A v 1 -banded vanishing line for the mod 2 Moore spectrum
AA v -banded vanishing line for the mod 2 Moorespectrum Kevin Chang
Abstract
The mod 2 Moore spectrum C (2) is the cofiber of the self-map S → S . Building on work of Burklund, Hahn, and Senger, we prove that abovea line of slope , the Adams spectral sequence for C (2) collapses at its E -page and characterize the surviving classes. This completes the proofof a result of Mahowald, announced in 1970, but never proven. The sphere spectrum S is perhaps the fundamental object of homotopy theory,and our understanding of its homotopy groups is a good measure of our under-standing of homotopy theory. The Adams spectral sequence is the primary toolused to compute these groups. Computing the Adams spectral sequence for S is an extremely difficult problems; in fact, it is only well-understood in finiteranges.The mod 2 Moore spectrum C (2) is defined as the cofiber of the map S → S . Understanding its homotopy groups and Adams spectral sequence would bea significant step towards understanding these objects for S .Due to [3], the Adams E -page for C (2) does possess nice structure in aninfinite range: above a line of slope , there is a periodicity isomorphism v : E s,t ( C (2)) ∼ = −→ E s +4 ,t +122 ( C (2)) . We will refer to the Adams spectral sequencein this range as the v -periodic Adams spectral sequence . Understanding the v -periodic Adams spectral sequence for C (2) is the first in an infinite sequence ofsteps towards understanding the entire structure of its Adams spectral sequence.In [11], Mahowald announced a theorem bounding the differentials and de-scribing the E ∞ -page in the v -periodic Adams spectral sequence for C (2) . How-ever, he never published a proof. In [6, Proposition 15.8], Burklund, Hahn, andSenger prove a weaker bound on the differentials and the E ∞ -page than Ma-howald envisioned. In this paper, we complete the proof of Mahowald’s result.Our main result is the following: Theorem 1 ([11, Theorem 5]) . Let E s,tr ( C (2)) denote the H F -Adams spectralsequence for C (2) . Then(1) There exist constants c and v such that E s,t ∼ = E s,t ∞ when s ≥ ( t − s ) + c and t − s ≥ v . 1 a r X i v : . [ m a t h . A T ] S e p
2) We can pick the constants c and v in part (1) such that for t − s ≥ v , theinclusion of Adams filtrations F ( t − s ) − . π t − s C (2) (cid:44) −→ F ( t − s )+ c π t − s C (2) is an isomorphism. In Section 2, we explain some prerequisite material on synthetic spectra. Sec-tion 2.1 is a brief introduction to Pstrągowski’s category of synthetic spectra.Section 2.2 introduces v -banded vanishing lines, which will allow us to refor-mulate Theorem 1 more compactly.In Section 3, we state and prove a stronger version of Theorem 1. We alsoinclude a computation of the homotopy of the K (1) -local Moore spectrum, whichis highly relevant to our proof of Theorem 1. This computation is not availableanywhere in the literature, to the best of our knowledge. Section 3 is the meatof this paper. All homology H ∗ written without coefficients is assumed to be taken with co-efficients in F . All Adams spectral sequences will be H F -Adams spectral se-quences unless stated otherwise. When we have a filtered abelian group F • A and r ∈ R , F r A means F (cid:100) r (cid:101) A . For an H F -nilpotent complete spectrum X , F • π ∗ X will denote the H F -Adams filtration on the homotopy of X . I’d like to thank Robert Burklund for suggesting and mentoring this project.I’d also like to thank Slava Gerovitch, Ankur Moitra, and David Jerison fororganizing MIT’s Summer Program in Undergraduate Research (SPUR), wherethis research was conducted.
In order to prove facts about ordinary spectra, we will pass to the more refined ∞ -category Syn H F , whose objects are called synthetic spectra. Rather than ex-plain what synthetic spectra are, we will describe some of their basic properties,as developed by Pstrągowski in [13], and then explain why they are useful to ourwork. Our treatment will be extremely brief, so for a more detailed introductionto synthetic spectra, see [6, Section 9, Appendix A].2 onstruction 1 (Pstrągowski) . There is a symmetric monoidal stable ∞ -category Syn H F equipped with a symmetric monoidal functor ν : Sp → Syn H F that preserves filtered colimits. Objects of Syn H F are called H F -syntheticspectra , although we will refer to them simply as synthetic spectra . Definition 1 ([13, Definition 4.6, Definition 4.9]) . The bigraded sphere S n,n isdefined to be ν S n . Since Syn H F is stable, we can define the bigraded sphere S a,b to be Σ a − b S b,b . For any synthetic spectrum X , the bigraded homotopy groups π a,b ( X ) are defined to be the abelian groups Hom( S a,b , X ) . Definition 2 ([13, Definition 4.27]) . Since Σ is defined as a pushout and ν isa pointed functor, there is a canonical natural transformation Σ ◦ ν → ν ◦ Σ .Applying this natural transformation to the spectrum S − , we get a naturalcomparison map S , − (cid:39) Σ( ν S − ) → ν (Σ S − ) (cid:39) S , . We denote this comparison map τ . We denote the cofiber of τ by Cτ .A synthetic spectrum is said to be τ -invertible if the map τ : Σ , − X → X is an equivalence. Theorem 2 (Pstrągowski) . (1) The localization functor τ − is symmetricmonoidal.(2) The full subcategory of τ -invertible synthetic spectra is equivalent to thecategory of spectra.(3) The composite τ − ◦ ν is equivalent to the identity. Remark 1.
Intuitively, Theorem 2 says that the only difference between ordi-nary spectra and synthetic spectra is the presence of τ . In particular, for anordinary spectrum X , the τ -torsion in the bigraded homotopy groups of νX contains information not captured by the homotopy groups of X . As we willsee (Theorem 3), this information is related to the differentials in the Adamsspectral sequence for X .The next few properties begin to show the relationship between syntheticspectra and Adams spectral sequences. Lemma 1 ([13, Lemma 4.23]) . Let A → B → C be a cofiber sequence of ordinary spectra. Then νA → νB → νC
3s a cofiber sequence of synthetic spectra if and only if → H ∗ A → H ∗ B → H ∗ C → is a short exact sequence. Remark 2.
A short exact sequence → H ∗ A → H ∗ B → H ∗ C → induces a long exact sequence of Adams E -pages: · · · → Ext s,tA ∗ ( F , H ∗ A ) → Ext s,tA ∗ ( F , H ∗ B ) → Ext s,tA ∗ ( F , H ∗ C ) → Ext s +1 ,tA ∗ ( F , H ∗ A ) → Ext s +1 ,tA ∗ ( F , H ∗ B ) → Ext s +1 ,tA ∗ ( F , H ∗ C ) → · · · . As we will see in Lemma 3, this is the long exact sequence in homotopy associ-ated to the cofiber sequence Cτ ⊗ A → Cτ ⊗ B → Cτ ⊗ C. Lemma 2 ([6, Lemma 9.15]) . If a map f : X → Y of ordinary spectra hasAdams filtration at least k , then there exists a factorization Σ , − k νYνX νY τ k (cid:101) fνf Remark 3.
Although the lift (cid:101) f in Lemma 2 is not canonical in general, it ispossible to pick a canonical lift for k = 1 . Suppose f : X → Y is a map ofordinary spectra with Adams filtration at least 1; in other words, suppose itinduces the 0 map on homology. If we take cofibers Y → Z → Σ X, then it follows that → H ∗ Y → H ∗ Z → H ∗ (Σ X ) → is a short exact sequence. Hence, Lemma 1 implies that applying ν to thissequence gives us a cofiber sequence νY → νZ → Σ , νX. Extending this to the right and suspending gives us a map (cid:101) f : Σ , νX → νY. It is shown in the proof of [6, Lemma 9.15] that τ (cid:101) f = f , so (cid:101) f is a canonical τ -division of νf . 4 xample 1. Consider the map S → S , and denote its cofiber by C (2) .This is the mod 2 Moore spectrum , the main object of study in this paper.Shifting the cofiber sequence S −→ S → C (2) to the right, we have a cofibersequence S → C (2) → S . This cofiber sequence induces a short exact sequence in homology: → H ∗ S → H ∗ C (2) → H ∗ S → . Thus, Lemma 1 implies that we get a cofiber sequence of synthetic spectra S , → νC (2) → S , . It can be shown (see [6, Lemma 15.4]) that the going-around map to the leftof this cofiber sequence is a map (cid:101) S , → S , such that ν (2) = τ (cid:101) . Inparticular, it turns out that (cid:101) is the τ -division provided by Remark 3. Hence, νC (2) (cid:39) C ( (cid:101) .The existence of such a map (cid:101) agrees with Lemma 2, since 2 induces the 0map on homology. Example 2.
Consider the Hopf fibration η : S → S . Note that η induces the0 map on homology. With similar reasoning to Example 1, it is possible to showthat νC ( η ) (cid:39) C ( (cid:101) η ) for a map (cid:101) η : S , → S , such that ν ( η ) = τ (cid:101) η .Since ν is symmetrical monoidal (Construction 1), we have an equivalence ν ( C (2) ⊗ C ( η )) (cid:39) C ( (cid:101) ⊗ C ( (cid:101) η ) . We will denote these objects Y := C (2) ⊗ C ( η ) and (cid:101) Y := C ( (cid:101) ⊗ C ( (cid:101) η ) .Despite the abstractness of the properties above, synthetic spectra are usefulin a very concrete way to our work. In particular, synthetic spectra provide aclarifying perspective on Adams spectral sequences. This connection is summa-rized in Lemma 3, Theorem 3, and Corollary 1. Lemma 3 ([13, Lemma 4.56]) . For any ordinary spectrum X , there is a naturalisomorphism π t − s,t ( Cτ ⊗ νX ) ∼ = Ext s,tA ∗ ( F , H ∗ X ) . In other words, π ∗ , ∗ ( Cτ ⊗ νX ) is precisely the Adams E -page of X . Lemma 4.
For each nonnegative integer i , there is a natural cofiber sequence Σ , − i Cτ → Cτ i +1 → Cτ i . Proof.
We have a commutative diagram S , − i − S , Cτ i +1 S , − i S , Cτ i Σ , − i Cτ , − i Cτ τ i +1 τ (cid:39) τ i Cτ i +1 → Cτ i → Σ , − i Cτ is a cofiber sequence as well. We get our desired cofiber sequence by extendingthis cofiber sequence to the left: Σ , − i Cτ → Cτ i +1 → Cτ i . Definition 3.
We call the going-around map Cτ i → Σ , − i Cτ from Lemma 4the τ -Bockstein . Theorem 3 ([6, Theorem 9.19]) . Let X be an H F -nilpotent complete spectrumwith strongly convergent Adams spectral sequence. Let x denote a class inthe E s,t -term of the Adams spectral sequence for X . Then the following areequivalent:(1) The differentials d , . . . , d r vanish on x .(2) x , as an element of π t − s,t ( Cτ ⊗ νX ) , lifts to π t − s,t ( Cτ r ⊗ νX ) .(3) x admits a lift to π t − s,t ( Cτ r ⊗ νX ) whose image under the τ -Bockstein Cτ r ⊗ νX → Σ , − r Cτ ⊗ νX is − d r +1 ( x ) .If x is a permanent cycle, then there exists a lift along the map π t − s,t ( νX ) → π t − s,t ( Cτ ⊗ X ) . For such a lift (cid:101) x , the following statements hold:(1) If x survives to the E r +1 -page, then τ r − (cid:101) x (cid:54) = 0 .(2) If x survives to the E ∞ -page, then the image of (cid:101) x in π t − s X is of Adamsfiltration s and is detected by x in the Adams spectral sequence for X .Furthermore, there always exists a choice of lift (cid:101) x satisfying the following:(1) If x is hit by a d r +1 -differential, then we can pick (cid:101) x to be τ r -torsion.(2) If x survives to the E ∞ -page and detects a class α ∈ π t − s X , then we canpick (cid:101) x ∈ π t − s,t ( νX ) mapping to α when we invert τ . Corollary 1 ([6, Corollary 9.21]) . Let X be an H F -nilpotent complete spec-trum with strongly convergent Adams spectral sequence. Then the decreasingfiltration of π t − s X given by F s π t − s X := im( π t − s,t ( νX ) → π t − s X ) coincides with the Adams filtration. 6 efinition 4. Given a synthetic spectrum Y , we will denote the filtration fromCorollary 1 by F • π ∗ ( τ − Y ) . Corollary 1 shows that this coincides with theAdams filtration on π ∗ X when Y (cid:39) νX , so there is no notational conflict. Remark 4.
The results above generalize Adams differentials in ordinary spectrato τ -Bocksteins in synthetic spectra. Even though classes in π ∗ , ∗ ( Cτ ⊗ Y ) nolonger correspond to classes in an Adams spectral sequence when Y is not of theform νX , it will still make sense to talk about classes in π ∗ , ∗ ( Cτ ⊗ Y ) killing otherclasses via τ -Bocksteins. One can think of these τ -Bocksteins as differentials ina modified Adams spectral sequence for τ − Y . This will be important in theproof of Theorem 1.We finish this section by discussing an additional property of synthetic spec-tra called τ -completeness. This property generalizes H F -nilpotent completenessfor ordinary spectra. Definition 5 ([6, Definition A.10]) . A synthetic spectrum X is τ -complete ifthe natural map X → lim ←− i Cτ i ⊗ X is an equivalence, where the maps Cτ i +1 ⊗ X → Cτ i ⊗ X are those fromLemma 4. In other words, X is τ -complete if the τ -Bockstein tower is conver-gent. Proposition 1 ([6, Proposition A.11]) . Let X be an ordinary spectrum. Then X is H F -nilpotent complete if and only if νX is τ -complete. Corollary 2. C ( (cid:101) and (cid:101) Y are τ -complete. Proof.
Both C (2) and Y are H F -nilpotent complete (see [14, Lemma 2.1.15]).Since C ( (cid:101) (cid:39) νC (2) and (cid:101) Y (cid:39) νY , Proposition 1 implies that C ( (cid:101) and (cid:101) Y are τ -complete. Proposition 2.
Limits of τ -complete synthetic spectra are τ -complete. Proof.
Since Cτ is dualizable (see [6, Remark 9.6]), the functor Cτ ⊗− commuteswith limits. Then the proposition follows from the fact that limits commute withlimits. v -banded vanishing lines The statement of our main result (Theorem 1) is quite unwieldy. We wouldlike to prove the existence of a line in the Adams spectral sequence for C (2) above which the Adams spectral sequence is well-behaved in sufficiently hightopological degree. Being “well-behaved” consists of the following key qualities:(1) Above some line of slope less than , the spectral sequence collapses atsome finite page. 72) The only classes that survive to the E ∞ -page live inside a band boundedby lines of slope . These classes are v -periodic .In order to prove our main result, we will want to analyze similar behaviorin the bigraded homotopy of synthetic spectra that do not come from ordinaryspectra. For these synthetic spectra, it is not possible to describe this behaviorin terms of Adams spectral sequences. We make this phenomenon precise byusing the following definition. Definition 6 ([6, Definition 13.6]) . Let X be a synthetic spectrum. We saythat X has a v -banded vanishing line with • band intercepts b ≤ d • range of validity v • line of slope m < and intercept c • torsion bound r if the following conditions hold:(1) Every τ -power torsion class in π t − s,t X is τ r -torsion for s ≥ m ( t − s ) + c and t − s ≥ v .(2) The natural map F ( t − s )+ b π t − s ( τ − X ) → F m ( t − s )+ c π t − s ( τ − X ) is an isomorphism for t − s ≥ v .(3) The composite F ( t − s )+ b π t − s ( τ − X ) → π t − s ( τ − X ) → π t − s ( L K (1) τ − X ) is an isomorphism for t − s ≥ v .(4) π t − s,t X ∼ = 0 for s > ( t − s ) + d .More concisely, we will say that X has a v -banded vanishing line with param-eters ( b ≤ d, v, m, c, r ) . Remark 5.
Given an H F -nilpotent complete ordinary spectrum X , we willsay that the Adams spectral sequence of X has a v -banded vanishing line withparameters ( b ≤ d, v, m, c, r ) when the synthetic spectrum νX has one.8igure 1: A cartoon of the E r +1 -page of the Adams spectral sequence for a H F -nilpotent complete spectrum X admitting a v -banded vanishing line withparameters ( b ≤ d, v, m, c, r ) . Here, k := t − s is the topological degree. Thisalso appears as Figure 1 in [6]. v -banded vanishing line for C (2) We recall the statement of our main result, Theorem 1.
Theorem.
Let E s,tr ( C (2)) denote the H F -Adams spectral sequence for C (2) .Then(1) There exist constants c and v such that E s,t ∼ = E s,t ∞ when s ≥ ( t − s ) + c and t − s ≥ v .(2) We can pick the constants c and v in part (1) such that for t − s ≥ v , theinclusion of Adams filtrations F ( t − s ) − . π t − s C (2) (cid:44) −→ F ( t − s )+ c π t − s C (2) is an isomorphism.In other words, part (1) of Theorem 1 states that the v -periodic Adamsspectral sequence for C (2) has no differentials past d , d , d . Part (2) states thatthe only surviving classes above this line fall between the lines s = ( t − s ) − . and s = ( t − s ) .In [6], Burklund, Hahn, and Senger prove part (2) of Theorem 1 and comeclose to a proof of part (1). The authors prove that the synthetic spectrum C ( (cid:101) has a v -banded vanishing line with parameters ( b ≤ d, v, m, c, r ) = (cid:18) − . ≤ ,
28 + 13 , . , , (cid:19) . E -page of the Adams spectral sequence for C (2) . We prove thatto the all the classes in between lines B and C and to the right of line D vanishdue to d , d , d . The surviving classes above line C and to the right of line D are precisely the “lightning flashes” between lines A and B (see [3, Theorem 5.4]for a proof of this periodicity).We will complete the proof of Theorem 1 by proving the following strongerresult: Theorem 4.
The synthetic spectrum C ( (cid:101) admits a v -banded vanishing linewith parameters ( b ≤ d, v, m, c, r ) = ( − . ≤ , , . , , . K (1) -local Moore spectrum We will use the following information about the homotopy groups of L K (1) C (2) .This information can be obtained by examining the long exact sequence in homo-topy associated to the cofiber sequence L K (1) S −→ L K (1) S → L K (1) C (2) . Whilethe homotopy of L K (1) S is well-known, the computation is not readily available10n the literature, so we provide one below. We use the following characterizationof L K (1) S as a homotopy fixed point spectrum. Theorem 5 ([10, Proposition 7.1]) . Let (cid:100) KU denoted 2-completed complex K -theory. This spectrum carries an action of the Morava stabilizer group G ∼ = Z × ,where for i ∈ Z × the action ψ i : (cid:100) KU → (cid:100) KU is the classical i th Adams operation. Then the K (1) -local sphere L K (1) S ishomotopy equivalent to the homotopy fixed point spectrum (cid:100) KU h Z × . Corollary 3.
The K (1) -local sphere L K (1) S has the following homotopy groups: π i L K (1) S ∼ = Z if i = − Z / v if i = 8 k − , k = 2 v m, m odd Z / ⊕ Z if i = 0 Z / if i = 8 k, k (cid:54) = 0 Z / ⊕ Z / if i = 8 k + 1 Z / if i = 8 k + 2 Z / if i = 8 k + 30 if i ∈ { k + 4 , k + 5 , k + 6 } . Here, Z denotes the 2-adic integers. Proof sketch.
There is a splitting of topological groups Z × ∼ = {± }× (1+4 Z ) × ,so Theorem 5 implies that L K (1) S (cid:39) (cid:100) KU h Z × (cid:39) (cid:18) (cid:100) KU h {± } (cid:19) h (1+4 Z ) × (cid:39) (cid:100) KO h (1+4 Z ) × , where (cid:100) KO is 2-completed real K-theory. The fact that KO (cid:39) KU hC is well-known to follow from [5] and a computation using a homotopy fixed point spec-tral sequence (see [15, Proposition 5.3.1] for a proof).The group (1 + 4 Z ) × has 5 as a topological generator, so there is a cofibersequence (cid:100) KO h (1+4 Z ) × → (cid:100) KO − ψ −−−→ (cid:100) KO.
We can therefore use the long exact sequence in homotopy groups associated tothe cofiber sequence L K (1) S → (cid:100) KO − ψ −−−→ (cid:100) KO to compute π ∗ ( L K (1) S ) .This finishes the proof of the theorem for all i except i ≡ , inwhich case there is an extension problem → Z / → π i L K (1) S → Z / → .
11e can deduce that π i ( L K (1) S ) by examining the Adams spectral sequence for S . The following pattern is present in the K (1) -local part of the Adams spectralsequence for the sphere (see [9, Corollary 1.3]). For each integer k > , thedotted line indicates a possible hidden 2-extension in π k +1 L K (1) S from theclass in (8 k + 1 , k ) to the class in (8 k + 1 , k + 1) . The diagram below showsthe pattern for k = 3 .
25 26 2712131415 h P c P h P h We claim that this hidden 2-extension from h P k − c to P k h does notoccur. Suppose it does. Then we have a relation in homotopy [ P k h ] = 2[ h P k − c ] . Then η [ P k h ] = 2 η [ h P k − c ] = 0 , a contradiction. This shows that no extension occurs, and we have π k +1 L K (1) S ∼ = Z / ⊕ Z / . Corollary 4.
The homotopy groups of L K (1) C (2) have the following orders: i (mod 8) (cid:12)(cid:12) π i ( L K (1) C (2)) (cid:12)(cid:12) roof. This follows from considering the long exact sequence in homotopy as-sociated to the cofiber sequence L K (1) S −→ L K (1) S → L K (1) C (2) . This is a cofiber sequence because localization preserves cofiber sequences.There are a few extension problems to solve before we know the group struc-ture of π ∗ L K (1) C (2) . They are the following: → Z / → π k L K (1) C (2) → Z / → → Z / ⊕ Z / → π k +1 L K (1) C (2) → Z / → → Z / → π k +2 L K (1) C (2) → Z / ⊕ Z / → → Z / → π k +3 L K (1) C (2) → Z / → Here, the extension for k is trivial for k = 0 , since Corollary 3 implies that π L K (1) S ∼ = Z / ⊕ Z / .That said, we will need only the orders of these groups, not the groupsthemselves, to prove Theorem 1. After proving Theorem 1, however, we will beable to provide full descriptions of the homotopy groups. In [6], the authors deduce a v -banded vanishing line for C ( (cid:101) from a v -banded vanishing line for (cid:101) Y := C ( (cid:101) ⊗ C ( (cid:101) η ) . This last fact translates to a v -banded vanishing line in the Adams spectral sequence for C (2) ⊗ C ( η ) , since (cid:101) Y (cid:39) ν ( C (2) ⊗ C ( η )) (see [6, Lemma 15.4]). The authors then use the followinglemma to go from (cid:101) Y to C ( (cid:101) through cofiber sequences. Lemma 5 ([6, Proposition 13.11]) . Let A → B → C be a cofiber sequence ofsynthetic spectra such that the following conditions hold: • A has a v -banded vanishing line with parameters ( b A ≤ d A , v A , m, c A , r A ) . • C has a v -banded vanishing line with parameters ( b C ≤ d C , v C , m, c C , r C ) .Then B has a v -banded vanishing line with parameters ( b B ≤ d B , v B , m, c B , r B ) ,where b B = min( b A , b C − r ) d B = max( d A , d C ) v B = max (cid:18) v A + 1 , v C , c B − b B / − m (cid:19) c B = max( c A + r A , c C ) r B = r A + max (cid:18) r C , (cid:22) max( d A , min( d A + r C , d C )) − b C − (cid:23)(cid:19) .
13n [6], the authors start with the following theorem, which is proven usingthe computation of the v -periodic Adams spectral sequence of Y = C (2) ⊗ C ( η ) in [8] and Miller’s method for computing differentials in [12]. Theorem 6 ([6, Theorem 14.1]) . The synthetic spectrum (cid:101) Y = C ( (cid:101) ⊗ C ( (cid:101) η ) hasa v -banded vanishing line with parameters ( − . ≤ , , . , . , .The authors then apply Lemma 5 to the following cofiber sequences (see [6,Proposition 15.8]): Σ , C ( (cid:101) ⊗ C ( (cid:101) η ) → C ( (cid:101) ⊗ C ( (cid:101) η ) → C ( (cid:101) ⊗ C ( (cid:101) η ) (1) Σ , C ( (cid:101) ⊗ C ( (cid:101) η ) → C ( (cid:101) ⊗ C ( (cid:101) η ) → C ( (cid:101) ⊗ C ( (cid:101) η ) (2)Finally, they use the splitting (see [6, Lemma 15.7]) C ( (cid:101) ⊗ C ( (cid:101) η ) (cid:39) C ( (cid:101) ⊕ Σ , C ( (cid:101) to deduce a v -banded vanishing line for C ( (cid:101) .The main content of Theorem 1 lies in decreasing r (cid:101) Y . We will do so byincreasing b (cid:101) Y and then applying Lemma 5 to cofiber sequence 2. Since theformula for r B in Lemma 5 contains − b C , raising b (cid:101) Y will allow us to lower r (cid:101) Y .In concrete terms, this means raising the bottom edges of the bands obtainedfor (cid:101) Y and (cid:101) Y in [6]. This entails showing that classes on and slightly above thesebottom edges are killed by differentials. We prove that these differentials arenonzero by examining all possible classes that occur and exploiting previouslyobtained vanishing lines to show that these classes must be killed.We first deduce a v -banded vanishing line for (cid:101) Y := C ( (cid:101) ⊗ C ( (cid:101) η ) . Ap-plying Lemma 5 and Theorem 6 yields a v -banded vanishing line for (cid:101) Y withparameters ( − . ≤ . , , . , . , . We wish to raise b from − . to − . . Proposition 3. (cid:101) Y admits a v -banded vanishing line with parameters ( − . ≤ . , , . , . , . Proof.
It is enough to show that for
12 ( t − s ) − . ≤ s <
12 ( t − s ) − . with t − s ≥ , the classes in π t − s,t ( Cτ ⊗ (cid:101) Y ) that lift to π t − s,t (cid:101) Y (we will callthese permanent cycles ) have lifts that are killed by some power of τ ( eventualboundaries ). Since we already have a v -banded vanishing line with r = 2 , infact all eventual boundaries will lift to τ -torsion.Our terminology comes from the relationship between bigraded homotopygroups and Adams spectral sequences. Theorem 3 is a precise formulation ofthis relationship for synthetic spectra obtained by applying ν to an ordinaryspectrum. For (cid:101) Y , which is not ν applied to an ordinary spectrum, it may behelpful to think of π t − s,t ( Cτ ⊗ (cid:101) Y ) as the E s,t -term of a modified Adams spectralsequence for τ − (cid:101) Y , if the reader is familiar with this notion.14 a Figure 3: The bigraded homotopy groups π t − s,t ( Cτ ⊗ (cid:101) Y ) . The line is the bottomedge of the band of Theorem 6. The exact homotopy groups are computed nearthe band in [6, Section 14].
28 29 30 31 32 331213141516 b Figure 4: An upper bound on π t − s,t ( Cτ ⊗ (cid:101) Y ) . Black dots indicate possibleclasses in the image of π ∗ , ∗ (Σ , Cτ ⊗ (cid:101) Y ) → π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) , whereas the whitedots indicate possible classes that have nonzero images along the map π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) → π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) . The solid line is the bottom edge of our initial band, andwe want to raise it to the dotted line. 15igures 3 and 4 require some explanation. A dot in position ( t − s, s ) corre-sponds to a copy of F in the bigraded homotopy group π t − s,t ( Cτ ⊗− ) . Figure 3is the bigraded homotopy of Cτ ⊗ (cid:101) Y in this range. Meanwhile, Figure 4 is onlyan upper bound obtained from the long exact sequence associated to the cofibersequence Σ , Cτ ⊗ (cid:101) Y → Cτ ⊗ (cid:101) Y → Cτ ⊗ (cid:101) Y .
We obtain Figure 4 from two copies of Figure 3, with the unfilled dotsbeing unmoved and the filled dots being shifted by (1 , (since they come from Σ , Cτ ⊗ (cid:101) Y → Cτ ⊗ (cid:101) Y ).The classes in π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) and, therefore, the upper bound on π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) (see Figure 4) are v -periodic, so they repeat in increments of (1 , . Hence, wewill only draw them in the indicated range.Observe that the only possible nonzero homotopy groups in the range
12 ( t − s ) − . ≤ s <
12 ( t − s ) − . with t − s ≥ are π k +1 , (2 k +1)+( k − ( Cτ ⊗ (cid:101) Y ) for k ≥ and that these groupsare either F or 0. Figures 3 and 4 illustrate the case k = 15 .For brevity, let i := 2 k + 1 and j := k − . In Figures 3 and 4, we have i = 31 and j = 13 .If π i,i + j ( Cτ ⊗ (cid:101) Y ) ∼ = 0 , then there exists no permanent cycle in π i,i + j ( Cτ ⊗ (cid:101) Y ) ,so there is nothing to check.Now suppose π i,i + j ( Cτ ⊗ (cid:101) Y ) ∼ = F with nonzero element b . Then by exact-ness, there is a unique element a ∈ π i,i + j (Σ , Cτ ⊗ (cid:101) Y ) mapping to b when we smash cofiber sequence (1) with Cτ . Because (cid:101) Y has a v -banded vanishing line with parameters ( − . ≤ , , . , . , , Σ , (cid:101) Y hasa v -banded vanishing line with parameters ( − ≤ . , , . , . , . Apply-ing Theorem 3 to Y shows that a is a permanent cycle, since all the Adamsdifferentials out of (2 k + 1 , k − are 0. Since
15 (2 k + 1) + 3 . ≤ k − <
12 (2 k + 1) ,a lies in the vanishing region of (cid:101) Y . As a permanent cycle in the vanishing region, a lifts to a τ -torsion class ˆ a ∈ π i,i + j (Σ , (cid:101) Y ) . Then the image of ˆ a under themap π i,i + j (Σ , (cid:101) Y ) → π i,i + j (cid:101) Y is a τ -torsion class lifting b .In the language of modified Adams spectral sequences, b is killed by a d induced by a d in the Adams spectral sequence for Y .Since there are no other possible permanent cycles in the range
12 ( t − s ) − . ≤ s <
12 ( t − s ) − . αγa Figure 5: An upper bound on π t − s,t (Σ , Cτ ⊗ (cid:101) Y ) . Black and white dots meanthe same thing as they did in Figure 4. The line is the bottom edge of the bandfrom Proposition 3. We will show that the red differentials are 0.with t − s ≥ , we have shown that (cid:101) Y admits a v -banded vanishing line withparameters ( − . ≤ . , , . , . , .Proposition 3 is good enough to establish the key part of Mahowald’s the-orem, which is that there exist no differentials higher than d in the Adamsspectral sequence for C (2) (i.e. r (cid:101) Y = 3 ). We can see this by applying Lemma 5to the cofiber sequence Σ , (cid:101) Y → (cid:101) Y → (cid:101) Y . However, we will go further. In the following proof, we use similar methodsto show an improved v -banded vanishing line for (cid:101) Y . Since (cid:101) Y splits into copiesof C ( (cid:101) , this will completely prove Theorem 4 and thus Theorem 1. Proof of Theorem 4.
Applying Lemma 5, Theorem 6, and Proposition 3 to thecofiber sequence Σ , (cid:101) Y → (cid:101) Y → (cid:101) Y , (3)we see that (cid:101) Y admits a v -banded vanishing line with parameters ( − . ≤ , , . , . , . All that remains is to raise b from − . to − . .Now recall cofiber sequence (2): Σ , (cid:101) Y → (cid:101) Y → (cid:101) Y .
We again get upper bounds on homotopy by exactness.We will show that all permanent cycles of (cid:101) Y in the range
12 ( t − s ) − . ≤ s <
12 ( t − s ) − . βb Figure 6: An upper bound on π t − s,t ( Cτ ⊗ (cid:101) Y ) . Dots indicate possible classesin the image of π ∗ , ∗ (Σ , Cτ ⊗ (cid:101) Y ) → π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) , whereas squares indi-cate possible classes that have nonzero images along the map π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) → π ∗ , ∗ ( Cτ ⊗ (cid:101) Y ) . The solid line is the bottom edge of our initial band, and we wantto raise it to the dotted line.
28 29 30 31 3212131415
Figure 7: The bigraded homotopy groups π t − s,t ( Cτ ⊗ (cid:101) Y ) . The line is the bottomedge of the band of Theorem 6. 18ith t − s ≥ are eventual boundaries. In particular, we will show that thesepermanent cycles lift to τ -torsion classes of π t − s,t (cid:101) Y . The groups π t − s,t ( Cτ ⊗ (cid:101) Y ) are 0 in this range, so exactness implies that all classes we have to consider arein the image of π t − s,t (Σ , Cτ ⊗ (cid:101) Y ) .As shown in Figure 6, the groups we have to consider are the potential copiesof F at (2 k, k − and at (2 k + 1 , k − for k ≥ . Figures 5, 6, and 7 illustratethis in the case k = 15 .First, consider the potential permanent cycle b ∈ π k, k +( k − ( Cτ ⊗ (cid:101) Y ) ∼ = F .For brevity, let i := 2 k and j := k − . In Figures 5, 6, and 7, i = 30 and j = 13 .If b is nonzero, it must be in the image of the composite π i,i + j (Σ , Cτ ⊗ (cid:101) Y ) → π i,i + j (Σ , Cτ ⊗ (cid:101) Y ) → π i,i + j ( Cτ ⊗ (cid:101) Y ) , and its preimage is uniquely determined. Since this preimage a ∈ π i,i + j (Σ , Cτ ⊗ (cid:101) Y ) lies in the vanishing region of Σ , (cid:101) Y and is a permanent cycle (see the proofof Proposition 3), it must lift to a τ -torsion class ˆ a ∈ π i,i + j (Σ , (cid:101) Y ) . Then theimage of ˆ a in π i,i + j (cid:101) Y is a τ -torsion lift of b . Thus, in the modified Adamsspectral sequence picture, b is killed by a d .The other potential permanent cycle in the range
12 ( t − s ) − . ≤ s <
12 ( t − s ) − . lies in π k +1 , (2 k +1)+( k − ( Cτ ⊗ (cid:101) Y ) ∼ = F . If such a class β exists, it has a uniquepreimage α ∈ π k +1 , (2 k +1)+( k − (Σ , Cτ ⊗ (cid:101) Y ) .For brevity, let u := 2 k + 1 and v := k − . Then in Figures 5, 6, and 7, u = 31 and v = 13 .We claim that α is a permanent cycle. We first note that Σ , (cid:101) Y is τ -complete. This is true because (cid:101) Y (cid:39) ν ( C (2) ⊗ C ( η )) is τ -complete (see [6,Proposition A.11]) and cofibers of τ -complete synthetic spectra are τ -complete(Proposition 2). It may help to think about this τ -completeness as saying thatthe modified Adams spectral sequence converges to π ∗ ( C (2) ⊗ C ( η )) .By the τ -completeness of Σ , (cid:101) Y , in order to show that α is a permanentcycle, it is enough to show that it admits compatible lifts along the maps π u,u + v (Σ , Cτ i +1 ⊗ (cid:101) Y ) → π u,u + v (Σ , Cτ i ⊗ (cid:101) Y ) for i ≥ . For this, it suffices to show that these maps are surjective. The i thmap is induced by the second map in the cofiber sequence Σ , − i Cτ ⊗ (cid:101) Y → Σ , Cτ i +1 ⊗ (cid:101) Y → Σ , Cτ i ⊗ (cid:101) Y obtained by smashing the cofiber sequence of Lemma 4 with Σ , (cid:101) Y . By exact-ness, the map π u,u + v (Σ , Cτ i +1 ⊗ (cid:101) Y ) → π u,u + v (Σ , Cτ i ⊗ (cid:101) Y )
19s surjective when the τ -Bockstein vanishes: π u,u + v (Σ , Cτ i ⊗ (cid:101) Y ) → π u,u + v (Σ , − i Cτ ⊗ (cid:101) Y ) . As we can see from Figure 5, this map automatically vanishes for i (cid:54) = 2 because its target is 0. Thus, it is enough to show that the following map( i = 2 ) vanishes: π u,u + v (Σ , Cτ ⊗ (cid:101) Y ) → π u,u + v (Σ , Cτ ⊗ (cid:101) Y ) . In the modified Adams spectral sequence picture, this means that the d from α vanishes. Suppose this d is nonzero. Then the class γ ∈ π u − ,u + v +2 ( Cτ ⊗ (cid:101) Y ) killed by this map lifts to a τ -torsion class of π u − ,u + v +2 (Σ , (cid:101) Y ) , which goesto 0 in the homotopy of τ − Σ , (cid:101) Y .However, we claim that this creates a contradiction. Since Σ , (cid:101) Y admits a v -banded vanishing line by Proposition 3, the map F k − π k ( τ − Σ , (cid:101) Y ) → π k L K (1) ( τ − Σ , (cid:101) Y ) ∼ = π k (Σ L K (1) ( C (2) ⊗ C ( η ))) is an isomorphism. By the τ -completeness of Σ , (cid:101) Y , the fact that γ is killed bythe τ -Bockstein implies that there is at most one surviving class contributingto F k − π k ( τ − Σ , (cid:101) Y ) (the one in (2 k, k − in Figure 5). Hence, | π k L K (1) (Σ C (2) ⊗ C ( η )) | ≤ . Moreover, we know from Corollary 4 that | π k L K (1) ( C (2) ⊗ C ( η )) | = | π k L K (1) ( C (2) ⊕ Σ C (2)) | = | π k L K (1) ( C (2)) || π k − L K (1) ( C (2)) | = 8 From the computation used to prove Theorem 6, we know that | π k L K (1) ( C (2) ⊗ C ( η )) | = 2 . Since · < , this contradicts the exactness of the sequence we get by applyingthe exact functor L K (1) ◦ τ − to cofiber sequence (2): π k L K (1) (Σ C (2) ⊗ C ( η )) → π k L K (1) ( C (2) ⊗ C ( η )) → π k L K (1) ( C (2) ⊗ C ( η )) . Therefore, the d vanishes, and α is a permanent cycle.Since α is in the vanishing region of Σ , (cid:101) Y given by Proposition 3, we canconclude that α lifts to τ -torsion as we did with a . Since α maps to β , β mustalso lift to τ -torsion.We finally have that (cid:101) Y admits a v -banded vanishing line with parameters ( − . ≤ , , . , . , . Since (cid:101) Y (cid:39) C ( (cid:101) ⊕ Σ , C ( (cid:101) , C ( (cid:101) admits a v -bandedvanishing line with parameters ( − . ≤ , , . , , . Since C ( (cid:101) (cid:39) νC (2) (see[6, Lemma 15.4]), this encodes a v -banded vanishing line in the Adams spectralsequence for C (2) and proves the theorem.20 .3 The homotopy groups of the K (1) -local Moore spec-trum Our proof of Theorem 4 shows that there is a v -banded vanishing line for C (˜2) ,where the bottom edge is the line s = ( t − s ) − . . By definition, this meansthat the following composite is an isomorphism for all sufficiently high i : F i − . π i C (2) (cid:44) −→ π i C (2) → π i L K (1) C (2) . Since we know what the Adams spectral sequence for C (2) looks like (Figure 2),we can read off the homotopy groups of L K (1) C (2) .As with the homotopy of the K (1) -local sphere, the following result is well-known to experts. However, we include the following proof because a proof isnot readily available in the literature and because this particular proof followsimmediately from Theorem 4. Theorem 7.
The homotopy groups of L K (1) C (2) are the following: i (mod 8) π i ( L K (1) C (2)) Z / ⊕ Z / Z / ⊕ Z / Z / ⊕ Z / Z / ⊕ Z / Z / Z / Proof.
We first recall that C (2) has a v self-map (see [2]), so that the self-map L K (1) v : Σ L K (1) C (2) → L K (1) C (2) is an equivalence. In particular, this map induces an isomorphism π ∗− L K (1) C (2) ∼ = π ∗ L K (1) C (2) , so the homotopy groups of L K (1) C (2) have period 8.We already know π i L K (1) C (2) for i ≡ , , , since the orders ofthese groups uniquely determine the groups. We know π L K (1) C (2) ∼ = Z / ⊕ Z / from the proof of Corollary 4, so our observation about v -periodicity impliesthat π i L K (1) C (2) ∼ = Z / ⊕ Z / for all i ≡ .We only have the following extension problems left to solve: → Z / ⊕ Z / → π k +1 L K (1) C (2) → Z / → → Z / → π k +2 L K (1) C (2) → Z / ⊕ Z / → → Z / → π k +3 L K (1) C (2) → Z / → We address them in order: 21 k + 1 : Pick k large enough that we have an isomorphism F (8 k +1) − . π k +1 C (2) (cid:44) −→ π k +1 C (2) → π k +1 L K (1) C (2) . Then as we can see from Figure 2, there is a 2-extension in F (8 k +1) − . π k +1 C (2) .Thus, π k +1 L K (1) C (2) ∼ = Z / ⊕ Z / . By v -periodicity, this determines π k +1 L K (1) C (2) for all k . • k +2 : This proceeds identically to the case k +1 . We use the 2-extensionin Figure 2 to deduce that π k +2 L K (1) C (2) ∼ = Z / ⊕ Z / . • k + 3 : Again, pick k large. Then looking at Figure 2, we want to ruleout the possibility of a hidden 2-extension in F (8 k +3) − . π k +3 C (2) . Ouranalysis in the case k + 2 shows that there is no hidden 2-extension be-tween the bottom two classes in F (8 k +2) − . π k +2 C (2) because this wouldcontradict the 2-extension above it. The η -multiplication from column k + 2 to column k + 3 then rules out the possibility that a 2-extensionoccurs in F (8 k +3) − . π k +3 C (2) . We conclude that π k +3 L K (1) C (2) ∼ = Z / ⊕ Z / . References [1] John F. Adams. On the structure and applications of the Steenrod algebra.
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