z + a z .In particular, by naturality, − a z + a z is a permanent cocycle. The argument is similar to [Die72, Satz 4] (see also [HKR00, Remark 6.15]).
Proof.
This map is a map of
TMF (3) B C -modules. Since z acts trivially on π ∗ TMF (3) , this means z tr 1 = 0 . Since A [ z ] is a UFD, we know that tr 1 must bea multiple of − a z + a z . Moreover, since it is equal to after modding out by z , the multiple must be . (cid:3) So after taking the cofiber by and tr , we get π ∗ TMF (3) B C = A { z, z } ,and the E page of the descent spectral sequence for π ∗ TMF B C is given by Ext Γ ( A, A { z, z } ) .3.2. Computing the cohomology mod . While the coaction itself is fairlycomplicated, there is a major simplification after we reduce mod .By computer calculation (Appendix A), we find that Lemma 3.2.
Let b = a z , b = a z ; | b i | = 2 i. Then A { z, z } = A { b , b } , and there is a short exact sequence of comodules → A { b } / → A { b , b } / → A { b } / → , where both ends are cofree on the indicated generator, inducing a long exact sequencein Ext .More precisely, the class b ∈ A { z, z } / is invariant, while ψ ( b ) − [1] b = [ r ] b . Thus, the connecting map of the long exact sequence in
Ext is given by b (cid:55)→ [ r ] b . For reference, we display the cohomology of A/ in Figure 2, as computed by[Bau08]. This chart is read as follows:– Each dot represents a copy of F ; h -multiplication and h -multiplicationare denoted by lines of slope and / respectively.– [ r ] represents the class x in bidegree (7 , . This class is uniquely charac-terized by the fact that a kills this class, coming from the cobar differential d( a ) = [ a r ] . – The long dotted lines denote the extension h = a ∆ − g .– The classes fading out continue a -periodically, and each “period” consistsof an infinite h tower. -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 11 x a xa x x ∆ g Figure 2.
Cohomology of A/
20 2 4 6 8 10 12 14 16 18 20 22 24 26 28 300123 b x , b Figure 3.
Connecting maps for
Ext Γ ( A, A { b , b } / − b a b x , √ ∆ b √ ∆ x , Figure 4.
Ext Γ ( A, A { b , b } / In Figure 3, we put two copies of this next to each other and draw the connectingdifferential. The resulting cohomology is in Figure 4. The hidden extensions followfrom a “multiplication by √ ∆ ” operator, which we shall next explain.Originally, we have an action of A [ z ] / (2 z − a z + a z ) on A { z, z } / . Sincewe have quotiented out by and − a z + a z , this reduces to an action by A [ z ] / (2 , a z + a z ) . Further, since we are acting on z -multiples only, this reducesto an action by A [ z ] / (2 , a + a z ) . In this ring, we have ∆ = a + a a = a + a a z = ( a (1 + a z )) . One can check via sage that √ ∆ = a (1 + a z ) is invariant in A [ z ] / (2 , z + a z ) ,so acts on Ext Γ ( A, A { z, z } / (see again Appendix A). For example, the survivingclass in bidegree (14 , is √ ∆ b = a z − a a z .3.3. -Bockstein spectral sequence. We now run the -Bockstein spectral se-quence, which we will find to degenerate on the E page. These Bockstein d ’sresemble the d ’s in the descent spectral sequence quite a bit. Thus, despite the factthat a lot of the differentials can be computed by writing down explicit cocycles, wetry our best to argue them formally so that the same argument can be applied tothe d ’s.Looking at the chart in Figure 4, it is not hard to see what to expect. Alldifferentials have bidegree ( − , , and we know that nothing above the zero linesurvives, since (Γ , A ) has no rational cohomology. Thus, for example, the class inbidegree (1 , must be hit by a differential from b . The main work to do is tomake sure nothing exotic happens with the highly a -divisible classes coming from ∆ division.To begin, recall that in the -Bockstein for Ext Γ ( A, A ) , we have d ( a ) = h . Lemma 3.3.
There are no non-zero classes of the form h a on the E page.Proof. If d ( h a ) (cid:54) = 0 , then it doesn’t survive. Otherwise, consider d ( a ) . This mustbe h torsion, so it is an h multiple. Then h d ( a ) = 0 . So d ( h a a ) = h a . (cid:3) The proof of the lemma is much more powerful than the conclusion itself. It tellsyou about what sort of classes can kill h x , and in certain bidegrees, it suffices tobe h divisible.In general, we let x t − s,s denote a class in the corresponding bidegree that generatesthe bidegree after modding out by a - and h -multiples, if this makes sense. Thisclass is well-defined up to a - and h -multiples. Lemma 3.4. d ( b ) = x , and d ( √ ∆ b ) = √ ∆ x , . Note that since x , is only well-defined up to a multiples, this is equivalent tosaying that d ( b ) and d ( √ ∆) b are not a divisible. Since x , is not well-defined,neither is √ ∆ x , , and we are not claiming that there is a single choice of x , forwhich both equations hold. Proof.
First observe that there is a choice of x , that is permanent. Indeed, forany choice of x , , the class d ( x , ) must be h -divisible, so it must be hit by a d from an a -multiple by Lemma 3.3, which we can add to x , , so that it survivesthe d . From the E page onwards, the target bidegree of the differential is byLemma 3.3 again.Now h x , must be hit by a d , and the source can only be h b + O ( a ) , orelse the differential would only hit highly a -divisible elements. So d ( b ) must hita version of x , .The case of √ ∆ b is analogous. (cid:3) Lemma 3.5. d ( a b ) = d ( a √ ∆ b ) = 0 . This implies a d ( b ) = h b . Proof.
Note that b and √ ∆ b generate the -line under a and ∆ ± , and d ( a b ) and d ( a √ ∆ b ) must be in the submodule generated by these and h . We firstshow that the values of the differentials must be a -divisible. Indeed, we cannothave d ( a b ) = h b + O ( a ) , because applying d again would imply that h x , + O ( a ) , -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 13 a contradiction. The argument for d ( a √ ∆ b ) is similar.Set ∆ = 1 , and let x = a b , y = a √ ∆ b . Then we can write d ( x ) = h ( f x + gy ) , d ( y ) = h ( hx + ky ) , for some f, g, h, k odd polynomials in a . Applying d again gives us equations f + f (cid:48) + gh = 0 k + k (cid:48) + gh = 0 gk + gf + g (cid:48) = 0 hf + hk + h (cid:48) = 0 Adding the first two equations together tells us ( f + k ) = ( f + k ) (cid:48) . Since squaringincreases degree while differentiating decreases it, any polynomial that squares toits derivative must vanish. Thus f + k = 0 . Then the last two equations tell us g (cid:48) = h (cid:48) = 0 . But g and h are odd. So they must be . Then the first two terms tellus f = f (cid:48) and k = k (cid:48) , so they must both be zero too. (cid:3) Remark.
We can in fact write down explicit lifts of a b and a √ ∆ b , namely a b + 2 a z, a √ ∆ b + 2 a z, whose coboundary vanishes mod . However, the proofs above will be used for d ’sin the descent spectral sequence too, and we cannot write down explicit cocycles forthat.With ∆ ± and g periodicity, this gives all d ’s. No classes are left in positive s so we are done. The resulting the E page of the descent spectral sequence of TMF B C has a fairly regular pattern, which we exhibit in Figure 5. The names areintentionally left off; they can be found in Figure 6. Figure 5. E page of ANSS4. Differentials in the DSS
We have now computed the E page of the descent spectral sequence of TMF B C .The goal of this section is to compute the differentials.The main difficulty in computing the descent spectral sequence differentials istranslational invariance — the E page is ∆ -invariant, but the E ∞ page will onlybe ∆ -invariant. If we had a connective version, then the leftmost class must bepermanent since there is nothing to hit. Since we do not, we need external means ofdetermining that certain classes are permanent. Once we do so, we can use standardtechniques in homotopy theory to compute the remaining differentials. We begin by computing the d ’s, where most of the hard work lies in. We depictthe end result in Figure 6 for reference. x , = t h tx , x , √ ∆ t h √ ∆ t x , x , ∆ t gx , gt gx , Figure 6. E page of the descent spectral sequenceTo compute the d ’s, we have to show that x , is permanent by explicitlyconstructing a homotopy class t , while √ ∆ x , supports a d . The rest then followsformally using η = 0 . Along the way, we will find a hidden ν -extension from h √ ∆ t to h gt , which will be useful later on. Theorem 4.1.
There is a choice of x , that survives and has order . We call thisclass t . In fact, all choices survive and have order , but we will only get to see this aftercomputing the spectral sequence fully. Proof.
We define t to be the composition t : ΣTMF (cid:44) → TMF ∧ RP ∞ + = TMF hC Nm −→ TMF B C → TMF B C / (1 , tr) , where the first map is the inclusion of the -cell. We claim this this has ANSSfiltration and is non- v -divisible. Then it must be detected by a choice of x , .To do so, consider the composite t (cid:48) : ΣTMF t → TMF B C / (1 , tr) → TMF hC / (1 , tr) = TMF RP ∞ + / (1 , tr) → TMF RP / (1 , tr) . It suffices to prove the same properties for t (cid:48) .The key fact from equivariant homotopy theory we use is the following: let X bea spectrum with trivial C action. Then the cofiber of the composition X ∧ RP n + → X ∧ RP ∞ + = X hC Nm → X hC = X RP ∞ + → X RP m + is Σ X ∧ P n − m − , where P n − m is the stunted projective space.Take n = 1 , so that RP n + = S ∨ S . The cell diagram of Σ P − , is given by − − -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 15 where as usual the attaching maps of degree , , are , η, ν respectively. In thisdiagram, we think of each cell as a TMF -module cell, i.e. a copy of
TMF .We can read off all the information we need from this diagram. We start with
TMF RP , which is the bottom three cells in the diagram. We first understand whathappens when we mod out and tr .Recall that is the global sections of the projection map E [2] → M , which issplit by the identity section M → E [2] . The global sections of the identity sectionis the inclusion of the fixed points
TMF B C → TMF hC → TMF by construction ofequivariant TMF. That is, is a section of the projection TMF RP → TMF ontothe -cell. Thus, quotienting out by kills off the -cell, and we are left with thebottom two cells.By construction, tr is the attaching map of the -cell. Thus, further quotientingby tr adds the -cell, and TMF RP / (1 , tr) is the question mark complex, i.e. thesubcomplex consisting of the ( − -, ( − - and -cell.Finally, t (cid:48) is the attaching map of the -cell. It must factor through the bottomcell since π TMF /η = 0 , and the diagram tells us this map is ν on the bottom cell,as desired. (cid:3) Corollary 4.2.
There is a choice of x , with d ( x , ) = h t .For any choice of x , , d ( x , ) = O ( v ) (i.e. it is divisible by v ). In fact, we will later see that d ( x , ) = 0 . Proof.
We must have η t = 0 , which forces the first part. For the second, we onlyhave to be concerned by the possibility that d ( x , ) = h t + O ( v ) . However, thisis prevented by g -multiplication, since gx , is h -divisible. (cid:3) Our next goal is to show that √ ∆ t is not permanent, and instead supports a d .We can think of this as a d on the hypothetical √ ∆ (which, if existed, must supporta d since √ ∆ supports a non- -divisible d ). The proof is somewhat roundabout.Since t is -torsion, we get a map ΣTMF / → TMF B C picking out t . Thehomotopy groups of tmf / up to the th stem are depicted in Figure 7. We namethese classes as follows — if y ∈ π ∗ TMF is -torsion, we let ˜ y ∈ π ∗ TMF / be theclass that is y on the top cell. This is well-defined up to an element in the image of π ∗ TMF . In particular, we are interested in the following classes:(1) κ ∈ π TMF is well-defined, while ¯ κ ∈ π TMF is well-defined mod .(2) ˜ ν ∈ π TMF / ∼ = Z / is the unique non-zero element in this degree.(3) ˜ κ ∈ π TMF / is well-defined up to ηκ . Thus, ν ˜ κ is well-defined. ν κ ˜ κ η ¯ κ ¯ κ Figure 7. E ∞ page of the ANSS of TMF / Lemma 4.3. In π ∗ TMF / , we have η ¯ κ = ν ˜ κ + κ ˜ ν . Proof.
We start with the Adams spectral sequence for π ∗ tmf , which is depictedin Figure 8. This may be computed by the May spectral sequence or a computer.The only possible d ’s in this range are the ones we have drawn, and any of thedifferentials implies all others by the Leibniz rule. We can get these via the factthat v ν = 0 , for example.From this, Moss’ convergence theorem [Mos70] tells us η ¯ κ = (cid:104) κ, , ν (cid:105) ∈ π ∗ TMF with no indeterminacy. By definition, the right-hand side is given by the composite κ ν Let ν : Σ TMF / → TMF / be the map that first projects to the top TMF -cell,and then maps via ˜ ν . Then ν − ν maps trivially to the top cell, so factors throughthe bottom cell. This is a valid choice of “ ν on the bottom cell”. So η ¯ κ = ( ν − ν )˜ κ = ν ˜ κ − ν ˜ κ. Finally, note that ν ˜ κ = ˜ ν κ , and that everything is -torsion, so we can drop thesigns. (cid:3) c v κ ¯ κ Figure 8. E page of Adams spectral sequence for tmf In this diagram, the spectrum in the middle is (a shift of)
TMF / and the ones at the end are TMF . The map on the left is any map such that if you project onto the top cell of
TMF / , thenthe map is κ (“ κ on the top cell”); the map on the right is any map such that the restriction tothe bottom cell of TMF / is ν (“ ν on the bottom cell”). The composite of any two such choicesgiven an element in the Toda bracket, and vice versa. -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 17 Corollary 4.4. h √ ∆ t represents ˜ κt , and in particular is permanent. Further,there is a hidden ν extension from h √ ∆ t to h gt .Proof. The class t gives a map ΣTMF / → TMF B C . Thus, the previous lemmagives ν ˜ κt = η ¯ κt + ˜ ν κt ∈ π ∗ TMF B C We know that ˜ ν t ∈ π TMF B C has very high ANSS filtration (at least 7) becausethere is nothing in lower degrees. So ν ˜ κt = η ¯ κt + higher filtration .η ¯ κt is represented by h gt , so ˜ κt must be detected by a permanent class withfiltration at most , which must be h √ ∆ t (this can alternatively follow fromcalculating the products in Ext , but we spare ourselves the trouble). (cid:3) If z is a cocycle in the E page, we let [ z ] denote any element of π ∗ that isrepresented by z . Corollary 4.5.
There is a hidden ν extension from ∆ k h √ ∆ t to ∆ k h gt for every k . That is, if ∆ k h √ ∆ t is permanent, then ν [∆ k h √ ∆ t ] is detected by ∆ k h gt .Proof. We work in BP -synthetic spectra, and identify TMF B C with its syntheticanalogue ν TMF B C . We can rephrase the previous result as ν ˜ κt = τ η ¯ κt .Suppose ∆ k h √ ∆ t is permanent. Let α ∈ π k, TMF B C be a class whoseimage in TMF B C /τ is ∆ k h √ ∆ t . Consider its image in TMF B C /τ . Since ∆ survives to the E -page, we know that ∆ lifts to π , TMF /τ (uniquely, since thereis nothing else in the bidegree). Since ˜ κt represents h √ ∆ t , we can write α = ∆ k ν ˜ κt ∈ π k, TMF B C /τ ∼ = Z / . So we know that να = ∆ k ν ˜ κt = τ ∆ k η ¯ κt ∈ π ∗ , ∗ TMF B C /τ . So in π ∗ , ∗ TMF B C , we know that να = τ [∆ k h √ ∆ t ] + O ( τ ) . (cid:3) Lemma 4.6. √ ∆ t does not survive to the E page. Note that if √ ∆ t survived to the E ∞ page, then η ¯ κt would be ν -divisible.However, ν is η -divisible in π ∗ TMF / , and there is no candidate for the η division of η ¯ κt . The proof runs this argument in synthetic spectra to get thestronger claim that it doesn’t survive to E . Proof.
We again work in synthetic spectra.Suppose √ ∆ t survived to the E page. Then it lifts to a class in π , (TMF B C /τ ) ,which we shall call √ ∆ t again. Then ν √ ∆ t = τ η ¯ κt (cid:54) = 0 ∈ π , TMF B C /τ . Now note that √ ∆ t = 0 ∈ π , TMF B C /τ , since it is true algebraically, andthere are no τ multiples in the bidegree. So we get a map of synthetic spectra Σ , TMF / → TMF B C /τ picking out √ ∆ t .From Figure 2, we can read that ν = η ˜ ν ∈ π , TMF / , noting that thereare no τ -divisible classes in that bidegree. Thus, ν √ ∆ t = η ˜ ν √ ∆ t . However, ˜ ν √ ∆ t ∈ π , TMF B C /τ = 0 , which is a contradiction ( τ h = 0 , so the h towerscannot contribute). (cid:3) Since we do not know that Cτ n is a ring, we have to interpret multiplication as compositionhere. Corollary 4.7. d ( √ ∆ t ) = x , .Proof. This is equivalent to saying that d ( √ ∆ t ) is not v -divisible. If it were, thenLemma 3.3 tells us d ( √ ∆ t + O ( v )) = 0 , which the previous argument forbids. (cid:3) Corollary 4.8. d vanishes on any class in bidegree (8 k, .Proof. Pick x , and x , to be such that they generate the -line under v and ∆ ± . Define x , = v x , . Observe that d ( x , ) and d ( x , ) are both v -divisible,so the argument of Lemma 3.5 shows that the d ’s must both in fact be . Theresult follows. (cid:3) This concludes the calculation of the E page.The E page (with the ko -like patterns omitted) is shown in Figure 9. Thedifferentials come from applying the Leibniz rule with d (∆) = h g . We then have d s in Figure 10 that are forced by the hidden ν extensions (it is easy to check thatthere cannot be differentials from the ko -like classes since the possible targets arenon- g -torsion). The E page is then depicted in Figure 9, which is still ∆ -invariant.We will show that the greyed out classes do not survive, while the black ones do.Afterwards, all the remaining differentials are long differentials that kill off high ¯ κ powers. These are shown in Figure 12, and the E ∞ page is shown in Figure 13. Inthe rest of the section, we shall show that the long differentials that occur are indeedwhat we indicated. We then conclude the calculation using that ∆ ± is permanent.We start with the observation that Lemma 4.9.
There are no classes above the s = 24 -line that survive.Proof. Any permanent class above the line is divisible by ¯ κ = 0 . (cid:3) The hardest part is to show that ∆ t is permanent. The difficulty here is againtranslational invariance. Our starting piece of knowledge is that t is permanent,and we want to somehow deduce that ∆ t is permanent too. However, we must notallow ourselves to repeat the argument, using that ∆ t is permanent to deduce that ∆ t is, because it is not.The key property we can make use of is the fact that the class t extends to a mapfrom TMF ∧ RP ∞ . Our job would be easy if ∆ of the bottom cell is permanentin TMF ∧ RP ∞ , but that’s not true. However, we can get by with the followingversion: Lemma 4.10.
The class t : ΣTMF → TMF B C extends to a map from TMF ∧ L . Recall that L is the dual of DL , as in the statement of the main theorem(Theorem 1.1). Its cell diagram is depicted in Figure 14. Proof.
First of all, it extends to
ΣTMF / since t = 0 . The obstruction to extendingto the -cell is (cid:104) η, , t (cid:105) . Since t comes from restricting the norm map TMF hC → TMF B C , we know it extends to a map RP → TMF B C . Let y ∈ π ∗ TMF B C bethe image of the -cell. Then the cell structure of RP (Figure 14) tells us (cid:104) η, , t (cid:105) = 2 y. But all possible images of y are -torsion. So (cid:104) η, , t (cid:105) = 0 . Finally, the obstructionto extending to all of L is (cid:104) ν, η, , t (cid:105) , which is defined since (cid:104) ν, η, (cid:105) = 0 with noindeterminacy. However, the only possible class is a ν -multiple, hence is in theindeterminacy. So ∈ (cid:104) ν, η, , t (cid:105) , and we can extend to L . (cid:3) Remark.
A posteriori , we expect such a map to exist. We know that
TMF B C =TMF ∧ DL , and this is the map TMF ∧ L → TMF ∧ DL whose cofiber is KO . -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 19 t ∆ t ∆ t ∆ t ∆ t Figure 9. E page of the descent spectral sequence t ∆ t ∆ t Figure 10. E page of the descent spectral sequence -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 21 t w t w t w t w t w t ∆ t w ∆ t w t w t w t w t w ∆ t w ∆ t w ∆ t w ∆ t w t w t w t w t ∆ t w ∆ t w ∆ t w ∆ t w t w t w t w t Figure 11. E page of the descent spectral sequence w t w t w t w t w t w t w t w t w t w t w t w t w t w t w t w t w t w t w t w t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t Figure 12.
Remaining long differentials -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 23 w t w t w t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t w ∆ t Figure 13.
The E ∞ page of the descent spectral sequence w w w w νη L RP Figure 14.
Cell diagrams of L and RP Let w k be the k -cell of L . Theorem 4.11.
In the Adams–Novikov spectral sequence of
TMF ∧ L , the class ∆ w survives and has order .Proof. It suffices to prove this for tmf ∧ L instead.We do not know of a way to compute the E page of the Adams–Novikov spectralsequence for tmf ∧ L , as the attaching maps are filtration but non-injective inhomology, so there is no long exact sequence. To remedy this problem, we use amodified Adams spectral sequence via the technology of synthetic spectra.First observe that → tmf is in fact injective in BP -homology, since BP ∧ tmf is non-torsion (see e.g. [Mat16, Corollary 5.2]). So ν (tmf /
2) = ( ν tmf) / .We next construct synthetic ν tmf -modules (cid:101) Q, (cid:101) L by the cofiber sequences Σ , tmf ν (Σtmf / (cid:101) Q Σ , tmf (cid:101) Q (cid:101) L [ ηw ][ νw ] The universal property of the cofiber gives us natural comparison maps (cid:101) Q → ν (tmf ∧ Q ) and (cid:101) L → ν (tmf ∧ L ) . For example, the second map is obtained fromthe first via Σ , tmf ˜ Q (cid:101) L Σ , tmf = ν (Σ tmf) ν (tmf ∧ Q ) ν (tmf ∧ L ) . [ νw ] ν ([ νw ]) Here the top row is a cofiber sequence in the category of synthetic spectra, and thebottom row is ν applied to a cofiber sequence in the category of spectra. The onlything to check is that the left-hand square commutes, which is true since every map S k, → νZ is uniquely of the form νf ; there are no τ -torsion classes in this bidegreesince these would have to be hit by a differential from below the -line.Given this, it suffices to show that ∆ w survives in (cid:101) L . To understand (cid:101) L , westart with Σtmf / , whose ANSS is shown in Figure 15 (with ko -like terms omittedas usual). The only differential to justify is d (∆ w ) = g xw . This is in fact true,but instead of justifying this, let us just observe that if this differential were , then ∆ w would survive in Σtmf / , hence in (cid:101) L .The synthetic cofiber sequence Σ , tmf → ν Σtmf / → (cid:101) Q tells us the E page ofthe ANSS for ˜ Q sits in a long exact sequence between that of Σ , tmf and ν Σtmf / .This is displayed in Figure 16, where the blue differentials are the connecting map. -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 25 The crucial claim in this diagram is that there is a hidden extension η [ νx ] = xw on the E page. Then since (cid:101) L is obtained by killing [ νx ] , we are done.To see this hidden extension, note that x ∈ π ∗ tmf / detects ν on the top cell.If we quotient out the bottom cell in ˜ Q , then we can write the class of interest as η [ νx ] = η (cid:104) ν, η, w (cid:105) = (cid:104) η, ν, η (cid:105) w = ν w , as desired.Finally, it is straightforward to check that there are no classes above ∆ w , so itmust have order . (cid:3) Corollary 4.12.
The class ∆ t in the descent spectral sequence of TMF B C ispermanent and has order .Proof. We previously constructed a map
TMF ∧ L → TMF B C where the bottomcell hits t . Applying ν to this, we get a map t : ν (TMF ∧ L ) → ν TMF B C wherethe bottom cell hits τ t . Now consider t (∆ w ) . This is a permanent class, andsince ∆ ∈ TMF /τ , after modding out by τ , we know that it must hit τ ∆ t . So t (∆ w ) is detected by ∆ t . (cid:3) For the rest of the section, let z = t or ∆ t . It remains to consider the “ w -chains”starting from z . There is a partially defined multiplication-by- w operation on the E page, where w increases stem by . To formally define this, define w to be equalto g , and manually define w z = ˜ κz, w z = ∆ h z, w z = ∆ h z. Corollary 4.13.
The w chain starting from z is permanent.Proof. The argument of Corollary 4.4 shows that w z is permanent. Since w =[ h ∆] is permanent, we know that w z is also permanent. This leaves the w terms,before we can conclude by g = w -periodicity. We observe that ν z = 0 , since z isin the image of tmf ∧ L and ν z = 0 over there. So w z detects (cid:104) ¯ κ, ν , z (cid:105) and ispermanent. (cid:3) Corollary 4.14.
There is a differential d ? (∆ w k z ) = w k +19 z for all k . Here the length of the differential depends on the value of k (mod 4) , which canbe read off the charts. The precise values are, however, unimportant. Proof.
This follows from ¯ κ = 0 and g -division, since these are the only classes thatcan hit them. (cid:3) Identification of the last factor
To identify
TMF B C ∼ = TMF ∧ DL , we map DL in by obstruction theory, andshow it is an isomorphism after base change to TMF (3) . To do so, we need tounderstand the TMF (3) -homology of DL . Lemma 5.1.
We can choose classes y − , y − , y − ∈ π ∗ TMF (3) ∧ DL such that y − k ∈ π − k TMF (3) ∧ DL ; y − is the bottom cell of DL ; { y − , y − } generates π ∗ TMF (3) ∧ DL as a free π ∗ TMF (3) -module; and y − = v − ( a y − + 2 y − ) + O (2 ) ,d ( y − ) ≡ ψ ( y − ) − [1] y − = [ r ] y − + O (2 ) . The choice of y − and y − is pretty much arbitrary. Other choices will result inslightly different formulas. These are chosen to simplify the ensuing calculation. − w [ h w ] x w h g w g x w ∆ w ∆ w Figure 15.
Adams–Novikov spectral sequence for (cid:102) C -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 27 − w x w g x w w h w Figure 16.
Adams–Novikov spectral sequence for (cid:101) Q Proof.
We carefully construct
TMF ∧ DL in the category of TMF -modules. Westart with the bottom cell and attach y − to kill [ r ] y − . The class y − is onlywell-defined up to multiples of y − , which in this case is integral multiples of a y − .The coboundary of a y − is [12 r ] y − , and so ψ ( y − ) = y − + [ kr ] y − , where k ≡ .The next cell will kill of the cocycle { h y − } = [ s ] y − − [ k ( a r + t )] y − − ?([ s ] a − [12 t ]) y − . Here ? is either or , noting that twice the class is a coboundary. There is exactly onechoice of ? for which this cocycle is permanent, since there is a d ([ s ] a − [12 t ]) = h .On the other hand, we know that y − is not entirely well-defined, and we can absorbthe term into y − (and redefine k ). We choose y − so that ? = 1 .We now set ψ ( y − ) = y − + { h y − } . and then the class y − + a y − − ka y − − ( a − a ) y − . is a cocycle, which the top cell kills off. So we get the relation y − + a y − − ( a − a ) y − + O (2 ) , as desired. (cid:3) We now construct a map f : TMF ∧ DL → TMF B C . This is constructed viaobstruction theory. The relevant homotopy groups are in the range [ − , , whichare depicted in Figure 17. In this range, all the homotopy groups come from the ko -like patterns. − − − − − − Figure 17.
Homotopy groups of
TMF B C with ko -like termsIn general, let z − k be the images of y − k under f (after base change to TMF (3) ).In constructing f , the first step is to pick the image of the bottom cell, i.e. thevalue of z − . This lives in π − (TMF B C ) , which is the direct sum of infinitely manycopies of Z . Choosing z − requires a bit of care, but once we have chosen it, we canalways extend it to a full map f . Indeed, the obstructions are νx − , , (cid:104) η, ν, x − , (cid:105) , (cid:104) , η, ν, x − , (cid:105) , which all vanish because π − TMF B C = π − TMF B C = π − TMF B C = 0 . While the extension always exists, it is not unique. Specifically, the extensionto the second cell is not unique. This results in z − being well-defined up to apermanent class in π − TMF B C , which is again infinitely many copies of Z . Afterextending to the second cell, there is a unique way to extend all the way to DL ,because π − TMF B C = π − TMF B C = 0 . -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 29 We are now ready to choose z − . We definitely don’t want this to be v -divisible,but this only defines z − up to v -multiples. After some experimentation, we settleon the following choice: Lemma 5.2.
There is a permanent cocycle of the form z − = v − [ a ( z − a − a z ) + 2 z ] + O (2 ) This will be our choice of z − . This is a lift of ∆ − a x , . Proof.
Computer calculation (see Appendix A) verifies that this is a cocycle mod .In the bidegree ( − , , there are no -Bockstein differentials and DSS differentials,so any cocycle mod n would lift to a permanent cocycle. (cid:3) Given the relation between the y − k , we know there must be some g such that z − = z − a − a z + 2 g + O (2 ) z − = z + a g + O (2) . The indeterminacy tells us g is well-defined up to a permanent class. Lemma 5.3.
There is a choice of f such that g = 0 mod 2 .Proof. In TMF (3) ∧ DL , we have a cobar differential d ( y − ) = [ r ] y − + O (2 ) . So we find that d ( z − ) = [ r ] z − + O (2 ) . On the other hand, by computer calculation, we find that (see Appendix A) d ( z − a a − z ) = [ r ] z − + O (2 ) . Comparing the two, we must have d (2 g ) = O (2 ) . So g is a cocycle mod . Since there are no -Bocksteins, g lifts to an actual cocycle ˜ g ,and g is permanent. So we can subtract g from z − so that g is now . (cid:3) Corollary 5.4.
This choice of f is an equivalence, and hence TMF B C ∼ = TMF ∧ DL .Proof. It suffices to show that f is an equivalence after base change to TMF (3) .Moreover, since we -complete, we can further reduce mod . Both surjectivity andinjectivity in π ∗ are easy linear algebra. (cid:3) Further questions – Is there a “geometric” description of the piece
TMF ∧ DL , similar to therepresentation-theoretic description of KO B G ? Can such a descriptionstreamline the calculations in the paper? (e.g. avoid the need of computersand “explain” the formula for z − ) In particular, the equation − v y − + a y − + 2 y − in TMF (3) ∧ DL looks remarkably similar to the definingequation a z − a z + 2 z in TMF (3) B C .– The initial calculation of the Hopf algebroid is made possible by the factthat the inversion map of a Weierstrass elliptic curve has a simple formula.Is there a similar trick available for C ? Or must we bite the bullet andcompute [3] by hand? Appendix A. Sage scripts
This appendix contains the sage script used to perform the computer calculations. Γ[ z ] / (2 z − a z + a z ) -EQUIVARIANT TOPOLOGICAL MODULAR FORMS 31 a z − a z + 2 z = 0 to get rid of all ∆ = 1 and drop constantsresult = substitute_full ( result , *( SUBS [4]))result = result . quo_rem (zp )[0] * zp assert diff ( a_3p * zp ^3 - a_1p * zp) == 0 √ ∆ is invariantassert reduce ( diff ( a_3p ^2 * (1 + a_1p * zp )) * z, 1) == 0 References [Bau08] Tilman Bauer. “Computation of the homotopy of the spectrum tmf ”.In:
Groups, homotopy and configuration spaces . Vol. 13. Geom. Topol.Monogr. Geom. Topol. Publ., Coventry, 2008, pp. 11–40.[BHS19] Robert Burklund, Jeremy Hahn, and Andrew Senger.
On the boundariesof highly connected, almost closed manifolds . 2019. arXiv: .[BR19] Scott M. Bailey and Nicolas Ricka. “On the Tate spectrum of tmf at theprime 2”. In:
Math. Z.
Math. Z.
126 (1972), pp. 31–39.[GM20] David Gepner and Lennart Meier.
On equivariant topological modularforms . 2020. arXiv: .[HKR00] Michael J. Hopkins, Nicholas J. Kuhn, and Douglas C. Ravenel. “Gener-alized group characters and complex oriented cohomology theories”. In:
J. Amer. Math. Soc.
Algebraic Topology: TheAbel Symposium 2007 . Ed. by Nils Baas et al. Berlin, Heidelberg: SpringerBerlin Heidelberg, 2009, pp. 219–277.[Lur12] Jacob Lurie.
Higher Algebra . eng. 2012.[Lur18a] Jacob Lurie. “Elliptic Cohomology I: Spectral Abelian Varieties”. eng. In:(2018).[Lur18b] Jacob Lurie. “Elliptic Cohomology II: Orientations”. eng. In: (2018).[Lur19] Jacob Lurie. “Elliptic Cohomology III: Tempered Cohomology”. eng. In:(2019).[Mat16] Akhil Mathew. “The homology of tmf ”. eng. In:
Homology, homotopy,and applications
Journal ofPure and Applied Algebra
Journal of Topology
Math. Z.
115 (1970), pp. 283–310.[MR08] Mark Mahowald and Charles Rezk.
Topological Modular Forms of Level3 . 2008. arXiv: .[Pst18] Piotr Pstrągowski.