THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m)

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1 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) DOUGLAS C. RAVENEL Abstract. There are p-local spectra T (m) with BP (T (m)) = BP [t,..., t m]. In this paper we determine the first nontrivial differential in the Adams Novikov spectral sequence for each of them for p odd. For m = 0 (the sphere spectrum) this is the Toda differential, whose source has filtration and whose target is the first nontrivial element in filtration p +. The same goes for m =, and for m > the target is v times the first such element. The proof uses the Thomified Eilenberg-Moore spectral sequence. We also establish a vanishing line and the behavior near it in the Adams Novikov spectral sequence for T (m). This paper concerns the Adams Novikov spectral sequence for the spectra T (m) introduced in [Rav86, 6.5]. We begin by recalling their basic properties. For each prime p and natural number m there is a p-local spectrum T (m) such that BP (T (m)) = BP [t,..., t m ] BP (BP ) as a comodule algebra over BP (BP ). It is a summand of the p-localization of the Thom spectrum of the stable bundle induced by ΩSU(k) ΩSU = BU for any k satisfying p m k < p m+. These Thom spectra figure in the proof of the Nilpotence Theorem of [DHS88]. The T (m) themselves figure in the method of infinite descent, the technique for calculating the stable homotopy groups of spheres described in [Rav86, Chapter 7], [Rav04, Chapter 7] and [Rav0]. In particular T (0) is the p-localized sphere spectrum. T () is the p-localization of the Thom spectrum of the bundle induced by the map ΩS p BU obtained using the loop space structure of BU to extend the map S p BU representing the generator of the homotopy group π p (BU). Let (A, Γ) denote the Hopf algebroid (BP, BP (BP )); see [Rav86, A] for more information. A change-of-rings isomorphism identifies the Adams-Novikov E -term for for T (m) with () Ext Γ(m+) (A, A) where Γ(m + ) = Γ/(t,..., t m ) = BP [t m+, t m+,...] This Hopf algebroid is cocommutative below the dimension of t m+, so its Ext group (and the homotopy of T (m)) in this range is relatively easy to deal with. We will denote this Ext group by Ext Γ(m+) for short. 99 Mathematics Subject Classification. Primary 55Q0, 55T5; Secondary 55Q45, 55Q5. The author acknowledges support from NSF grant DMS

2 DOUGLAS C. RAVENEL The following was proved in [Rav86, and 6.5.]. Theorem A. Description of Ext 0 Γ(m+). For each m 0 and each prime p, Ext 0 Γ(m+) = Z (p) [v,..., v m ], and we denote this ring by A(m). Each of these generators is a permanent cycle, and there are no higher Ext groups below dimension v m+. Hence π (T (m)) = A(m) in this range. Our next result concerns Ext Γ(m+). Before stating it we need some chromatic notation. Consider the short exact sequences of Γ-comodules (and hence of Γ(m+)- comodules) () 0 N 0 M 0 N 0 and (3) 0 N M N 0 where N 0 = BP, M 0 = p BP = Q BP, Let N = BP /(p ) = Q/Z (p) BP M = v /(p ), and N = BP /(p, v ). δ 0 : Ext s Γ(m+)(N ) Ext s+ Γ(m+) and δ : Ext s Γ(m+)(N ) Ext s+ Γ(m+) (N ) denote the associated connecting homomorphisms. We will write elements in N and M as fractions x pc where c is a p-local integer and x BP ( x v BP for M ) is not divisible by p, i.e., the fraction has been reduced to lowest terms. If pc is divisible by p e and not by p e+, then it is understdood that p e kills this element. Similarly elements in N are written as fractions x pcv e with c Z (p), e > 0 and x BP is nontrivial mod I. This element is killed by (pc, v e ). The long exact sequence of Ext groups associated with () has a surjective connecting homomorphism Ext 0 Γ(m+)(N ) Ext Γ(m+). The algebraic statement in the following was proved in [Rav86, 6.5.] while the topological part is proved in [Rav0]. From now on we will denote v m+i by v i for i > 0 and p m by ω.

3 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) 3 Theorem B. Description of Ext Γ(m+). In all cases except m = 0 and p =, Ext Γ(m+) is isomorphic (via δ 0 ) to the A(m)-submodule of N generated by the set { } v i ip : i > 0. Each of these elements is a permanent cycle, and there are no higher Ext groups below dimension p v. We will denote δ 0 ( v p ) by α. For m = 0 we will use similar notation without the hats. For higher Ext groups in low dimensions, we need the following notation for elements in Ext Γ(m+). In each case the indicated element of N is invariant, i.e., it lies in Ext 0 Γ(m+)(N ). Let ( ) b,0 = β v = δ 0 δ pv ( ) v i β i = δ 0 δ pv (4) ( b, = β v p p/p = δ 0 δ pv p ( v 3 θ = δ 0 δ ) pv v v p pv +p θ = δ 0 δ ( v v 3 pv v v v p pv +p ) + vpω v p v + vpω v p v Again for m = 0 we will use similar notation without the hats. The elements θ and θ are not defined. The element θ is in the Massey product α, pι, θ where ι π 0 (T (m)) denotes the fundamental class. Note that even though the preimage of θ has order p in N, θ itself has order p since the preimage of p θ is the coboundary (in the chromatic cobar complex [Rav86, 5..0]) of v vpω v p Let B denote the A(m)-submodule of N generated by b, and θ, and let C be the one generated by { } v j (5) vi : 0 < i p, 0 j p pi. ipv For the -line and above, we have the following, essentially proved as [Rav86, Theorem 7..3], [Rav04, Theorem 7..6] and [Rav0, Theorem 4.5]. Each of them determines Ext,t Γ(m+) for t < p v. This is extended to t < (p + p) v in [NR, Theorem 7.], where θ and θ are denoted by θ 0 and v θ0 respectively.. )

4 4 DOUGLAS C. RAVENEL Theorem C. Description of Ext Γ(m+) in low dimensions. For m > 0, Ext,t Γ(m+) for t p p + p v is isomorphic (via δ 0 δ ) to B C as defined above in (5). Moreover Ext +ε+k Γ(m+) is isomorphic to α b ε k,0c. We are now ready to state the main result of this paper. in this range (for ε = 0, and k + ε > 0) Theorem. The first differential for T (m) for m > 0. The first (lowest dimensional) nontrivial differential in the Adams Novikov spectral sequence for the spectrum T () at an odd prime p is d p ( θ) = α bp,0 where α is an in Theorem B, and b,0 and θ are as in (4). For m > the first nontrivial differential in the Adams Novikov spectral sequence for the spectrum T (m) at an odd prime p is d p ( θ ) = v α bp,0. Corollary. The first nontrivial group extension in T (m) for m >. For each m > the first nontrivial group extension in the passage from the E -term of the Adams Novikov spectral sequence to π (T (m)) is up to unit scalar multiplication. p θ = v bp,0. For m = 0, the corresponding statement is the Toda differential, d p (β p/p ) = α β p = α b p,0 originally proved by Toda in [Tod68] and [Tod67] and stated as [Rav86, Theorem 4.4.]. Our proof is the opposite of Toda s in the following sense. For all m the first differential occurs in the smallest dimension where there is a potential source and target with compatible bidegrees. Toda shows that his potential target, α β p, vanishes in homotopy, the first instance of his important relation. It follows that the corresponding element in Ext must be killed by a differential, and there is only one possible source. We show that for m > 0 our source cannot represent a homotopy element. It follows that it supports a nontrivial differential, and there is only one possible target. Toda s target is on the vanishing line described below in Theorem 3, i.e., it is the first element in filtration p +. The same is true for our target when m =, but for m > the first such element does not lie above an element on the -line with the right bidegree. Its product with v is the first one that does and is therefore the first potential target. For m = the dimension of the element θ (the source of the differential) is 54 for p = 3 and 38 for p = 5. For p = 3 this Ext group and the first differential are illustrated for m = and m = in Figures and respectively. The following vanishing line result is needed to prove Theorem and may be of independent interest.

5 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) 5 Theorem 3. The Adams-Novikov vanishing line for T (m). In the Adams Novikov spectral sequence for T (m), E s,t = 0 for { ps for s even t < (pω ) (ps p + ) for s odd. Moreover each element for s and { (ps + ) for s even t < (pω ) (ps p + 4) for s odd. is the product of a monomial in α and b,0 with an element in A(m)/I. In particular the first nontrivial element in E s, [s/] is b,0 αs [s/]. This will be proved at the end of the paper. The first element in Ext Γ(m+) not having the form described above is the image of v v pv, which is in the Massey product α, pι, b,0. To prove Theorem we use certain cohomology operations in T (m)-theory for m > 0, i.e., maps T (m) r r Σ t p m T (m) and T (m) Σ p t m T (m) derived from the splitting of T (m) T (m). They have properties similar to Steenrod and Quillen operations, but they commute with each other. These operations do not exist for m = 0, so the method used here cannot be used to establish the Toda differential. Lemma 4. Two commuting Quillen operations in T (m)-theory. Let I = (i, i,..., i m ) be a sequence of m nonnegative integers, and let t I = t i... tim m. There are operations r I T (m) ti (T (m)) dual to the elements where t I T (m) ti (T (m)) T (m) (T (m)) = π (T (m))[t,..., t m ], and t i maps to the element of the same name in BP (BP ). In particular let r and r p m be the duals of t and t p m respectively. Then r r p m = r p m r. Proof. We can compute T (m) (T (m)) using the Atiyah Hirzebruch spectral sequence with E = H (T (m); π (T (m))) = π (T (m))[t,..., t m ].

6 6 DOUGLAS C. RAVENEL 7 6 b3,0 5 4 b,0 b,0 β s 3 β b,0 b, θ 0 α t s Figure. The Adams-Novikov E -term for T () at p = 3 in dimensions 54, showing the first nontrivial differential. Elements on the 0- and -lines divisible by v are not shown. Elements on the -line and above divisible by v are not shown. Vertical, horizontal and diagonal lines indicate multiplication by p, v and α respectively. We know from Theorem A that π (T (m)) is concentrated in even dimensions below dimension v. It follows that each t i is a permanent cycle and the spectral sequence collapses, so the Hopf algebroid T (m) (T (m)) is as claimed. The map T (m) (T (m)) BP (BP ) is monomorphic through dimension t m, so the coproduct on t i T (m) (T (m)) is determined by the coproduct on t i BP (BP ). The commuting of the two specified operations follows from the fact that the only t I having either t t p m or t p m t in its coproduct expansion is t t p m, which

7 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) b3,0 5 4 b,0 β b,0 s 3 b,0 β b, θ θ α t s Figure. The Adams-Novikov E -term for T () at p = 3 in dimensions 530. Elements on the 0- and -lines divisible by v or v are not shown. Elements on the -line and above divisible by v or v 3 are not shown except for v α b3,0, the target of the first differential. Vertical, horizontal and diagonal lines indicate multiplication by p, v and α respectively. has both terms, i.e., (t t p m) = t t p m + t t p m + t t p m + t p m t The action of these operations in Ext has the following interpretation. Consider the Hopf algebroid extension (this term is defined in [Rav86, A..5]) (A(m), G(, m )) (BP, Γ) (BP, Γ(m + )),

8 8 DOUGLAS C. RAVENEL where G(, m ) = A(m)[t,..., t m ]. Then Ext Γ(m+), which is the Adams- Novikov E -term for T (m) by (), is a G(, m )-comodule by [Rav86, A.3.4 (a)]. Lemma 5. Relation between Quillen operations and comodule structure. The G(, m )-comodule structure on Ext(BP (T ())) described above is dual to the action of the operations of Lemma 4. Proof. The operation r i of Lemma 4 is induced by the composite map η R T (m) T (m) T (m) Σ ti T (m). Applying the functor Ext Γ (BP ( )) and the change-of-rings isomorphism of () we get η R Ext Γ(m+) Ext Γ(m+) (BP [t,..., t m ]) Σ ti Ext Γ(m+). The middle term is G(, m ) A(m) Ext Γ(m+) = A(m)[t,..., t m ] A(m) Ext Γ(m+), and the map induced by η R is the G(, m )-comodule structure map. The following lemmas concern the action of specific operations in the relevant dimensions. Lemma 6. Action of Quillen operations in the Adams Novikov spectral sequence for T (m). In the E -term of the Adams Novikov spectral sequence for T (m) we have For m = we also have r I r ( θ ) = v p m θ, r I r p m ( θ ) = v θ = v b,, r p m (v p m θ) = 0, r p m (v v p m b p,0 ) = 0, and r (v b, ) = 0. r ( θ) = 0 and r p ( θ) = b,. Lemma 7. Action of Quillen operations in π (T (m)). In π (T (m)), r p m (v p m θ) = 0 and r (v b, ) = v bp,0 up to unit scalar multiplication. For m =, r ( b, ) = ± b p,0. We will prove these two lemmas below. To prove Theorem we will assume that θ represents a homotopy element x and show that (8) r r p m (x) r p m r (x),

9 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) 9 which contradicts that fact that r and r p m commute. This means that θ cannot survive to represent a homotopy element. The indicated differential d p ( θ ) = v α bp,0 is the only one it can support because the indicated target is the only element in an appropriate bidegree by Theorem 3. When m = we can make a similar calculation on θ and conclude that d p ( θ) = α bp,0. As before let x denote a hypothetical homotopy element represented by θ. Lemma 6 determines r (x) modulo elements of higher filtration, so we have r (x) = v p m( θ + cv bp,0 ) for some scalar c. Note that b p,0 is the first element in filtration p, so below its dimension, computing in Ext is equivalent to computing in homotopy. It follows that On the other hand we have r p m r (x) = r (v p m θ + cv v p m b p,0 ) = 0. r r p m (x) = r ( v b, ) = v bp,0 up to unit scalar multiplication. This completes the proof of Theorem, modulo Lemmas 6 and 7. For Corollary, note that in Ext Γ(m+) θ α, pι, θ. Since θ is not a homotopy element, the Toda bracket corresponding to the Massey product above must not be defined. Since we know that α represents an element of order p, it follows that the homotopy element represented by θ does not have order p. Hence p times it must have filtration p, and the only possibility there is v bp,0. Remark: Possible higher differentials. We conjecture that d pm ( b,m ) = α m bp,0 for all m > 0 (we prove this here for m = ), where for m > b,m = δ 0 δ ( v ω/p 3 pv ω/p vω/p v ω pv (p+)ω/p + vω v ω/p pv (p+)ω/p Computations similar to those to be described below show that if y is a homotopy element representing b,m, then r m r p m(y) = b ω,0 but r p mr m (y) = 0, which leads to a contradiction as above. The element α m bp,0 is in the right dimension to be a target; one has to show that this differential is not preempted by an earlier one in the same dimension. If we let m go to as in [Rav00], this element disappears. This suggests we should look at a Hopf algebroid graded over Q[ω] rather than just over Z Zω. Proof of Lemma 6. We can compute the action of r and r p m on these elements by computing the actions of the corresponding Quillen operations on the numerators ).

10 0 DOUGLAS C. RAVENEL of the chromatic fractions modulo the ideals associated with the denominators. For example we need to know r I (v v v p ) modulo (p, vp+ ), and so on. We have ( ) r p m ( θ v v v p ) = r p m pv +p and ( ( r ( θ ) = δ 0 δ r v v 3 pv v v v p pv p+ = δ 0 δ ( r ( v ) v 3 pv + v r ( v 3 ) pv = δ 0 δ ( = v p m θ. r (v ) v v p pv p+ + r (v pω ) v p v = v v p pv p = v b,, v r ( v ) v p pv p+ )) + vpω v p v v v r ( v p ) pv p+ + vpω r ( v ) ( p + v pω v v r p v vp m v 3 pv v v p pv + vp v v p pv p+ + v v p m v p pv p+ )) vpω vp m v p v We leave the additional computations for m = as an exercise. ) vpω v pv Proof of Lemma 7. We need the Thomified Eilenberg-Moore spectral sequence of [MRS0]. Given a fibration of spaces X E B with a stable vector bundle over E, we get a spectral sequence converging to the homotopy of Y, the Thom spectrum for the induced bundle over X. If the fibration is ΩSU(pω ) ΩSU B with the evident vector bundle over ΩSU = BU, we get the usual Adams Novikov spectral sequence for X(pω ), the Thom spectrum associated with ΩSU(pω ), which has T (m) as a summand. If we take the Cartesian product of this fibration with Ω S pω pt. ΩS pω, the E -term is a subquotient of the tensor product of the one above with H (Ω S pω ) equipped with the Eilenberg-Moore filtration, and the spectral sequence converges to π (Ω S pω X(pω )). The map Ω S pω ΩSU(pω ) sends this homology to E( α, h m+,,...) P ( b,0, b,,...)

11 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) in the Adams Novikov spectral sequence for T (m). Thus we have a map to the Adams Novikov spectral sequence for T (m) from a spectral sequence converging to T (m) (Ω S pω ). In E this gives us where E is a subquotient of E Ext Γ(m+) (9) E( α, h m+,,...) P ( b,0, b,,...) Ext Γ(m+). We want to deduce the lemma from the fact that in H (Ω S pω ), P ( b, ) = b p,0 and P p m ( b, ) = 0. For this note that the following diagrams commute. T (m) r Σ p T (m) T (m) r p m Σ p tm T (m) ι H/(p) P ι Σ p H/(p) and ι H/(p) χ(p p m ) ι Σ p tm H/(p), where H/(p) denotes the mod p Eilenberg Mac Lane spectrum, the map ι : T (m) H/(p) is the bottom mod p cohomology class, and P and χ(p p m ) denote the conjugates of the Steenrod operations dual to ξ and ξ p m respectively. It follows that in E we have r ( b, ) = b p, where the missing terms involve elements with lower dimensional first factors. However this operation must respect the Snaith splitting of Ω S pω, which means there are no lower dimensional first factors, so r ( b, ) = b p,0 and r ( b, ) = b p,0 π (T (m)) precisely. For m > it follows that r (v b, ) = v bp,0 as required. A similar argument shows that r p m ( b, ) = 0, and the result follows. Remark: Why this method does not apply to the original Toda differatial. The method described here cannot be used to establish the Toda differential of [Tod67] and [Tod68], i.e., d(b, ) = α b p,0, because we do not have the operation r acting in stable homotopy. However as in the proof of Lemma 7 an essential fact is the action of the Steenrod algebra in H (Ω S p ), in particular the fact that P (b, ) = b p,0. As above one can use the Thomified Eilenberg-Moore spectral sequence to show that the element b, Ext Γ cannot be a permanent cycle. Proof of Theorem 3. Let { ps for s even f m (s) = (pω ) (ps p + ) for s odd,

12 DOUGLAS C. RAVENEL so the theorem says that Ext s,t Γ(m+) = 0 for t < f m(s). Let g m (s) be the corresponding function for the actual vanishing line, i.e., the smallest number such that Ext s,gm(s) Γ(m+) 0. We will use the small descent spectral sequence of [Rav0,.7]. For each h 0 there is a T (m)-module spectrum T (m) h with In particular BP (T (m) h ) = BP (T (m)){t j m+ : 0 j h}. T (m) = T (m) 0 and T (m + ) = lim T (m) h. The small descent spectral sequence computes Ext Γ (BP (T (m) p i )) in terms of Ext Γ (BP (T (m) p i+ )). Its E -term is where the elements E,s = E(ĥ,i) P ( b,i ) Ext s Γ(BP (T (m) p i+ )) ĥ,i E,0 with ĥ,i = p i (pω ) and b,i E,0 with b,i = p i+ (pω ) are permanent cycles. Letting i go to, we can conclude that is a subquotient of This means that Ext Γ(m+) = Ext Γ (BP (T (m))) E( α, ĥ,,... ) P ( b,0, b,,... ) Ext Γ(m+). g m (s) = min(f m (s), g m+ (s)), i.e., that Ext Γ(m+) has the desired vanishing line if Ext Γ(m+) does. We will now use downward induction on m, i.e., the method of infinite descent, as follows. We know that T (m ) is equivalent to BP below dimension v m +. It follows that for any fixed t, Ext s,t Γ(m +) vanishes for s > 0 and m sufficiently large. This means that for a fixed value of s there is an m > m such that g m (s) f m (s). This implies that g m (s) = min(f m (s), g m (s)) f m (s), so g m (s) = min(f m (s), g m (s)) f m (s) if m m, and so on, leading to the desired conclusion that g m (s) = f m (s). The second assertion of the theorem concerns elements for which t < f m (s)+ v. We will look at the small descent spectral sequence converging to Ext Γ(m+) with E = E( α ) P ( b,0 ) Ext Γ (BP (T (m) p )). The following differentials occur in the range in question. d ( v ) = p α, d (ĥ,) = p b,0 and d (ĥ,0) = v b,0.

13 THE FIRST ADAMS-NOVIKOV DIFFERENTIAL FOR THE SPECTRUM T (m) 3 These imply that b [s/],0 and hence b,0 αs [s/] for each s are annihilated by I and by nothing else in this range, and the result follows. References [DHS88] Ethan S. Devinatz, Michael J. Hopkins, and Jeffrey H. Smith. Nilpotence and stable homotopy theory. I. Ann. of Math. (), 8():07 4, 988. [MRS0] Mark Mahowald, Douglas C. Ravenel, and Paul Shick. The Thomified Eilenberg-Moore spectral sequence. In Cohomological methods in homotopy theory (Bellaterra, 998), volume 96 of Progr. Math., pages 49 6, Basel, 00. Birkhäuser. [NR] H. Nakai and D. C. Ravenel. The method of infinite descent in stable homotopy theory II. To appear, preprint available online at second author s website. [Rav86] Douglas C. Ravenel. Complex cobordism and stable homotopy groups of spheres, volume of Pure and Applied Mathematics. Academic Press Inc., Orlando, FL, 986. Errata and second edition available online at author s website. [Rav00] Douglas C. Ravenel. The microstable Adams-Novikov spectral sequence. In Une dégustation topologique [Topological morsels]: homotopy theory in the Swiss Alps (Arolla, 999), volume 65 of Contemp. Math., pages 93 09, Providence, RI, 000. Amer. Math. Soc. [Rav0] Douglas C. Ravenel. The method of infinite descent in stable homotopy theory. I. In Recent progress in homotopy theory (Baltimore, MD, 000), volume 93 of Contemp. Math., pages 5 84, Providence, RI, 00. Amer. Math. Soc. Available online at [Rav04] D. C. Ravenel. Complex Cobordism and Stable Homotopy Groups of Spheres, Second Edition. American Mathematical Society, Providence, 004. Available online at author s website. [Tod67] Hirosi Toda. An important relation in homotopy groups of spheres. Proc. Japan Acad., 43:839 84, 967. [Tod68] Hirosi Toda. Extended p-th powers of complexes and applications to homotopy theory. Proc. Japan Acad., 44:98 03, 968. University of Rochester, Rochester, NY 467