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

There are p-local spectra T (m) with BP∗(T (m)) = BP∗[t1, . . . , tm]. 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 2 and whose target is the first nontrivial element in filtration 2p + 1. The same goes for m = 1, and for m > 1 the target is v2 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∗[t1, . . . , tm] ⊂ 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 ≤ k < p. 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 [Rav02]. In particular T (0) is the p-localized sphere spectrum. T (1) is the p-localization of the Thom spectrum of the bundle induced by the map ΩS2p−1 → BU obtained using the loop space structure of BU to extend the map S2p−2 → BU representing the generator of the homotopy group π2p−2(BU). Let (A,Γ) denote the Hopf algebroid (BP∗, BP∗(BP )); see [Rav86, A1] for more information. A change-of-rings isomorphism identifies the Adams-Novikov E2-term for for T (m) with (1) ExtΓ(m+1)(A, A) where Γ(m+ 1) = Γ/(t1, . . . , tm) = BP∗[tm+1, tm+2, . . .] This Hopf algebroid is cocommutative below the dimension of t2m+2, 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 ExtΓ(m+1) for short. 1991 Mathematics Subject Classification. Primary 55Q10, 55T15; Secondary 55Q45, 55Q51. The author acknowledges support from NSF grant DMS-9802516. 1 2 DOUGLAS C. RAVENEL The following was proved in [Rav86, 6.5.9 and 6.5.12]. Theorem A. Description of ExtΓ(m+1). For each m ≥ 0 and each prime p, ExtΓ(m+1) = Z(p)[v1, . . . , vm], 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 |vm+1|−1. Hence π∗(T (m)) ∼= A(m) in this range. Our next result concerns ExtΓ(m+1). Before stating it we need some chromatic notation. Consider the short exact sequences of Γ-comodules (and hence of Γ(m+1)comodules)