Calculs de groupes d'homotopie stables de la sph�re, par la suite spectrale d'Adams-Novikov

The triviality of the 61-stem in the stable homotopy groups of spheres

“…In the 1980's, using the Adams-Novikov spectral sequence, Ravenel [47] computed up to the 108-stem at the prime 3, and the 999-stem at the prime 5. Previously, the computation was due independently to Nakamura [41] and Tangora [55] up to the 103-stem at the prime 3, and to Aubry [3] up to the 760-stem at the prime 5.…”

“…The first claim of Theorem 5.16 follows from the fact that π 3 (CP 3 1 ) = 0. In fact, in the E 1 -page of the Atiyah-Hirzebruch spectral sequence of CP 3 1 , there is only one candidate that lies in the degree that converges to π 3 : η [2]. However, because of the attaching map in CP 2 1 , we have an Atiyah-Hirzebruch differential…”

“…This follows from the fact that π 5 (CP 3 1 ) = Z/2, generated by ν [2]. In fact, because of the attaching maps in CP 3 1 , we have the Atiyah-Hirzebruch differentials 1 [6] →2ν [2] 2 [6] →4ν [2] η[4] →η 2 [2], which leave ν [2] as the only nontrivial element in the Atiyah-Hirzebruch E ∞ -page that converges to π 5 (CP 3 1 ). Next, we consider the following commutative diagram of cofiber sequences…”

On the existence of the self map 𝑣⁹₂ on the Smith-Toda complex 𝑉(1) at the prime 3

“…We only need the E 2 term of the Adams spectral sequence through the dimension of v 9 2 , which is 144. Since the dimension of v 4 is 160, the cohomology of Λ (2) coincides with H * (P * ) ⊗ P [v 3 ] in the range we are interested in. Figure 3.1 displays H * (P * ).…”

Book Review: Complex cobordism and stable homotopy groups of spheres

“…It takes a certain amount of youthful zeal to tabulate the 5-primary groups 7rf, 5 x in dimensions up to 1000, the previous record being 760 by Aubry [2]. In contrast, a smooth picture of 7rf/ % follows from the Adams-Novikov spectral sequence in dimensions up to 2p(p 2 -1) for odd primes (in dimensions up to 25 for p = 2).…”