aa r X i v : . [ m a t h . A T ] J a n R -MOTIVIC STABLE STEMS EVA BELMONT AND DANIEL C. ISAKSEN
Abstract.
We compute some R -motivic stable homotopy groups. For s − w ≤
11, we describe the motivic stable homotopy groups π s,w of a completion ofthe R -motivic sphere spectrum. We apply the ρ -Bockstein spectral sequenceto obtain R -motivic Ext groups from the C -motivic Ext groups, which are well-understood in a large range. These Ext groups are the input to the R -motivicAdams spectral sequence. We fully analyze the Adams differentials in a range,and we also analyze hidden extensions by ρ , 2, and η . As a consequence ofour computations, we recover Mahowald invariants of many low-dimensionalclassical stable homotopy elements. Introduction
The goal of this article is to compute the stable homotopy groups of the R -motivic sphere spectrum in a range. These stable homotopy groups are the mostfundamental invariants of the R -motivic stable homotopy category, and thus leadto a deeper understanding of many of the computational aspects of R -motivic ho-motopy theory. More specifically, we work in cellular R -motivic stable homotopytheory, completed appropriately at 2 and η so that the R -motivic Adams spectralsequence converges.Our main tool is the R -motivic Adams spectral sequence, which takes the form E = Ext A ( M , M ) = ⇒ π ∗∗ . Here A is the R -motivic Steenrod algebra, M is the R -motivic cohomology of apoint, and π ∗ , ∗ is the bigraded homotopy groups of the R -motivic sphere (completedat 2 and η ). We obtain complete results about π s,w for s − w ≤
11. This approachfollows [11], which computed π s,w for s − w ≤ R -motivic Adams charts. These charts are an essentialcompanion to this manuscript. In a sense, this manuscript consists of a series ofarguments for the computational facts displayed in the Adams charts.1.1. The ρ -Bockstein spectral sequence. The first step in an Adams spectralsequence program is to obtain the algebraic E -page. We study this computationin Sections 5, 6, and 7. We use the ρ -Bockstein spectral sequence, which takes theform Ext A C ( M C , M C )[ ρ ] = ⇒ Ext A ( M , M ) . Here A C is the C -motivic Steenrod algebra, and M C is the C -motivic cohomologyof a point. Mathematics Subject Classification.
Key words and phrases. motivic stable homotopy group, motivic Adams spectral sequence, ρ -Bockstein spectral sequence, Mahowald invariant, root invariant. The ρ -Bockstein spectral sequence is a tool that passes from C -motivic Extgroups to R -motivic Ext groups. We discuss the general properties of this spec-tral sequence in Section 5, and we describe an unexpectedly effective strategyfor computing differentials. The key idea is to compute the ρ -periodic groupsExt A ( M , M )[ ρ − ] in advance. Then naive combinatorial considerations force avery large number of Bockstein differentials. We discuss specific Bockstein differ-ential computations in Section 6.Having obtained the E ∞ -page of the ρ -Bockstein spectral sequence, we do notyet have a complete knowledge of Ext A ( M , M ). It remains to resolve extensionsthat are hidden by the ρ -Bockstein filtration. There is an unmanageable quantityof hidden extensions, so we do not attempt to analyze them completely, not evenin a range. Nevertheless, we do analyze all extensions by h and h in the rangeunder consideration. These computations are carried out in Section 7.1.2. The R -motivic Adams spectral sequence. Having obtained the E -pageof the R -motivic Adams spectral sequence, the next step is to determine Adamsdifferentials. We carry out these computations in Section 8. These differentialscan be obtained by a variety of techniques. One important technique is the use ofthe Moss Convergence Theorem 8.2 to compute Toda brackets, which determinethat certain elements are permanent cycles. Another technique is comparison topreviously established computations in the C -motivic and classical computations.See Section 1.3 for more discussion of these comparisons.After computing Adams differentials and obtaining the Adams E ∞ -page, thereare once again hidden extensions to resolve. As in the algebraic case, there aretoo many extensions to study exhaustively, but we do consider all extensions by ρ , h , and η exhaustively (where ρ , h , and η are stable homotopy elements detectedby ρ , h , and h respectively). These computations are carried out in Section 9.Once again, the key techniques are shuffling relations involving Toda brackets andcomparison to the C -motivic and classical cases.1.3. Comparison of homotopy theories.
An essential ingredient in our com-putations is comparison between the R -motivic, C -motivic, C -equivariant, andclassical stable homotopy theories, as depicted in the diagram(1.1) R -motivic realization / / extension of scalars (cid:15) (cid:15) C -equivariant forgetful (cid:15) (cid:15) C -motivic realization / / classical.The horizontal arrows labelled “realization” refer to the Betti realization functorsthat take a variety over C (resp., over R ) to the space (resp., C -equivariant space)of C -valued points. The vertical arrow labelled “extension of scalars” refers to thefunctor that takes a variety over R and views it as a variety over C . The verticalarrow labelled “forgetful” refers to the functor that takes a C -equivariant objectto its underlying non-equivariant object.Our philosophy in this article is to accept computational information about the C -motivic and classical stable homotopy groups as given, and to use this informationto study the R -motivic stable homotopy groups. See [18] for an extensive summaryof computational information about the C -motivic and classical Adams spectralsequences. The presence of the C -equivariant stable homotopy category in this -MOTIVIC STABLE STEMS 3 diagram is relevant for our consideration of Mahowald invariants, to be discussedbelow in Section 1.4.There is a surprising connection between C -motivic and R -motivic that enablesmany of our detailed computations. Namely, Theorem 3.4 shows that the C -motivicstable homotopy groups are isomorphic to the R -motivic homotopy groups of thecofiber S/ρ of ρ . This means that the structure of C -motivic stable homotopygroups governs both the cokernel and the kernel of multiplication by ρ . This allowsus to deduce many R -motivic computational facts with relative ease from known C -motivic information.1.4. Mahowald invariants.
Let α be a non-zero classical stable homotopy ele-ment. The Mahowald invariant (or root invariant) R ( α ) is a non-zero equivalenceclass of classical stable homotopy elements in a stem that is higher than the stemof α . One source of interest in Mahowald invariants is that R ( α ) appears to havegreater chromatic complexity than α . Thus one can construct more exotic stablehomotopy elements out of elements that are better understood [20].Bruner and Greenlees reformulated the definition of the Mahowald invariant interms of C -equivariant stable homotopy groups [9]. Although we do not study C -equivariant homotopy groups directly, we have indirectly obtained informationabout them because the R -motivic and C -equivariant stable homotopy groups areisomorphic in a range [6]. In Section 4, we show how many Mahowald invariantscan be immediately deduced from our R -motivic computations. While these resultsonly recover previously known Mahowald invariants [20] [4], we believe that ourtechniques can be extended into uncharted territory without much more effort. Theorem 1.5.
Table 1 gives some values of the Mahowald invariant.
Table 1: Some Mahowald invariantsstem α R ( α ) indeterminacy0 2 η η η η ν ν , 4 ν η ν ν σ σ , 4 σ , 8 σ ν ησ ǫ ν η σ ηǫ ν σ κ σ σ σ η ηρ σ ηη νκ , η ρ ησ ν ν , 4 ν ǫ σ η σ νν ηκ Proof.
Theorem 4.10 reduces the computation to an R -motivic Mahowald invariant,as defined in Section 4.3. Table 3 gives the values of the R -motivic Mahowald EVA BELMONT AND DANIEL C. ISAKSEN invariant. Finally, Table 17 gives the Betti realizations of the R -motivic Mahowaldinvariants. (cid:3) See Examples 4.9 and 4.11 for detailed illustrations of how this technique playsout in practice.We have computed the Mahowald invariant of most, but not every, α throughthe 11-stem. In particular, we do not compute the Mahowald invariants of 2 k for k ≥
4, 8 σ , ηǫ , µ , ηµ , nor ζ and its multiples. In these cases, the problem is thatthe inequality of Theorem 4.10 does not apply, so our R -motivic computations donot determine C -equivariant behavior.2. Notation
We write M for the R -motivic homology of a point with coefficients in F . Recallthat M is isomorphic to F [ ρ, τ ], where ρ and τ have degrees ( − , −
1) and (0 , − A for the R -motivic dual Steenrod algebra. Recall that A is describedby the equations A = M [ τ , τ , . . . , ξ , ξ , . . . ] / ( τ k = τ ξ k +1 + ρτ k +1 + ρτ ξ k +1 ) η L ( τ ) = τ, η R ( τ ) = τ + ρτ , η L ( ρ ) = η R ( ρ ) = ρ ∆( τ k ) = τ k ⊗ X ξ i k − i ⊗ τ i ∆( ξ k ) = X ξ i k − i ⊗ ξ i , where τ i and ξ k have degrees (2 i +1 − , i −
1) and (2 i +1 − , i −
1) respectively[27].We write M C for the C -motivic homology of a point with coefficients in F , andwe write A C ∗ for the C -motivic dual Steenrod algebra. These objects are easilydescribed in terms of M and A . Namely, they are the result of setting ρ equal tozero.We write A cl ∗ for the classical dual Steenrod algebra, which can be obtained from A by setting ρ and τ to be 0 and 1 respectively.We write Ext or Ext R for Ext A ( M , M ), i.e., the cohomology of the R -motivicSteenrod algebra. We write Ext C and Ext cl for the cohomologies of the C -motivicand classical Steenrod algebras respectively.We write π p,q or π R p,q for the stable homotopy groups of the R -motivic spherespectrum. Similarly, we write π C p,q for the stable homotopy groups of the C -motivicsphere spectrum. We adopt the usual motivic grading convention, so that π p,q X denotes maps out of S p,q , where S p,q is the smash product of p − q copies of thesimplicial sphere and q copies of A − π C p,q for the stable homotopy groups of the C -equivariant sphere spec-trum. We use an equivariant grading convention that is compatible with the motivicgrading convention, so that π p,q X denotes maps out of S p,q , where S p,q is the one-point compactification of R p , with C acting by negating the last q coordinates.Betti realization takes R -motivic S p,q to C -equivariant S p,q .We write π p for the classical stable homotopy groups. -MOTIVIC STABLE STEMS 5 All stable homotopy groups are suitably completed so that Adams spectral se-quences converge. Classically, this means completion at 2. In the motivic cases,this means completion at 2 and η [17]. Grading conventions.
Following [18] and [11], we use the following grading con-vention for the motivic Adams spectral sequence: s denotes the stem, f denotesthe Adams filtration, and w denotes the motivic weight. Then the internal degreeis s + f . In this grading, Adams differentials take the form d r : E s,f,wr → E s − ,f + r,wr . The coweight of an element in degree ( s, f, w ) is defined to be s − w . Note that ρ has coweight 0. In particular, an element x and its ρ -multiple ρx lie in the samecoweight. This makes coweights particularly useful in the ρ -Bockstein perspectivethat we adopt.2.1. Stable homotopy elements.
We adopt conventional notation, as used (forexample) in [18] [19], for the names of elements in the classical stable homotopygroups π ∗ and the C -motivic stable homotopy groups π C ∗ , ∗ .Table 9 gives the notation that we use for elements of π R ∗ , ∗ . We define theseelements in terms of the elements of the Adams E ∞ -page that detect them. Thesedefinitions have indeterminacy parametrized by elements of the Adams E ∞ -page inhigher Adams filtration. As a general rule, this indeterminacy does not matter toour computations. It is possible to use Toda brackets, or geometric constructions(see [10]), to eliminate the indeterminacy in many cases. Remark 2.2.
We use the symbol h to denote an element of π , that is detected by h . The symbol stands for “hyperbolic” because it corresponds to the hyperbolicplane in the Grothendieck-Witt group interpretation of π , [22, Remark 6.4.2].(Alternatively, it can also stand for “Hopf”, since h is the zeroth Hopf map.) Bewarethat h does not equal 2; in fact, 2 = h + ρη . Remark 2.3.
The element σ requires more discussion. We write σ for an elementof π , that is detected by h . There are 256 possible choices for σ , because ofthe presence of elements in higher Adams filtration. One such element in higherfiltration is ρc . Lemma 7.19 shows that τ h · ρc equals ρ d . Therefore, somepossible choices of σ have the property that τ ν · σ is detected by ρ d in π , , whileother possible choices of σ have the property that τ ν · σ is zero. (The elements τ h · τ P h and ρh · τ h · τ P h are not relevant, by comparison to kq as in Remark8.15.)We will need to use the relation τ ν · σ = 0 in later computations, so we mustassume that our choice of σ satisfies this condition. Remark 2.4.
In some cases, we have chosen names for elements of π R ∗ , ∗ that reflectthe values of the extension of scalars functor given in Table 17. For example, wewrite τ σ for an element of π R , that is detected by ρh , since this element mapsto τ σ in π C , . Remark 2.5.
Beware that our use of the symbol κ is inconsistent with its usage in[18]. In this manuscript, τ κ refers to a non-zero element of π C , that is detectedby τ g . The symbol κ is used in [18] for the same element. EVA BELMONT AND DANIEL C. ISAKSEN
Remark 2.6.
Occasionally we refer to stable homotopy elements that have nostandard name. In these cases, we use the symbol { x } to indicate a stable homotopyelement that is detected by an element x of an Adams E ∞ -page.3. Comparison between R -motivic and C -motivic homotopy We first discuss the relationship between R -motivic and C -motivic stable homo-topy theory. We will use these ideas frequently in later sections to obtain R -motivicinformation from known C -motivic information.Consider the cofiber sequence S − , − ρ / / S , / / S/ρ.
The cofiber
S/ρ of ρ is a 2-cell complex whose structure governs multiplication by ρ in the R -motivic stable homotopy groups, in a sense to be made precise in thissection. In addition, we will draw an unexpected connection between the R -motivichomotopy groups of S/ρ and C -motivic stable homotopy groups.As shown in diagram (1.1), there is an extension of scalars functor from R -motivic stable homotopy theory to C -motivic stable homotopy theory, and a Bettirealization functor from C -motivic stable homotopy theory to classical stable ho-motopy theory. These functors take Eilenberg-Mac Lane spectra to Eilenberg-MacLane spectra, and thus interact nicely with Adams spectral sequences. In par-ticular, they induce highly structured morphisms of Adams spectral sequences.We will frequently use these comparison functors to deduce information about the R -motivic Adams spectral sequence from already known information about the C -motivic and classical Adams spectral sequences. See [18] for an extensive summaryof computational information about the C -motivic and classical Adams spectralsequences.Extension of scalars takes the element ρ of π − , − to zero. In particular, itinduces the map M → M C that takes ρ to zero, and it similarly induces the map A → A C ∗ that takes ρ to zero.For an R -motivic spectrum, we write Ext R ( X ) for the E -page of the R -motivicAdams spectral sequence that converges to π ∗ , ∗ ( X ), i.e., for Ext A ( M , H ∗ , ∗ ( X )),and similarly for Ext C ( X ).Extension of scalars induces a diagram / / Ext R ( S − , − ) ρ / / (cid:15) (cid:15) Ext R ( S , ) / / (cid:15) (cid:15) Ext R ( S/ρ ) (cid:15) (cid:15) / / / / Ext C ( S − , − ) / / Ext C ( S , ) / / Ext C ( S , ∨ S − , − ) / / . Because ρ becomes zero after extension of scalars, the bottom row of the diagramsplits. The map Ext R ( S/ρ ) → Ext C ( S , ∨ S − , − ) lifts to a map Ext R ( S/ρ ) → Ext C ( S , ) that makes the diagramExt R ( S , ) / / (cid:15) (cid:15) Ext R ( S/ρ ) x x ♣♣♣♣♣♣♣♣♣♣♣ Ext C ( S , )commute. -MOTIVIC STABLE STEMS 7 Proposition 3.1.
The map
Ext R ( S/ρ ) → Ext C ( S , ) is an isomorphism.Proof. Let C ∗ R and C ∗ C be the cobar complexes for Ext R ( S , ) and Ext C ( S , ) respec-tively. Note that C ∗ C is isomorphic to C ∗ R /ρ . Because multiplication by ρ is injectiveon C ∗ R , this is also isomorphic to the cobar complex that computes Ext R ( S/ρ ). (cid:3) Remark 3.2.
Because of the isomorphism of Proposition 3.1, the object Ext C isa module over Ext R . By careful inspection of definitions, this module action iseasy to describe. Using the ρ -Bockstein spectral sequence notation from Section5, a typical element of Ext R is of the form ρ k x , where x belongs to Ext C . TheExt R -module action on Ext C is described by ρ k x · y = (cid:26) k > xy if k = 0 , where the last expression xy is to be interpreted as the usual Yoneda product ofelements in Ext C . Remark 3.3.
Proposition 3.1 implies that there is a long exact sequence · · · / / Ext R ρ / / Ext R i / / Ext C p / / Ext R ρ / / Ext R / / · · · of Ext R -module maps, where Ext C is an Ext R -module as in Remark 3.2. If x is apermanent cycle in the ρ -Bockstein spectral sequence, then the map i takes x inExt R to the element of Ext C of the same name.Now consider the diagram(3.1) π R ∗ +1 , ∗ +1 ρ / / π R ∗ , ∗ / / (cid:15) (cid:15) π R ∗ , ∗ ( S/ρ ) z z ✉✉✉✉✉✉✉✉✉ π C ∗ , ∗ , in which the diagonal arrow exists because ρ maps to zero in π C ∗ , ∗ . Theorem 3.4.
The map π R ∗ , ∗ ( S/ρ ) → π C ∗ , ∗ is an isomorphism.Proof. Proposition 3.1 shows that there is an isomorphism of E -pages of Adamsspectral sequences, so the targets of the spectral sequences are also isomorphic. (cid:3) Corollary 3.5.
Let α be an element of π R ∗ , ∗ . Extension of scalars takes α to zeroin π C ∗ , ∗ if and only if α is divisible by ρ .Proof. Chase the diagram (3.1), using that the diagonal map is an isomorphism. (cid:3)
Remark 3.6.
Corollary 3.5 has a C -equivariant analogue, as stated later in Propo-sition 4.2. Remark 3.7.
The isomorphism of Theorem 3.4 can be strengthened to an equiv-alence of categories [5, Corollary 8.6]. Namely, the 2-complete C -motivic cellularstable homotopy category is equivalent to the homotopy category of S/ρ -modulesin the 2-complete R -motivic cellular stable homotopy category. Corollary 3.8.
There is a long exact sequence · · · / / π R s +1 ,w +1 ( S ) ρ / / π R s,w ( S ) / / π C s,w ( S ) / / π R s,w +1 ( S ) / / · · · . EVA BELMONT AND DANIEL C. ISAKSEN
Proof.
This is the long exact sequence in homotopy for the fiber sequence S ρ / / S / / S/ρ in R -motivic spectra, after applying the identification in Theorem 3.4. (cid:3) Mahowald invariants
The goal of this section is to use R -motivic computations to recompute someMahowald invariants. See [4, Section 4] for a careful discussion of the definition,using Lin’s theorem that R P ∞−∞ is equivalent to S − .4.1. C -equivariant homotopy theory and Mahowald invariants. Using C -equivariant homotopy theory, Bruner and Greenlees [9] gave an alternative defini-tion of the Mahowald invariant. We will summarize this definition, but first weneed some background on C -equivariant homotopy theory.Let S a,b be the one-point compactification of R a , where C acts by negating thelast b coordinates. Then ρ : S , → S , is the inclusion of fixed points. Note thatthe cofiber of this map is Σ( C ) + , i.e., the suspension of the based free C -space.We use the same notation ρ for the map S − , − → S , in the C -equivariantstable homotopy group π C − , − . The identification of the cofiber of ρ leads imme-diately to the following proposition, whose short proof appears in [12, Proposition11.2]. Proposition 4.2.
Let α be a C -equivariant stable homotopy element. The under-lying classical stable homotopy element U ( α ) of α is zero if and only if α is divisibleby ρ . Geometric fixed points gives a map π C a,b → π a − b , and this map takes ρ to 1. The ρ -periodic groups π C ∗ , ∗ [ ρ − ] are isomorphic to π ∗ ⊗ Z [ ρ ± ], i.e., to the classical stablehomotopy groups with ρ and ρ − adjoined [8, Proposition] [2, Proposition 7.0].With this background on C -equivariant stable homotopy groups, we now givethe Bruner-Greenlees definition of the Mahowald invariant. Start with a classicalstable homotopy element α in π n , which we identify with the obvious element of π ∗ ⊗ Z [ ρ ± ] in degree (0 , − n ). Using the isomorphism π ∗ ⊗ Z [ ρ ± ] ∼ = π C ∗ , ∗ [ ρ − ] , write α = ρ k β for some β in π C ∗ , ∗ and some integer k , with k maximal. Finally, theMahowald invariant R ( α ) is the underlying classical stable homotopy element U ( β )of β .Note that the Mahowald invariant is not strictly defined; it is a set of classicalstable homotopy elements. While the choice of k is unique, the choice of β is not.Different choices of β can lead to different values of U ( β ).Also note that U ( β ) is necessarily non-zero by Proposition 4.2. The point is that β is not divisible by ρ , since k was chosen to be maximal.4.3. R -motivic homotopy theory and Mahowald invariants. We will nowadapt the framework of Bruner and Greenlees [9] from the C -equivariant to the R -motivic settings. In order to carry this out, we need to observe some key R -motivicproperties.First, the ρ -periodic groups π R ∗ , ∗ [ ρ − ] are isomorphic to π ∗ ⊗ Z [ ρ ± ], i.e., tothe classical stable homotopy groups with ρ and ρ − adjoined [11]. See also [3] -MOTIVIC STABLE STEMS 9 for a more structured version of this isomorphism. Second, Corollary 3.5 relates ρ -divisibility to the kernel of the extension of scalars map. Definition 4.4.
Let α be a classical stable homotopy element in π n . The R -motivicMahowald invariant R R ( α ) is defined as follows. Identify α with the obvious elementof π ∗ ⊗ Z [ ρ ± ] ∼ = π R ∗ , ∗ [ ρ − ]in degree (0 , − n ). Write α = ρ k β for some β in π R ∗ , ∗ and some integer k , with k maximal. Define R R ( α ) in π C ∗ , ∗ to be the extension of scalars of β . Remark 4.5.
As for the traditional Mahowald invariant, the R -motivic Mahowaldinvariant is not strictly defined. Different choices of β can have different values in π C ∗ , ∗ under extension of scalars. Remark 4.6.
As for the traditional Mahowald invariant, the R -motivic Mahowaldinvariant is always non-zero by Corollary 3.5. The point is that β is not divisibleby ρ , since k was chosen to be maximal. Remark 4.7.
See [24] [25] for a different consideration of Mahowald invariants inthe motivic context. Our construction does not compare directly.
Theorem 4.8.
Some values of the R -motivic Mahowald invariant are given inTable 3.Proof. This follows immediately from the computations carried out later in thearticle. In particular, one needs the values of the extension of scalars map, asshown in Table 17 and discussed in Section 10 (cid:3)
Example 4.9.
We illustrate Theorem 4.8 by describing the computation of M R ( σ ).The element σ in π is identified with the element α of π R ∗ , ∗ ⊗ Z [ ρ ± ] in degree (0 , − ρ h . Then α equals ρ β , where β is detected by ρh . Finally,Table 17 shows that the realization of β is τ σ in π C , .In general, the relationship between R ( α ) and R R ( α ) is not obvious. The choicesinvolved in the definitions are not necessarily compatible. For example, it is possiblethat an element β in π R ∗ , ∗ is not divisible by ρ , while its realization in π C ∗ , ∗ is divisibleby ρ .The main result of [6] tells us that the R -motivic and C -equivariant stablehomotopy groups agree in a range. In this range, R ( α ) and R R ( α ) are easier tocompare. Theorem 4.10.
Let R R ( α ) belong to π C s,w , and Suppose that w − s < . Then R ( α ) equals the Betti realization of R R ( α ) .Proof. The isomorphism between R -motivic and C -equivariant stable homotopygroups [6] implies that the choice of β in the definition of R R ( α ) realizes to thechoice of β in the definition of R ( α ). By the commutativity of the diagram (1.1),the realization of R R ( α ) equals R ( α ). (cid:3) Example 4.11.
We showed in Example 4.9 that R R ( σ ) equals τ σ in π C , . Thenumerical condition of Theorem 4.10 is satisfied. It follows that R ( σ ) equals σ in π , since σ is the realization of τ σ . Remark 4.12.
Theorem 4.10, together with our computations of R -motivic stablehomotopy groups, can be used to compute the Mahowald invariants R ( α ) for most α up to the 11-stem. The exceptions are 2 k for k ≥
4, 8 σ , ηǫ , µ , ηµ , and ζ and its multiples. In these cases, R R ( α ) can still be computed as shown in Table3. However, the numerical condition of Theorem 4.10 does not hold, so we cannotdraw a conclusion about R ( α ) in these cases.5. The ρ -Bockstein spectral sequence We briefly recall some background on the ρ -Bockstein spectral sequence thatcomputes the cohomology of the R -motivic Steenrod algebra. See [16] and [11] foradditional details.Begin with the observation that the C -motivic cohomology of a point M C equals M /ρ , and the C -motivic dual Steenrod algebra A C ∗ equals A /ρ . Then filter thecobar complex by powers of ρ to obtain the ρ -Bockstein spectral sequence(5.1) E = Ext ∗∗A C ∗ ( M C , M C )[ ρ ] = ⇒ Ext ∗∗A ( M , M ) . Our goal is to analyze the ρ -Bockstein spectral sequence (5.1) in computationaldetail in a range of degrees. We recall some structural results about this spectralsequence from [11]. Proposition 5.1. [11, Lemma 3.4] If d r ( x ) is nontrivial in the ρ -Bockstein spectralsequence, then x and d r ( x ) are both ρ -torsion free on the E r -page. Recall that A cl ∗ is the classical dual Steenrod algebra. Proposition 5.2. [11, Theorem 4.1]
There is an isomorphism
Ext A cl ∗ ( F , F )[ ρ ± ] ∼ = Ext A ( M , M )[ ρ − ] that takes elements of degree ( s, f ) in Ext A cl ∗ ( F , F ) to elements of degree (2 s + f, f, s + f ) in Ext A ( M , M ) . In particular, the classical element h n corresponds tothe R -motivic element h n +1 . Moreover, the isomorphism is highly structured, i.e.,preserves products and Massey products. The point of Proposition 5.2 is that we a priori know the elements of Ext R that are ρ -periodic, in the sense that they support infinitely many non-zero multiplicationsby ρ . In the range considered in this manuscript, these ρ -periodic elements are h , h , h , h , c , h g , h g , as well as products of these elements. This corresponds tothe fact that through the 11-stem, Ext cl is generated by the classical elements h , h , h , h , c , P h , and P h . We may effectively ignore these ρ -periodic elementswhen analyzing the ρ -Bockstein spectral sequence, since they can be neither sourcenor target of any ρ -Bockstein differential.Let { x i } be an F -linear basis for Ext C , i.e., an F [ ρ ]-linear basis for the ρ -Bockstein E -page, excluding the ρ -periodic permanent cycles described in the pre-vious paragraph. For every i , either x i supports a differential, or ρ r x i is the targetof the d r differential for some r . In other words, the set { x i } may be partitionedinto pairs ( x i , x j ) such that d r ( x i ) = ρ r x j for some j . Actually, one must be some-what careful about the choice of basis in situations where two or more elements ofthe basis have the same degree. Nevertheless, it is always possible to change basisso that the basis elements can be partitioned into pairs. -MOTIVIC STABLE STEMS 11 The Bockstein differential d r : E s,f,wr → E s − ,f +1 ,wr preserves the quantity s + f − w , and ρ lies in a degree satisfying s + f − w = 0. Thus we may consider onevalue of s + f − w at a time when analyzing the ρ -Bockstein spectral sequence.We exploit this structure in the following strategy for analyzing the ρ -Bocksteinspectral sequence. Strategy 5.3. (1) Fix a value N = s + f − w .(2) Find an F [ ρ ]-basis B N for the part of the ρ -Bockstein E -page in degrees( s, f, w ) satisfying N = s + f − w .(3) Remove elements from B N that detect ρ -periodic elements of Ext R .(4) Use a variety of techniques, to be described below, to identify some differ-ential d r ( x i ) = ρ r x j , where x i and x j belong to B N .(5) Remove x i and x j from B N .(6) Repeat steps (4) and (5) until B N is empty.For this strategy to be effective, we need to know that the basis B N chosen instep 2 is finite. Lemma 5.4 establishes this fact. Lemma 5.4.
Let N be fixed. In degrees ( s, f, w ) satisfying N = s + f − w , the ρ -Bockstein E -page is a finitely generated F [ ρ ] -module.Proof. Recall that Ext C is non-zero only in degrees ( s, f, w ) satisfying s + f − w ≥ s + f − w ≥
12 ( s + f ) . In other words, we only need consider the part of Ext C in total degree at most2 N . (cid:3) One consequence of our strategy is that we do not compute the Bockstein dif-ferentials d r in order of increasing r . Rather, we obtain all differentials as part ofthe same process.Step (4) is the limiting factor in the practical effectiveness of our algorithm. Thead hoc arguments required to establish specific differentials become more difficultas the value of N increases. However, these difficulties increase at a surprisinglyslow rate, and we are able to carry out the computation remarkably far withoutmuch difficulty.Our goal is to compute the ρ -Bockstein spectral sequence through coweight 13.Unfortunately, infinitely many values of N in Step 1 are relevant in this range. Forexample, consider the elements h k of coweight 0, which belong to degrees satisfying s + f − w = k .Similarly, any h -periodic sequence of elements h k x of Ext C lies in degrees forwhich s + f − w is unbounded. Fortunately, it is only these h -periodic familiesthat are problematic. Lemma 5.5.
Let x be a non-zero element of Ext C of degree ( s, f, w ) whose coweightis at most k . Then:(1) x is an h -periodic element, in the sense that h i x is non-zero for all i ≥ ;or(2) s + f − w ≤ k + 3 . Proof.
If 2 f − s ≥
4, then x is h -periodic [14]. So we may assume that 2 f − s < s + f − w ≥
0. Combiningwith the assumption s − w ≤ k , we conclude that s + f − w = (2 f − s ) − ( s + f − w ) + 3( s − w ) < k = 3 k + 4 . (cid:3) As we wish to consider elements up to coweight 13, Lemma 5.5 suggests weneed to look at degrees satisfying the inequality s + f − w ≤
42, in addition tostudying h -periodic elements. However, inspection of elements in Ext C shows that s + f − w ≤
28 for all elements that are relevant in our range.The h -periodic elements of Ext C are well-understood [13]. Up to coweight 13,all such elements are of the form 1, P k h , P k c , P k d , P k e , P k c d , d , or c e , aswell as the h -multiples of these elements. Lemma 5.5 indicates that the behaviorof the ρ -Bockstein spectral sequence on these elements must be studied separately.See Proposition 6.2 for the analysis of these h -periodic elements.6. ρ -Bockstein differentials The goal of this section is to describe a variety of methods for determining ρ -Bockstein differentials. These methods are applied in Step (4) of Strategy 5.3.Taken together, these methods allow us to determine all ρ -Bockstein differentialsthrough coweight 13.We begin with a result that describes all ρ -Bockstein differentials on the elementsof Adams filtration zero. Proposition 6.1. [11, Proposition 3.2] (1) d ( τ ) = ρh .(2) d k ( τ k ) = ρ k τ k − h k for k ≥ . Next we consider h -periodic elements. These elements must be treated as specialcases because of Case (1) of Lemma 5.5. Proposition 6.2.
Table 4 gives some Bockstein differentials that are non-zeroafter inverting h . Through coweight , these are the only h -periodic ρ -Bocksteindifferentials. For legibility, we have not included powers of ρ in the values of the Bocksteindifferentials in Table 4. For example, the first row of the table is to be interpretedas d ( P h ) = ρ h c . Proof.
The differentials in the h -periodic ρ -Bockstein spectral sequence are com-pletely known [15]. For each h -periodic element x , this determines d r ( h k x ) forlarge values of k . However, it is possible that the elements h k x support shorterdifferentials for small values of k . By inspection, no such shorter differentials oc-cur. (cid:3) Remark 6.3.
The phenomenon considered at the end of the proof of Proposition6.2 turns out not to occur through coweight 13. However, it does occur in highercoweights.The following examples are representative arguments for establishing ρ -Bocksteindifferentials. In many situations, more than one argument leads to the same result. -MOTIVIC STABLE STEMS 13 Example 6.4.
Table 2 summarizes the analysis of Bockstein differentials in degrees( s, f, w ) satisfying s + f − w = 6. In these degrees, the E -page consists of ρ multiples of twenty elements. The first part of Table 2 lists the two elements thatare ρ -periodic, as in Proposition 5.2. They correspond to the classical elements h and h h .The second section of Table 2 lists some differentials that are easily deducedfrom Proposition 6.1 and the Leibniz rule.At this point, only the elements τ h and c remain unaccounted. The thirdsection of Table 2 gives the only possibility.Table 2: Bockstein differentials for s + f − w = 6coweight ( s, f, w ) x d r d r ( x )0 (6 , , h , , h h , , − τ d τ h , , − τ h d τ h , , − τ h d τ h , , − τ h d h , , − τ h h d τ h , , τ h d τ h h , , τ h h d h h , , − τ h d τ h , , − τ h d c Example 6.5.
In some situations, a more careful analysis of multiplicative struc-ture establishes a differential. For example, d ( f ) cannot equal ρh e because h f = 0 but ρh e is not zero.For a slightly more complicated example, consider the relation h · τ g = τ · h g .This implies that h · d ( τ g ) = d ( τ ) · h g = ρh g, so d ( τ g ) must equal ρh g . Example 6.6.
Sometimes, the multiplicative structure and an already known dif-ferential imply that a certain element is killed by ρ k . Then that element must bekilled by a differential d r with r ≤ k . For example, the element τ h h = ( τ h ) h is a permanent cycle because it is a product of permanent cycles. There are twopossible differentials that could hit a ρ -multiple of it: d ( τ h ) or d ( τ h ). Notethat τ h h is killed by ρ because of the differential d ( τ ) = ρ τ h . Therefore, ρ τ h h must be hit by a d r differential with r ≤
4. The only possibility is that d ( τ h ) = ρ τ h h .This differential can be obtained another way using the Leibniz rule, the multi-plicative relation τ h = τ · τ h · h , and the differential d ( τ ) = ρ τ h . Example 6.7.
Sometimes one must look ahead to larger values of s + f − w inorder to use multiplicative relations to rule out differentials. For example, in orderto show that d ( i ) = ρ h c e (in degrees satisfying s + f − w = 18), we firstuse other techniques to rule out possible differentials until it suffices to eliminatethe possibility that d ( τ P c ) might equal ρ h c e . But this would imply that d ( τ P h c ) equals h c e (in degrees satisfying s + f − w = 19), and this contra-dicts the h -periodic differential d ( P e ) = ρ h c e from Table 4. Example 6.8.
The Leibniz rule implies that certain elements survive at least toa certain page of the spectral sequence. For example, the element τ h cannot behit by a differential, so it must support a differential. There are two possibilities: d ( τ h ) might equal ρ τ h h , or d ( τ h ) might equal ρ τ c . The Leibniz ruleand the relation τ h = τ · τ h imply that d ( τ h ) = d ( τ ) · τ h = ρ τ h · τ h = 0 . Therefore, d ( τ h ) must equal ρ τ c . Example 6.9.
The multiplicative structure implies that certain elements do notsupport any differentials because they are the product of elements that do notsupport any differentials.Extending Example 6.6, sometimes the Massey product structure of Ext R impliesthat some element ρ k x must be zero. Then ρ k x must be the target of a Bockstein d r differential for r ≤ k . Through coweight 12, we apply this method only once in thefollowing Lemma 6.10. However, we anticipate that this approach will become moreand more important in higher coweights. Massey products in Ext R are discussedbelow in Section 7 and Table 6. Lemma 6.10. d ( τ g ) = ρ h f .Proof. Table 6 shows that h f equals the Massey product (cid:10) τ h , h , h (cid:11) in Ext R .Shuffle to obtain ρ (cid:10) τ h , h , h (cid:11) = (cid:10) ρ , τ h , h (cid:11) h , which equals zero because the last bracket is zero. Therefore, ρ h f is hit by a d or d differential, and the only possibility is that d ( τ g ) = ρ h f . (cid:3) Theorem 6.11 summarizes the results of the analysis of ρ -Bockstein differentials. Theorem 6.11.
Table 5 lists some values of the ρ -Bockstein d r differentials onmultiplicative generators of the E r -page. Through coweight , the d r differentialvanishes on all other multiplicative generators of the E r -page. For legibility, we have not included powers of ρ in the values of the Bocksteindifferentials in Table 5. For example, the first row of the table is to be interpretedas d ( τ ) = ρh .7. Hidden extensions in the ρ -Bockstein spectral sequence Section 6 explains how to obtain the E ∞ -page of the ρ -Bockstein spectral se-quence through coweight 12. As usual, this E ∞ -page is an associated graded objectof Ext R .We abuse notation and use the same name for generators of the ρ -Bockstein E ∞ -page and elements of Ext R that they represent. A generator of the ρ -Bockstein E ∞ -page can represent more than one element in Ext R , where the indeterminacyis parametrized by elements of the E ∞ -page in higher filtration. For example, theelement τ h of the E ∞ -page represents two elements of Ext R whose difference is ρ h . -MOTIVIC STABLE STEMS 15 We adopt the following convention in selecting generators in Ext R . We alwayschoose an element of Ext R that is annihilated by the same power of ρ as its repre-sentative in the E ∞ -page. For example, τ h is annihilated by ρ in the E ∞ -page.Therefore, we write τ h for the (unique) element of Ext R that is annihilated by ρ . (The other possible choice is ρ -periodic.)This convention concerning annihilation by powers of ρ eliminates much of theambiguity in passing from the E ∞ -page to Ext R . In some cases, our conventiondoes not eliminate all ambiguities. However, the remaining ambiguities make littlepractical difference.In order to recover the full structure of Ext R from the ρ -Bockstein E ∞ -page, wemust determine hidden multiplicative extensions. We adopt the precise definitionof a hidden extension given in [18, Section 4.1.1]. In this section, we will analyzeall hidden extensions by h and h through coweight 12.The ρ -Bockstein spectral sequence has numerous hidden extensions by otherelements. There are so many examples that it is not practical to enumerate themexhaustively. In practice, these other hidden extensions are occasionally useful, andwe treat them on an ad hoc basis as necessary. Definition 7.1.
A hidden a extension from x to y is decomposable if there existsa hidden a extension from u to v , and there exists z such that x = zu and y = zv in the E ∞ -page. Example 7.2.
There is a hidden h extension from τ h to ρτ h . Multiplicationby τ h gives the decomposable hidden h extension from τ h to ρτ h .Definition 7.1 allows us to focus only on the hidden extensions that are mostsignificant. In practice, decomposable hidden extensions are easy to understand,once the indecomposable hidden extensions have been studied. Remark 7.3.
The structure of the ρ -Bockstein spectral sequence guarantees thatthere are no hidden extensions by ρ . For degree reasons, if there is a possible hidden ρ extension from x to y , then in fact y is a multiple of ρ . According to the definitionof a hidden extension [18, Section 4.1.1], this means that y cannot be the target ofa hidden ρ extension.7.4. Massey products.
Our main tool for establishing hidden extensions is theMay Convergence Theorem [21, Theorem 4.1], restated here for convenience.
Theorem 7.5 (May Convergence Theorem) . Let α , α , and α be elements of Ext R such that the Massey product h α , α , α i is defined. For each i , let a i be apermanent cycle in the Bockstein E r -page that detects α i . Suppose further that:(1) there exist elements a and a in the Bockstein E r -page such that d r ( a ) equals a a and d r ( a ) equals a a ;(2) if either a or a has degree ( s, f, w ) and ρ -Bockstein degree m , and x isan element in degree ( s, f, w ) and ρ -Bockstein degree m ′ such that m ′ ≤ m ,then d t ( x ) = 0 for all t such that m ′ + t > ( m − m ′ ) + r .Then a a + a a is a permanent cycle in the ρ -Bockstein spectral sequence, andit detects an element of h α , α , α i in Ext R . We will often use Theorem 7.5 in the situation when a has ρ -Bockstein degree0 and a has negative ρ -Bockstein degree. Since the ρ -Bockstein spectral sequenceis zero in negative ρ -Bockstein degrees, condition (2) of Theorem 7.5 simplifies to the condition that no element in the same degree as a with ρ -Bockstein degree 0supports a longer differential. Proposition 7.6.
Table 6 lists some Massey products in
Ext R .Proof. Most of these Massey products are straightforward applications of the MayConvergence Theorem 7.5. In those cases, the sixth column of Table 6 gives the ρ -Bockstein differential that is relevant for computing the Massey product.In some cases, the Massey products follow by comparison to the C -motivic case.This is denoted by the word “ C -motivic” in the sixth column of Table 6. However,this only determines the Massey product up to multiples of ρ . These ambiguitiescan typically be eliminated by the multiplicative structure. In particular, if theMassey product h x, y, z i is defined and ρ a x and ρ b z are both zero, then ρ a + b h x, y, z i = ρ b h ρ a , x, y i z = 0 . The indeterminacies can be computed by inspection. (cid:3)
Table 6 is not meant to be an exhaustive list of Massey products. It merelyprovides an assortment of Massey products that are needed for various specificcomputations throughout the manuscript.7.7.
Hidden h extensions.Proposition 7.8. Table 7 lists all indecomposable hidden h extensions in the ρ -Bockstein spectral sequence, through coweight .Proof. All of the hidden h extensions in Table 7 are proved using a single technique,which was introduced in the proof of [11, Lemma 6.2]. To illustrate this technique,we will show that there is a hidden h extension from τ h c to ρ P h .First we show that the product h · τ h c is nonzero in Ext R . If not, thenthe Massey product (cid:10) ρ, h , τ h c (cid:11) would be defined in Ext R . The May Conver-gence Theorem 7.5, together with the ρ -Bockstein differential d ( τ ) = ρh , wouldthen imply that τ h c is a permanent cycle. But this contradicts the ρ -Bocksteindifferential d ( τ h c ) = ρ P h .This shows that there must be a hidden h extension on τ h c . The target ofthis hidden extension can only be ρ P h or τ P h . But the target must have higher ρ -Bockstein filtration than the source, which rules out τ P h .In some cases, one needs to use multiplicative relations to rule out possible hidden h extensions. For example, the target of a hidden h extension cannot support a ρ multiplication, since ρh = 0 in Ext R .We must also show that many elements do not support hidden h extensions. Inall cases through coweight 12, the non-existence follows from simple multiplicativerelations. For example, if x is already known to not support an h extension, thenthe product xy cannot support an h extension. Similarly, if h y or ρy is non-zero,then y cannot be the target of a hidden extension because of the relations h h = 0and ρh = 0 in Ext R . (cid:3) Hidden h extensions.Proposition 7.10. Table 8 lists all indecomposable hidden h extensions in the ρ -Bockstein spectral sequence, through coweight . -MOTIVIC STABLE STEMS 17 Proof.
Many of the extensions are established using the mapExt C p / / Ext R of Remark 3.3. To illustrate this technique, we will show that there is a hidden h extension from τ h c to ρP h . The relation h · τ c = τ h c in Ext C implies that h · p ( τ c ) = p ( τ h c ). Observe that p ( τ c ) = ρτ h · τ c and p ( τ h c ) = ρ P h .This shows that there is a hidden h extension from ρτ h c to ρ P h , and it followsthat there is also a hidden h extension from τ h c to ρP h .Several more difficult cases are established in the following lemmas.We must also show that many elements do not support hidden h extensions. Inmost cases through coweight 12, the non-existence follows from simple multiplica-tive relations. For example, if x is already known to not support an h extension,then the product xy cannot support an h extension. Similarly, if h y is non-zero, then y cannot be the target of a hidden h extension because of the relation h h = 0 in Ext R .Additionally, the map p : Ext C → Ext R can be used to detect the absence ofsome h extensions. (cid:3) Remark 7.11.
The first three extensions in Table 8 were established in [11].
Lemma 7.12.
There is a hidden h extension from τ h to ρ d .Proof. The element τ h of the ρ -Bockstein E ∞ -page detects the element τ h · τ h in Ext R . Table 8 shows that h · τ h = ρc , and h · τ h = ρ c . Therefore, h · τ h · τ h = ρ c · ρc = ρ h d . It follows that h · τ h · τ h equals ρ d . (cid:3) Lemma 7.13.
There is a hidden h extension from τ f to ρ τ h g .Proof. Table 6 shows that τ f belongs to the Massey product (cid:10) τ h , h , h h (cid:11) .Table 8 shows that there is a hidden h extension from τ h to ρ τ h . Therefore,we have h (cid:10) τ h , h , h h (cid:11) = (cid:10) ρ τ h , h , h h (cid:11) = ρ (cid:10) τ h , h , h h (cid:11) , where the equalities follow from inspection of indeterminacies. Table 6 shows thatthe element τ h g of the Bockstein E ∞ -page detects both elements of the Masseyproduct (cid:10) τ h , h , h h (cid:11) , so ρ τ h g is the target of the hidden h extension. (cid:3) Lemma 7.14. (1) There is a hidden h extension from τ h c to ρτ P h .(2) There is a hidden h extension from τ P h to ρ τ h d .(3) There is a hidden h extension from τ P h c to ρτ P h .(4) There is a hidden h extension from τ P h to ρ τ P h d .Proof. We will show that h · τ c equals ρ τ h d . This will establish the first twoextensions simultaneously.Table 6 shows that h · τ c equals the Massey product (cid:10) τ h · τ c , τ h , ρ (cid:11) . Byinspection of indeterminacies, h (cid:10) τ h · τ c , τ h , ρ (cid:11) = h (cid:10) h · τ h · τ c , τ h , ρ (cid:11) . This expression equals h (cid:10) ρτ P h , τ h , ρ (cid:11) , since Table 8 shows that there is ahidden h extension from τ h c to ρτ P h . By inspection of indeterminaciesagain, this also equals ρh (cid:10) τ P h , τ h , ρ (cid:11) .Now shuffle to obtain ρh (cid:10) τ P h , τ h , ρ (cid:11) = ρ (cid:10) h , τ P h , τ h (cid:11) . Finally, Table 6 shows that (cid:10) h , τ P h , τ h (cid:11) equals τ h d . This establishes thefirst two extensions.The argument for the last two extensions is essentially identical. The Masseyproduct (cid:10) τ h · τ P c , τ h , ρ (cid:11) equals h · τ P c . We have h (cid:10) τ h · τ P c , τ h , ρ (cid:11) = h (cid:10) h · τ h · τ P c , τ h , ρ (cid:11) , which equals h (cid:10) ρP h , τ h , ρ (cid:11) = ρh (cid:10) P h , τ h , ρ (cid:11) . Finally, shuffle to obtain ρh (cid:10) P h , τ h , ρ (cid:11) = ρ (cid:10) h , P h , τ h (cid:11) = ρ τ P h d . (cid:3) Lemma 7.15.
There is a hidden h -extension from τ c to ρ τ h c .Proof. Table 6 shows that τ c is contained in the Massey product (cid:10) ρ , τ h , τ c (cid:11) .Shuffle to obtain (cid:10) ρ , τ h , τ c (cid:11) h = ρ h τ h , τ c , h i . Table 6 shows that the element τ h c of the Bockstein E ∞ -page detects bothelements of h τ h , τ c , h i , so ρ τ h c is the target of the hidden h extension. (cid:3) Lemma 7.16. (1) There is a hidden h extension from τ h e to ρ j .(2) There is a hidden h extension from j to ρd .Proof. Table 8 shows that h · τ h = ρc , and h · τ e = h · ρτ h · d = ρ c d .Therefore, h · τ h · τ e = ρ c d = ρ h d . Both hidden extensions are immediate consequences. (cid:3)
Miscellaneous relations.
We briefly consider a few other types of hiddenextensions.In the Bockstein E ∞ -page, we have the relation h · τ h + ( τ h ) h = 0.However, in Ext R , it is possible that the sum h · τ h + ( τ h ) h equals a non-zeroelement that is detected in higher ρ -Bockstein filtration. Lemma 7.18 demonstratesthat this does in fact occur. It provides one additional piece of information aboutthe multiplicative structure of Ext R . Lemma 7.18. In Ext R we have the relation h · τ h + ( τ h ) h = ρ τ h h . Proof.
This follows by comparison along the map p : Ext C → Ext R of Remark3.3. The relation h · τ h = τ h in Ext C implies that h · p ( τ h ) = p ( τ h )in Ext R . Observe that p ( τ h ) = ρ τ h h and p ( τ h ) = ρ τ h h . This showsthat there is a hidden h extension from ρ τ h h to ρ τ h h , which implies thedesired relation. (cid:3) -MOTIVIC STABLE STEMS 19 Lemma 7.19.
There is a hidden τ h extension from c to ρ d .Proof. Table 8 shows that there are hidden h extensions from τ h to ρc , and from τ h to ρ d . Therefore, τ h · ρc = τ h · h · τ h = ρ d . (cid:3) Lemma 7.20.
There is a hidden h extension from h f to ρh h c .Proof. We use the map p : Ext C → Ext R of Remark 3.3. The relation h · τ g = τ h g in Ext C implies that h · p ( τ g ) = p ( τ h g ). Observe that p ( τ g ) = ρh f ,and p ( τ h g ) = ρ h h c .Therefore, there is a hidden h extension from ρh f to ρ h h c , and also ahidden h extension from h f to ρh h c . (cid:3) Adams differentials
Sections 6 and 7 describe how to compute Ext R , which serves as the E -pageof the R -motivic Adams spectral sequence. We now proceed to analyze Adamsdifferentials. We remind the reader of the notation for stable homotopy elementsdiscussed in Section 2.1 and Table 9.Recall from Section 3 that extension of scalars induces a map from the R -motivicAdams spectral sequence to the C -motivic Adams spectral sequence. We will fre-quently use these comparison functors to deduce information about the R -motivicAdams spectral sequence from already known information about the C -motivic andclassical Adams spectral sequences. See [18] for an extensive summary of compu-tational information about the C -motivic and classical Adams spectral sequences.8.1. Toda brackets.
The Moss Convergence Theorem 8.2 is a key tool for deter-mining Toda brackets [23] [18, Section 3.1]. We restate a version of the theoremhere for convenience.
Theorem 8.2 (Moss Convergence Theorem) . Let α , α , and α be elements ofthe R -motivic stable homotopy groups such that the Toda bracket h α , α , α i isdefined. Let a i be a permanent cycle on the Adams E r -page that detects α i for each i . Suppose further that:(1) the Massey product h a , a , a i E r is defined (in Ext R when r = 2 , or usingthe Adams d r − differential when r ≥ ).(2) if ( s, f, w ) is the degree of either a a or a a ; f ′ < f − r + 1 ; f ′′ > f ; and t = f ′′ − f ′ ; then every Adams differential d t : E s +1 ,f ′ ,wt → E s,f ′′ ,wt is zero.Then h a , a , a i E r contains a permanent cycle that detects an element of the Todabracket h α , α , α i . Theorem 8.3.
Table 10 lists some Toda brackets in π ∗ , ∗ .Proof. Most of these Toda brackets are straightforward applications of the MossConvergence Theorem 8.2. When a Massey product appears in the fifth column ofTable 10, the Toda bracket follows from the Moss Convergence Theorem 8.2 with r = 2. When an Adams differential appears in the fifth column of Table 10, theToda bracket follows from the Moss Convergence Theorem 8.2 with r >
2, and thegiven Adams differential is relevant for computing the Toda bracket.
In some cases, the Toda brackets follow by comparison along the extension ofscalars functor to the C -motivic case. This is denoted by the word “ C -motivic” inthe fifth column of Table 10.One slightly different case is handled below in Lemma 8.4. (cid:3) Table 10 is not meant to be exhaustive in any sense. It merely provides theToda brackets that are needed for various specific computations. Beware thatthese brackets have non-trivial indeterminacies, although we have not specified theindeterminacies because they are not generally relevant to our specific needs.Beware that some of the Toda brackets in Table 10 require knowledge of Adamsdifferentials that are established below in Section 8.5.
Lemma 8.4.
The Toda bracket (cid:10) ρ , τ η, ν (cid:11) is detected by τ h · h .Proof. Table 6 shows that τ h is contained in the Massey product (cid:10) ρ , τ h , h (cid:11) .By inspection of indeterminacies, τ h · h = (cid:10) ρ , τ h , h (cid:11) h = (cid:10) ρ , τ h , h h (cid:11) . The Moss Convergence Theorem 8.2 implies that τ h · h detects the correspondingToda bracket. (cid:3) Adams d differentials. We now proceed to analyze Adams differentials.
Theorem 8.6.
Table 12 lists some values of the R -motivic Adams d differential.Through coweight , the d differential is zero on all other multiplicative generatorsof the R -motivic Adams E -page.Proof. The multiplicative structure rules out many possible differentials. For ex-ample, d ( τ h ) cannot equal τ h · h because h · τ h = 0, while τ h · h isnon-zero.Other multiplicative generators are known to be permanent cycles, because theMoss Convergence Theorem 8.2 shows that they must survive to detect variousToda brackets. These instances are shown in Table 11. In one case, the element h · τ c must survive to detect the product σ · τ η , by comparison to the C -motivicstable homotopy groups.Many non-zero differentials follow by comparison to the C -motivic or classicalAdams spectral sequences.Several more difficult cases are established in the following lemmas. (cid:3) Remark 8.7.
Table 11 shows that τ h is a permanent cycle because it detectsthe Toda bracket (cid:10) ρ , τ ν, σ (cid:11) . We give an alternative proof that is geometricallyinteresting, following the method of [11, Lemma 7.3].There is a functor from classical homotopy theory to R -motivic homotopy theorythat takes the sphere S p to S p, . Let σ top : S , → S , be the image of the classicalHopf map σ : S → S under this functor.The cohomology of the cofiber of σ top is free on two generators x and y of degrees(8 ,
0) and (16 , ( x ) = τ y and Sq ( x ) = ρ y . The proof of theseformulas is essentially identical to the proof of [11, Lemma 7.4].This shows that τ h + ρ h is a permanent cycle in the Adams spectral sequence,since it detects the stabilization of σ top in π , . Also, ρ h is a permanent cyclebecause there are no possible values for differentials. Therefore, τ h is a permanentcycle. -MOTIVIC STABLE STEMS 21 Lemma 8.8. d ( τ h h ) = ρ h d .Proof. Table 12 shows that d ( e ) = h d . Therefore, d ( h · τ h h ) = d ( ρ e ) = ρ h d . It follows that d ( τ h h ) equals ρ h d . (cid:3) Lemma 8.9. d ( f ) = h e .Proof. Comparison to the C -motivic or classical case shows that d ( f ) equals either h e or h e + ρ h e . But h · f = 0 in the E -page, while h ( h e + ρ h e ) isnon-zero. The only possibility is that d ( f ) equals h e . (cid:3) Lemma 8.10. d ( τ f ) = h · τ e + ρ τ h · d .Proof. The C -motivic differential d ( τ f ) = τ h e implies that d ( τ f ) equalseither h · τ e or h · τ e + ρ τ h · d . We rule out the first possibility by notingthat ( h + ρ h ) · τ f = 0 in Ext R whereas ( h + ρ h ) · τ h e = ρ h c d . (cid:3) Lemma 8.11. d ( τ h g ) = ρ c d .Proof. Table 8 shows that h · τ h g = ρτ h · e . Therefore, h · d ( τ h g ) = ρτ h · d ( e ) = ρτ h · h d , which equals ρ h c d because Table 8 shows that h · τ h = ρc . (cid:3) Higher Adams differentials.
Theorem 8.6 completely describes the Adams d differential through coweight 12. From this information, one can compute theAdams E -page in a range. We now proceed to analyze higher differentials. Theorem 8.13.
Table 13 lists some values of the R -motivic Adams d differentialfor r ≥ . Through coweight , the d differential is zero on all other multiplicativegenerators of the R -motivic Adams E -page. Moreover, through coweight , thereare no higher differentials, and the R -motivic Adams E -page equals the R -motivicAdams E ∞ -page.Proof. As in the proof of Theorem 8.6, many multiplicative generators cannotsupport differentials because there are no possible targets. Comparison to the C -motivic and classical cases also determines some differentials. For example, d ( h h ) cannot equal h d .Other multiplicative generators are known to be permanent cycles, because theMoss Convergence Theorem 8.2 shows that they must survive to detect variousToda brackets. These instances are shown in Table 11.The multiplicative structure rules out additional cases. For example d ( ρh )cannot equal ρd because of the relation h · ρh = ρ · h h , together with the factthat d ( h h ) is already known to be zero.The harder cases are established in the following lemmas. (cid:3) Lemma 8.14. d ( ρ e ) = 0 .Proof. If d ( ρ e ) equaled ρh · τ h · τ P h , then ρ e would be a permanent cyclethat detected an element α of π , , and α could not be divisible by ρ . Therefore,by Corollary 3.5, α would map to a non-zero element β in π C , . Then β wouldhave to be detected by τ P h , so ηβ would also have to be non-zero in π C , .But ηα would be detected by ρ h e and would be divisible by ρ , so it wouldmap to zero in π C , . This contradicts that ηβ is non-zero. (cid:3) Remark 8.15.
Lemma 8.14 can also be proved using the R -motivic spectrum kq ,which is the very effective slice cover of the Hermitian K -theory spectrum KQ [1].The cohomology of kq is isomorphic to A // A (1), where A (1) is the M -subalgebraof the R -motivic Steenrod algebra that is generated by Sq and Sq .By a change-of-rings isomorphism, the homotopy of kq is computed by an Adamsspectral sequence whose E -page is Ext A (1) ( M , M ). This E -page was computedin [16], and also in [12, Section 6].The element ρτ h · τ P h · h maps to a non-zero permanent cycle inExt A (1) ( M , M ) , so it cannot be the target of a differential. Lemma 8.16. d ( h h ) = h d + ρh d Proof.
The classical differential d ( h h ) = h d implies that in the R -motivic case, d ( h h ) equals either h d or h d + ρh d .Note that τ h · h d = ρτ h · h d is non-zero on the E -page, but τ h · h h = ρτ h · h h is a permanent cycle, as shown in Table 11. Therefore, d ( h h ) cannotequal h d . (cid:3) Lemma 8.17. (1) d ( τ h · τ e ) = ρτ P h · d .(2) d ( ρj ) = τ P h · h d .Proof. Let α be an element of π , that is represented by τ P h · h d . By compar-ison of Adams spectral sequences, extension of scalars must take α to zero in π C , .Moreover, τ P h · h d cannot be the target of a hidden ρ extension. Therefore, byCorollary 3.5, τ P h · h d must be the target of an R -motivic Adams differential,and there is only one possible such differential. This establishes the second formula.The first formula follows immediately from the second one, using the relation h · τ h · τ e = ρc · τ e . (cid:3) Hidden extensions in the Adams spectral sequence
We have now obtained the Adams E ∞ -page through coweight 11. It remainsto determine hidden extensions that are hidden in the R -motivic Adams spectralsequence. As in Section 7, we use the precise definition of a hidden extension givenin [18, Section 4.1.1]. We will analyze all hidden extensions by ρ , h , and η throughcoweight 11.We begin by analyzing all hidden extensions by ρ . The main tools are Corollaries3.5 and 3.8. Proposition 9.1.
Table 14 lists all hidden ρ extensions in the Adams spectralsequence, through coweight .Proof. The long exact sequence of Corollary 3.8 gives short exact sequences0 → (coker ρ ) s,w → π C s,w → (ker ρ ) s,w +1 → . The rank of π C s,w , which is entirely known in our range [18] [19], severely constrainsthe possible ranks of coker ρ and ker ρ . From these constraints, we can generallydeduce the presence and absence of hidden ρ extensions, and there is typically onlyone possibility in each case in the range under consideration. The only exceptionis considered below in Lemma 9.2. (cid:3) -MOTIVIC STABLE STEMS 23 Lemma 9.2.
There is a hidden ρ extension from τ h c d to P h d .Proof. Table 16 shows that there is a hidden η extension from ρτ c · d to P h d .Therefore, there must be a hidden ρ extension from h · τ c · d to P h d . (cid:3) Theorem 9.3.
Table 15 lists all hidden h extensions in the R -motivic Adams spec-tral sequence, through coweight .Proof. The long exact sequence of Corollary 3.8 gives short exact sequences0 → (coker ρ ) s,w → π C s,w → (ker ρ ) s,w +1 → . Some of the extensions can be determined via these short exact sequences, usingknown 2 extensions in π C ∗ , ∗ . For example, the element ρ e in the R -motivic Adams E ∞ -page lies in (coker ρ ) , , and it maps to the element τ ζ in π C , that isdetected by τ P h . But 2 τ ζ is non-zero in π C , , so h α must also be non-zero.It follows that ρ e supports a hidden h extension.We must also show that many elements do not support hidden h extensions.In most of the cases through coweight 11, the non-existence follows from simplemultiplicative relations. For example, if x is a multiple of ρ or of h , then x cannotsupport a hidden h extension because of the relations ρ h = 0 and h η = 0. Similarly,if h y or ρy is non-zero, then y cannot be the target of a hidden h extension.The following lemmas handle a few additional more complicated cases. (cid:3) Lemma 9.4.
There is a hidden h extension from h f to ρc d .Proof. Table 10 shows that h f detects the Toda bracket h ρ, { h e } , η i . Shuffle toobtain h ρ, { h e } , η i h = ρ h{ h e } , η, h i . Table 10 shows that c d detects the latter bracket. (cid:3) Lemma 9.5.
There is no hidden h extension on τ h · h .Proof. The only possible target is ρτ c · d . Table 16 shows that ρτ c · d supportsa hidden η extension, so it cannot be the target of a hidden h extension. (cid:3) Lemma 9.6.
There is a hidden h extension from τ c · d to P h d .Proof. Let α be an element of π , that is detected by τ c , so τ c · d detects ακ .Table 14 shows that there is a hidden ρ extension from h · τ c · d to P h d , so P h d detects ρηακ . But ( h + ρη ) κ is zero, so ( h + ρη ) ακ must also be zero. Thisimplies that h ακ is also detected by P h d . (cid:3) Lemma 9.7.
There is no hidden h extension on h c .Proof. By comparison to the C -motivic (or classical) case, h c detects the product ση . By inspection, h η is zero in π , . (cid:3) Theorem 9.8.
Table 16 lists some hidden η extensions in the R -motivic Adamsspectral sequence, through coweight .Proof. The long exact sequence of Corollary 3.8 gives short exact sequences0 → (coker ρ ) s,w → π C s,w → (ker ρ ) s,w +1 → . Many of these extensions can be obtained by comparison to the C -motivic case,using these short exact sequences, as in the proof of Theorem 9.3. For example, the element ρτ h · τ P c detects an element α in (ker ρ ) , . The pre-image β of α in π C , is detected by τ P c . There is a C -motivic hidden η extension from τ h h to τ P c , so β is divisible by η . This implies that α is also divisible by η , and thatthere is an R -motivic hidden η extension from τ h · h h to ρτ h · τ P c .We must also show that many elements do not support hidden η extensions. Inall cases through coweight 11, the non-existence follows from simple multiplicativerelations. For example, if x is a multiple of h , then x cannot support a hidden η extension because of the relation h η = 0. Similarly, if h y is non-zero, then y cannot be the target of a hidden η extension. (cid:3) Lemma 9.9.
There is no hidden η extension on τ h .Proof. Table 10 shows that τ h detects the Toda bracket (cid:10) τ ν, σ, ν (cid:11) . Shuffle toobtain (cid:10) τ ν, σ, ν (cid:11) η = τ ν h σ, ν, η i . The latter bracket is zero. (cid:3)
Lemma 9.10.
There is no hidden η extension on τ c .Proof. The possible target ρh f is ruled out by the fact that ρh f supports an h extension, as shown in Lemma 7.20. The possible target τ h · d is ruled out bycomparison to the C -motivic case. (cid:3) Extension of scalars
We will now study the values of the extension of scalars map π R ∗ , ∗ → π C ∗ , ∗ . Corol-lary 3.5 tells us exactly which elements of π R ∗ , ∗ have non-trivial images in π C ∗ , ∗ . Thisinformation about extension of scalars is essential to our approach to the Mahowaldinvariant described in Section 4.For the most part, the extension of scalars map is detected by the map fromthe R -motivic Adams E ∞ -page to the C -motivic Adams E ∞ -page. For example,the element ( τ η ) of π R , is detected by τ h in the R -motivic Adams E ∞ -page, soits image in π C , must be τ η , which is detected by τ h in the C -motivic Adams E ∞ -page.However, there are a few values that are hidden by the Adams spectral sequence.In other words, there exist elements α in π R ∗ , ∗ such that the Adams filtration of α is strictly less than the Adams filtration of its image in π C ∗ , ∗ . Theorem 10.1.
Through coweight , Table 17 lists all hidden values of the ex-tension of scalars map π R ∗ , ∗ → π C ∗ , ∗ .Proof. We inspect all elements of the R -motivic Adams E ∞ -page that are not tar-gets of ρ extensions. Most of these elements map non-trivially to the C -motivicAdams E ∞ -page. For example, ( τ h ) maps to τ h .A few elements map to zero in the C -motivic Adams E ∞ -page. We treat theseelements individually. In some cases, there is only one possible target in sufficientlyhigh Adams filtration. The remaining cases are handled by the following lemmas. (cid:3) Lemma 10.2.
Extension of scalars takes elements detected by ρh to elementsdetected by τ h . -MOTIVIC STABLE STEMS 25 Proof.
Table 10 shows that ρh detects the Toda bracket (cid:10) ρ, h , σ (cid:11) . Extension ofscalars takes (cid:10) ρ, h , σ (cid:11) in π R , to (cid:10) , , σ (cid:11) in π C , , which equals { , τ σ } . Theonly non-zero value is τ σ , which is detected by τ h . (cid:3) Lemma 10.3.
Extension of scalars takes elements detected by ρf to elementsdetected by τ h d .Proof. Table 10 shows that ρf detects the Toda bracket h ρ, h , νκ i . Extension ofscalars takes h ρ, h , νκ i in π R , to h , , νκ i in π C , , which equals { , τ νκ } . Theonly non-zero value is τ νκ , which is detected by τ h d . (cid:3) Lemma 10.4.
Extension of scalars takes elements detected by ρ τ f to elementsdetected by τ h d .Proof. The long exact sequence of Corollary 3.8 gives a short exact sequence0 → (coker ρ ) , → π C , → (ker ρ ) , → . The group π C , is generated by an element of order 32, detected by τ h h , and anelement of order 2, detected by τ h d . Also (ker ρ ) , is generated by an elementof order 32, detected by τ h · h h . It follows that (coker ρ ) , maps onto anelement of order 2 that is detected by τ h d . (cid:3) Tables
Table 3: Some values of the R -motivic Mahowald invariant s α M R ( α ) indeterminacy0 2 k η k η ν ν , 4 ν η ν ν σ σ , 4 σ , 8 σ ν ησ ǫ ν η σ ηǫ ν σ κ σ τ σ σ η ηρ σ ηη η ρ , νκ σ η η η ρ ησ ν ν , 4 ν ǫ σ η σ νν τ ηκ ηǫ νσ τ η κ µ νκ νκ , 4 νκ ηµ ν · νκ ζ τ ν κ η ρ
11 2 ζ { h h g } η ρ
11 4 ζ η { h h g } η ρ
236 EVA BELMONT AND DANIEL C. ISAKSEN
Table 4: h -periodic Bockstein differentialscoweight ( s, f, w ) x d r d r ( x )4 (9 , , P h d h c , , P c d h d , , P h d h e
10 (22 , , P d d h c d
11 (25 , , P e d h c e
12 (25 , , P h d P h c
13 (30 , , P c d d h d Table 5: Bockstein differentialscoweight ( s, f, w ) x d r d r ( x )1 (0 , , − τ d h , , − τ d τ h , , − τ d τ h , , − τ h d τ h , , − τ h d c , , τ h h d h c , , P h d h c , , τ h d τ c , , τ h h d τ P h , , τ h c d P h , , τ c d d , , τ P h d h d , , τ h d d h d , , P c d h d , , − τ d τ h , , − τ h d τ h h , , − τ h d e , , − τ h h d h e , , τ P h d h e , , τ h h d h e , , P h d h e , , − τ h d τ h , , τ h h d f , , τ h d d τ P c , , τ g d h g
10 (6 , , − τ h d τ c
10 (9 , , − τ h h d τ e
10 (14 , , τ d d τ h e
10 (15 , , τ h h d τ P h
10 (17 , , τ P h c d P h
10 (20 , , τ g d τ h g
10 (22 , , P d d h c d -MOTIVIC STABLE STEMS 27 Table 5: Bockstein differentialscoweight ( s, f, w ) x d r d r ( x )11 (8 , , − τ h h d τ c
11 (14 , , τ h h d τ f
11 (17 , , τ e d τ h g
11 (20 , , τ h h e d c e
11 (23 , , τ h g d h h c
11 (23 , , i d h c e
11 (25 , , P e d h c e
12 (7 , , − τ h h d τ P h
12 (9 , , − τ P h d τ h d
12 (10 , , − τ P h d τ P c
12 (14 , , τ h d τ c
12 (15 , , τ h h d τ P h
12 (17 , , τ P h d τ P h d
12 (18 , , τ P h d P c
12 (23 , , τ h i d P h c
12 (25 , , P h d P h c
13 (14 , , τ h h d τ g
13 (17 , , τ e d τ h g
13 (18 , , τ h e d τ h h g
13 (20 , , τ h h e d j
13 (22 , , τ P h d d τ P c
13 (23 , , τ i d d
13 (25 , , τ P e d h d E VA B E L M O N T AN DD AN I E L C . I S A K S E N Table 6: Some Massey products in Ext R coweight ( s, f, w ) bracket contains indeterminacy proof used in3 (3 , , (cid:10) ρ , τ h , h (cid:11) τ h ρ h d ( τ ) = ρ τ h (cid:10) ρ , τ η, ν (cid:11) , Lemma 8.44 (8 , , h c , h , ρ i τ c ρτ h · h h d ( τ ) = ρh h ǫ, h , ρ i , , (cid:10) ρ , τ h , h (cid:11) τ h ρ h d ( τ ) = ρ τ h (cid:10) ρ , τ ν, σ (cid:11) , , (cid:10) τ h , h , h (cid:11) h f C -motivic Lemma 6.109 (21 , , h ρ, h e , h i h f ρ h g d ( τ g ) = ρh e h ρ, { h e } , η i
10 (18 , , (cid:10) τ h , h , h h (cid:11) τ f τ h · h h , ρ h c C -motivic Lemma 7.1310 (21 , , (cid:10) τ h , h , h h (cid:11) τ h g ρ h h c C -motivic Lemma 7.1311 (3 , , − (cid:10) ρ , τ h , h (cid:11) τ h d ( τ ) = ρ τ h (cid:10) ρ , τ η, ν (cid:11)
11 (9 , , − (cid:10) τ h · τ c , τ h , ρ (cid:11) h · τ c d ( τ ) = ρ τ h Lemma 7.1411 (11 , , (cid:10) ρ , τ h , P h (cid:11) τ P h ρ h g d ( τ ) = ρ τ h (cid:10) ρ , τ η, ζ (cid:11)
11 (14 , , (cid:10) h , τ P h , τ h (cid:11) τ h d C -motivic Lemma 7.1411 (17 , , (cid:10) τ h · τ P c , τ h , ρ (cid:11) h · τ P c d ( τ ) = ρ τ h Lemma 7.1411 (19 , , (cid:10) ρ, h , τ c (cid:11) τ c ρ τ h · h h d ( τ ) = ρh (cid:10) ρ, h , τ σ (cid:11)
11 (19 , , (cid:10) ρ , τ h , τ c (cid:11) τ c ρ τ h · h h d ( τ ) = ρ τ h Lemma 7.1511 (19 , , (cid:10) ρ , τ h , P h (cid:11) τ P h d ( τ ) = ρ τ h (cid:10) ρ , τ η, ζ (cid:11)
11 (22 , , h τ h , τ c , h i h · τ c ρh · τ c C -motivic Lemma 7.1511 (22 , , (cid:10) h , P h , τ h (cid:11) τ P h d C -motivic Lemma 7.1412 (20 , , (cid:10) ρ, τ h , ρ, h e (cid:11) τ g ρ h · τ c d ( τ ) = ρτ h , (cid:10) ρ, τ h , ρ, { h e } (cid:11) d ( τ g ) = ρh e -MOTIVIC STABLE STEMS 29 Table 7: Hidden h extensions in the ρ -Bockstein spectral sequencecoweight ( s, f, w ) source target1 (1 , , τ h ρτ h , , τ h h ρ h c , , h h ρ h c , , τ h ρ τ c , , τ c ρτ h c , , − τ h ρτ h , , τ h h ρ τ P h , , τ h c ρ P h , , τ P h ρτ P h , , τ h ρ τ h , , h d ρ h d , , − τ h h ρ e , , τ h h ρ h e , , τ P h h ρ h e , , h h ρ h e , , τ c ρτ h c , , τ h h ρ f , , τ h d ρ τ P c , , τ P c ρτ P h c , , − τ h ρτ h , , − τ h h ρ τ P h , , τ h h ρ τ e , , τ h c ρ τ P h , , τ P h ρτ P h , , τ h h ρ τ P h , , τ P h c ρ P h , , τ P h ρτ P h
10 (14 , , τ h h ρ τ f
10 (18 , , τ h f ρ τ h e
10 (20 , , τ h h e ρ c e
11 (3 , , − τ h h ρ τ h c
11 (7 , , − τ h h ρ τ P h
11 (11 , , τ P h h ρ τ P h c
11 (15 , , τ h h ρ τ P h
11 (19 , , τ c ρ τ h c
11 (19 , , τ P h h ρ P h c
11 (23 , , h i ρ P h c
12 (6 , , − τ h ρ τ c
12 (8 , , − τ c ρτ h c
12 (14 , , τ h h ρ τ g
12 (14 , , τ h d ρ τ P c
12 (16 , , τ P c ρτ P h c
12 (18 , , τ h f ρ τ h e
12 (20 , , τ h g ρ j Table 7: Hidden h extensions in the ρ -Bockstein spectral sequencecoweight ( s, f, w ) source target12 (22 , , τ P h d ρ τ P c
12 (24 , , τ P c ρτ P h c
12 (26 , , h j ρ h d Table 8: Hidden h extensions in the ρ -Bockstein spectral sequencecoweight ( s, f, w ) source target proof2 (0 , , − τ h ρτ h , , τ h ρ τ h , , τ h ρc , , τ h c ρP h , , − τ h ρτ h , , τ h ρ d Lemma 7.127 (14 , , τ h h ρ e , , τ h h ρ τ e , , τ h c ρτ P h , , τ P h c ρP h , , τ h e ρτ h d
10 (0 , , − τ h ρτ h
10 (14 , , τ h ρ τ c
10 (18 , , τ f ρ τ h g Lemma 7.1310 (19 , , τ c ρ τ h c
11 (3 , , − τ h ρ τ h
11 (6 , , − τ h ρτ c
11 (9 , , − τ h c ρτ P h Lemma 7.1411 (11 , , τ P h ρ τ h d Lemma 7.1411 (14 , , τ h d ρτ P c
11 (17 , , τ P h c ρτ P h Lemma 7.1411 (19 , , τ c ρ τ h c Lemma 7.1511 (19 , , τ P h ρ τ P h d Lemma 7.1411 (22 , , τ P h d ρP c
12 (21 , , τ h g ρτ h e
12 (22 , , τ P h d ρτ P h d
12 (23 , , τ h e ρ j Lemma 7.1612 (26 , , j ρd Lemma 7.16 -MOTIVIC STABLE STEMS 31
Table 9: Multiplicative generators of π R ∗ , ∗ coweight ( s, w ) element detected by0 ( − , − ρ ρ , h h , η h , τ η τ h , ν h , − τ h τ h , τ ν τ h , τ ν τ h , σ h , ǫ c , − τ h τ h , τ ǫ τ c , − τ η τ h , τ µ τ P h , ζ P h , − τ h τ h , κ d , τ σ τ h , τ ζ ρ e , τ σ ρh , ρ h h , η h h , − τ h τ h , τ ǫ τ c , τ σ τ h , τ η τ h · h , τ νκ ρf , ν h h ,
11) ¯ σ c , { h e } h e , − τ η τ h , τ µ τ P h , τ ζ τ P h , τ ηκ ρ τ e , τ µ τ P h , τ σ τ c , ζ P h , τ ηκ h f , νκ h g
10 (0 , − τ h τ h
10 (15 , τ ηκ ρ τ f
10 (18 , τ ν τ h · h
10 (19 , τ σ τ c
10 (20 , τ h κ h · τ e
02 EVA BELMONT AND DANIEL C. ISAKSEN
Table 9: Multiplicative generators of π R ∗ , ∗ coweight ( s, w ) element detected by10 (21 , τ νν τ h · h
11 (3 , − τ ν τ h
11 (6 , − τ ν τ h
11 (8 , − τ ǫ τ c
11 (11 , τ ζ τ P h
11 (15 , τ ρ τ h h
11 (17 , τ νκ τ h · τ e
11 (19 , τ σ τ c
11 (19 , τ ζ τ P h
11 (23 , ρ h i
11 (26 , τ ν κ ρh g
11 (28 , { h h g } h h g - M O T I V I C S T A B L E S T E M S Table 10: Some Toda brackets in π ∗ , ∗ coweight ( s, w ) bracket detected by proof used in3 (3 , (cid:10) ρ , τ η, ν (cid:11) τ h (cid:10) ρ , τ h , h (cid:11) Table 114 (8 , h ǫ, h , ρ i τ c h c , h , ρ i Table 117 (7 , (cid:10) ρ , τ ν, σ (cid:11) τ h (cid:10) ρ , τ h , h (cid:11) Table 117 (14 , (cid:10) ρ, h , σ (cid:11) ρh d ( h ) = h h Lemma 10.28 (8 , (cid:10) τ η, h ν, ν (cid:11) τ c C -motivic Table 118 (14 , (cid:10) τ ν, σ, ν (cid:11) τ h C -motivic Table 11, Lemma 9.98 (16 , (cid:10) σ , , τ η (cid:11) τ h · h d ( h ) = ( h + ρh ) h Table 118 (16 , h τ µ , h ν, ν i τ P c C -motivic Table 118 (17 , h ρ, h , νκ i ρf d ( f ) = h e Lemma 10.38 (18 , h ν, σ, h σ i h h d ( h ) = h h Table 119 (15 , h ρ, ρτ η, τ η · κ i ρ τ e d ( τ e ) = τ h d Table 119 (21 , h ρ, { h e } , η i h f h ρ, h e , h i Lemma 9.49 (21 , h{ h e } , η, h i c d C -motivic Lemma 9.410 (18 , (cid:10) ρ , τ η, ν (cid:11) τ h · h Lemma 8.4 Table 1110 (19 , (cid:10) τ ν, ησ, σ (cid:11) τ c C -motivic Table 1111 (3 , − (cid:10) ρ , τ η, ν (cid:11) τ h (cid:10) ρ , τ h , h (cid:11) Table 1111 (11 , (cid:10) ρ , τ η, ζ (cid:11) τ P h (cid:10) ρ , τ h , P h (cid:11) Table 1111 (19 , (cid:10) ρ , τ η, ζ (cid:11) τ P h (cid:10) ρ , τ h , P h (cid:11) Table 1111 (19 , (cid:10) ρ, h , τ ¯ σ (cid:11) τ c (cid:10) ρ, h , τ c (cid:11) Table 1112 (8 , − (cid:10) τ η, h ν, ν (cid:11) τ c C -motivic Table 1112 (16 , (cid:10) σ , , τ η (cid:11) τ h · h d ( h ) = ( h + ρh ) h Table 1112 (16 , (cid:10) τ µ , h ν, ν (cid:11) τ P c C -motivic Table 1112 (20 , (cid:10) ρ, τ h , ρ, { h e } (cid:11) τ g (cid:10) ρ, τ h , ρ, h e (cid:11) Table 1112 (24 , h τ µ , h ν, ν i τ P c C -motivic Table 11 Table 11: Some permanent cycles in the R -motivic Adams spectralsequencecoweight ( s, f, w ) element proof3 (3 , , τ h (cid:10) ρ , τ η, ν (cid:11) , , τ c h ǫ, h , ρ i , , τ h (cid:10) ρ , τ ν, σ (cid:11) , ρ e Lemma 8.148 (8 , , τ c (cid:10) τ η, h ν, ν (cid:11) , τ h (cid:10) τ ν, σ, ν (cid:11) , , τ P c h τ µ , h ν, ν i , , τ h · h (cid:10) σ , , τ η (cid:11) , , h h h ν, σ, h σ i , , ρ τ e h ρ, ρτ η, τ η · κ i
10 (18 , , τ h · h (cid:10) ρ , τ η, ν (cid:11)
10 (19 , , τ c (cid:10) τ ν, ησ, σ (cid:11)
11 (3 , , − τ h (cid:10) ρ , τ η, ν (cid:11)
11 (11 , , τ P h (cid:10) ρ , τ η, ζ (cid:11)
11 (19 , , τ c (cid:10) ρ, h , τ ¯ σ (cid:11)
11 (19 , , τ P h (cid:10) ρ , τ η, ζ (cid:11)
11 (23 , , h · τ c σ · τ η
12 (8 , , − τ c (cid:10) τ η, h ν, ν (cid:11)
12 (16 , , τ h · h (cid:10) σ , , τ η (cid:11)
12 (16 , , τ P c (cid:10) τ µ , h ν, ν (cid:11)
12 (20 , , τ g (cid:10) ρ, τ h , ρ, h e (cid:11)
12 (24 , , τ P c h τ µ , h ν, ν i Table 12: Adams d differentialscoweight ( s, f, w ) x d ( x ) proof7 (15 , , h h h classical7 (17 , , e h d classical7 (14 , , τ h h ρ h d Lemma 8.88 (18 , , f h e Lemma 8.99 (17 , , τ e ( τ h ) d classical10 (18 , , τ f τ h e + ρ τ h · d Lemma 8.1010 (21 , , τ h g ρ c d Lemma 8.1111 (23 , , h i P h d classical11 (27 , , h g h h c C -motivic12 (26 , , j P h · d classical -MOTIVIC STABLE STEMS 35 Table 13: Adams d differentialscoweight ( s, f, w ) x d r ( x ) proof7 (15 , , h h h d + ρh d Lemma 8.1612 (23 , , τ h · τ e ρτ P h · d Lemma 8.1712 (25 , , c · τ e τ P h · h d Lemma 8.17Table 14: Hidden ρ extensions in the R -motivic Adams spectralsequencecoweight ( s, f, w ) source target7 (15 , , h h ρ h e , , h d τ h · h d , , ρτ h · h h · τ h , , ρ f τ h · d
10 (15 , , ρ τ h · h τ h · h h
10 (15 , , ρ τ f τ h · d
10 (23 , , h · τ c · d P h d
11 (15 , , τ h · h h τ h d
11 (17 , , τ h · τ e τ h · h d
11 (18 , , ρ f · τ h h · τ h · τ e
11 (23 , , h i τ P h d Table 15: Hidden h extensions in the R -motivic Adams spectralsequence coweight ( s, w ) source target7 (11 , ρ e τ h · P h , h f ρc d , h h g h c d
10 (22 , τ c · d P h d
11 (23 , τ h · h g τ P h · d Table 16: Hidden η extensions in the R -motivic Adams spectralsequence coweight ( s, f, w ) source target7 (15 , , h h ρ h e , , τ h · h h ρτ h · τ P c , , h f c d
10 (20 , , h · τ e ρτ c · d
10 (21 , , ρτ c · d P h d
11 (15 , , τ h · h h τ P c
11 (23 , , h i P c
06 EVA BELMONT AND DANIEL C. ISAKSEN
Table 17: Hidden values of extension by scalarscoweight ( s, f, w ) source target7 (11 , , ρ e τ P h , , ρh τ h k, k, k ) ρ h k +21 e P h k c , , ρf τ h d , , ρ τ e τ h d
10 (15 , , ρ τ f τ h d
10 (22 , , τ c · d P d
10 (23 , , h · τ c · d P h d
11 (20 , , τ h · ρf τ h g
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Department of Mathematics, Northwestern University, Evanston, IL 60208
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