An extension in the Adams spectral sequence in dimension 54
AAN EXTENSION IN THE ADAMS SPECTRAL SEQUENCE INDIMENSION 54
ROBERT BURKLUND
Abstract.
We establish a hidden extension in the Adams spectral sequence con-verging to the stable homotopy groups of spheres at the prime 2 in the 54-stem. Thisextension is exceptional in that the only proof we know proceeds via Pstragowski’scategory of synthetic spectra. This was the final unresolved hidden -extension inthe Adams spectral sequence through dimension 80. We hope this provides a concisedemonstration of the computational leverage provided by F -synthetic spectra. The determination of the graded ring of stable homotopy groups of spheres is oneof the most concrete and difficult questions in stable homotopy theory. Recent theo-retical advances going via motivic homotopy theory beginning with [DI10], [Lev15] and[GWX18] have culminated in [IWX20] which constitutes the largest single leap forwardin the number of stems we understand, thus far.Though [IWX20] provides near complete information through the 90-stem severaluncertainties remain. The first of these is in the 54-stem where, as pointed out by JohnRognes, the argument in [Isa19] regarding the 2-extension contains a mistake whichleaves open whether κ ¯ κ xor η(cid:15)θ . + κ ¯ κ is divisible by 2 [IWX20, Remark 7.11]. Thisarticle is intended to be the first of several which resolve these uncertainties. Theorem 1.
The element κ ¯ κ in π is divisible by 2. In order to prove theorem 1 we will lift it to a statement in the category of F -syntheticspectra where it becomes easier to prove. The category of F -synthetic spectra, con-structed in [Pst18], is a stable presentably symmetric monoidal ∞ -category whose ob-jects constitute “formal Adams spectral sequences” in the sense of the E -model-categoryoriginally studied by Dwyer, Kan and Stover in [DKS93]. For a proper introduction tosynthetic spectra we direct the reader to [Pst18] and to [BHS19] for a more computa-tional viewpoint. In this paper we will make use of three main properties of syntheticspectra. • Each synthetic spectrum X has bigraded homotopy groups π a,b ( X ) and themonoidal unit S has the property that its bigraded homotopy groups, whichwe denote simply π a,b , form a ring. Further, this ring encodes all informationpresent in the Adams spectral sequence in a sense made precise in [BHS19,Theorem 9.19]. A weak form of this correspondence is demonstrated in fig. 1. • There is an element τ ∈ π , − with the property that Cτ is a ring and anidentification of homotopy rings π t − s,t ( Cτ ) ∼ = Ext s,t A ( F , F ) . The unit providesa graded ring map π t − s,t → Ext s,t A ( F , F ) . • There is a topological realization to the category of spectra given by inverting τ . Again, the unit provides a graded ring map π t − s,t → π t − s . The maps we have described provide a more direct link between topoology and extover the Steenrod algebra than previous techniques allowed. The main reason we use
Date : May 19, 2020. a r X i v : . [ m a t h . A T ] M a y ROBERT BURKLUND
The F -Adams spectral sequence and the F -synthetic homotopy of the sphere
53 54 55 56 5768101214 h Ch Pd MP ∆ h d Gh i ∆ h gmd g il ∆ h d e h Pe h jQ gt ∆ h h e g
53 54 55 56 57 h Ch Pd MP h h i ∆ h d d g il h Q gt Figure 1.
Left: the F -Adams spectral sequence near κ ¯ κ . Right: the bigradedhomotopy groups of the F -synthetic sphere in the same region. Black dots denote τ -torsion-free classes, red dots denote τ -torsion classes, blue, green and magentadots denote τ , τ and τ torsion classes respectively. In order to reconstruct thegroup in a given bidegree one must examine all degrees lying above it. For example π , ∼ = Z / ⊕ Z / with generators τ { il } and τ ρ respectively. In orderto translate from the left picture to the right we remove all boundaries and colorthe permanent cycles by the length of Adams differential that hits them. Thistranslation is made precise in [BHS19, Theorem 9.19]. F -synthetic spectra in this paper over other choices of Adams type homology theory isthe availability of large-scale machine calculations of the ext ring of the Steenrod algebrafrom [Bru97].Breaking with the notation from [BHS19, Theorem 9.19] we will use the followingnotation for elements of F -synthetic homotopy: Given a permanent cycle a , what pre-viously would have been called (cid:101) a we now call { a } . Given a element α ∈ π ∗ , whatpreviously would have been called (cid:101) α we now call α , with the exception of (cid:101) ∈ π , .These changes were suggested by Isaksen and Xu. We hope they better match withexisting conventions in the literature. Proof.
We will prove the theorem by showing that (cid:101) τ { h h i } = κ ¯ κ τ . Using [BHS19,Theorem 9.19(4)] and the Adams spectral sequence calculations from [IWX20] we learn N EXTENSION IN THE ADAMS SPECTRAL SEQUENCE IN DIMENSION 54 3 that there are unique coefficients a i ∈ { , } such that, (cid:101) { h h i } = a η { h P d } + a η { M P } τ + a κ ¯ κ τ ¯ κ { h h i } = a ν { ∆ h g } τ + a η { M P } ¯ κτ + a κ ¯ κ τ (cid:101) κ { h h i } = a ν { ∆ h g } + a η { M P } ¯ κτ + a κ ¯ κ τ • Since h P d is hit by a d differential, { h P d } τ = 0 . • In Cτ we can read off from [Bru97] that h h i = h P d h . This implies a = 1 . • From [IWX20, Lemma 7.21] we know that { h h i } is nonzero after inverting τ . Thus, at least one of a and a is nonzero. • From [BHS19, Theorem 9.19] and [IWX20] we can compute that π , has no τ -torsion, therefore ¯ κ { h P d } = 0 . This allows us to conclude that a = 0 , a = a and a = a . • Similarly, we can compute that π , ∼ = Z / ⊕ Z / , therefore κ ¯ κ = 0 . • Using the fact that one of a and a is nonzero we learn that (cid:54) = (cid:101) κ { h h i } = (cid:101) a ν { ∆ h g } τ + a η { M P } ¯ κτ + a κ ¯ κ τ ) = a (cid:101) ν { ∆ h g } τ, which implies a = 1 . • In Cτ we can read off from [Bru97] that h h ∆ h g = 0 . This implies a = 0 . Altogether, we may conclude that, (cid:101) { h h i } = η { h P d } + κ ¯ κ τ (cid:3) We end by commenting on why this argument cannot be run in the ordinary categoryof spectra. The key point is that η(cid:15)θ . ¯ κ = 0 which means that after multiplying by ¯ κ we’ve lost the ability to distinguish between the two possible extensions. In the Adamsspectral sequence h gM P is hit by a d differential which translates into the productbeing τ -torsion, but still non-trivial synthetically. Then, we are able to resolve theextension because its source jumps filtration in a way that makes it possible to read a off from homological algebra. This same strategy is not viable in the C -motivic worldbecause the differential killing M h d is shorter than the extension. Acknowledgments.
We thank Dan Isaksen for comments on a draft. We thank ZhouliXu for comments, suggesting this question and for introducing us to the joy of compu-tational stable homotopy theory.
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Department of Mathematics, MIT, Cambridge, MA, USA
E-mail address : [email protected] We follow [IWX20, Table 4] in using the notation ∆ h g for Bruner’s element x ,34