The Adams Spectral Sequence and the Triple Transfer

The "hit" problem of five variables in the generic degree and its application

“…2 ) be the subspace of H * (F s 2 ) consisting of all elements that are annihilated by all positive degree Steenrod squares, then there is an induced action of GL s on P H * (F s 2 ), and we have an F 2 -linear map from the coinvariant elements of P H * (F s 2 ) to mod-2 cohomology group Ext s,s+ * A (F 2 , F 2 ) of the Steenrod algebra, which is induced over the E 2 -term of the Adams spectral sequence by the geometrical transfer map Σ ∞ (B(F s 2 ) + ) −→ Σ ∞ (S 0 ) in stable homotopy theory (see also Mitchell [18]). These transfers can be played a key role in the study of the Kervaire invariant one problem (see Browder [3], Mahowald [15], Minami [16,17]). The Kervaire invariant was first introduced by Browder's work [3], where he shows that the classes h 2 j ∈ Ext 2,2 j+1 A (F 2 , F 2 ) are the permanent cycles in the classical Adams spectral sequence at the prime 2, if and only if smooth framed manifolds of Kervaire invariant one exist only in dimensions 2 j+1 − 2.…”

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“…The Conner-Floyd conjecture was proved in the early 80's: [29,32], and it turns out that, together with detailed information about the p-series, the above description of I n leads in fact to a full description of the (additive) structure of the Brown-Peterson homology of (BZ/ p) ∧n [20,21]. The relevance of such a calculation has been confirmed by Minami's work [25][26][27][28] on the possible existence of framed manifolds of Kervaire invariant 1 (that is, on the basic problem of understanding stable homotopy classes of spheres detected in the 2-line of the classical Adams spectral sequence). Now, in view of the basic role the p-series played in the above development, it would be interesting to see to what extent the information in this paper for the p k -series can be used in a calculation of BP * (BZ/ p k 1 × • • • × BZ/ p k n ), as well as its implications in the stable homotopy groups of spheres.…”

Transfert algébrique et action du groupe linéaire sur les puissances divisées modulo 2

“…(Comme le terme « triple » a été désigné au transfert de degré trois [36], le transfert de degré quatre sera désormais appelé transfert quadruple.) Signalons qu'une large partie de notre théorème a été découverte dans [8,17,18], à savoir : …”

“…Le transfert algébrique T r k est induit « au niveau E 2 » par le transfert homotopique π S * (B(Z/2) k + ) −→ π S * (S 0 ) [4,14,31,39,47]. Une analyse de son comportement apportera sans doute des informations importantes à la théorie de l'homotopie, comme l'ont montré les travaux de Minami [36,38]. La nécessité d'une telle analyse contribuera, nous l'espérons, à justifier la raison d'être de nos présents travaux.…”