THE KERVAIRE INVARIANT ONE ELEMENT AND THE DOUBLE TRANSFER-f

The Kervaire invariant one problem has been one of the most fundamental and challenging problems in topology [6,8,9, 18, 19,231. Of course, the pivotal work was [S], which translated the original geometric problem [18, 191 into the problem of the stable homotopy groups of the sphere: Is It; E Ext;“” (2/2,;2/2) a permanent cycle in the Adams spectral sequence of the sphere? The traditional belief [23,28,37] is that, for each j, hf is a permanent cycle represented by OjE 7~;,+,~2 (SO), which factors through the double transfer P A P%S”. Here 1: P -+ So is the Kahn-Priddy map [17], and the double transfer lift of 0j is forced to have x2,_1 @ .x2,_ 1 E Hz,* t _ 2(P A P) as its stable mod-2 Hurewicz image. The probability of such a double transfer factorization was primarily supported by the Kahn-Priddy theorem [17] and unpublished calculations of Mark Mahowald. And there was a more general conjecture of Mahowald [28] which would imply that any Kervaire invariant one element factors through the double transfer. Though Singer [38] disproved the naive conjecture for n = 5, which states that ?+(A”P)+ EF$“+*(S’) is onto (where the target is associated with the classical Adams spectral sequence of the sphere Cl]), it did not contradict this conjecture of Mahowald, at least on the nose. Now, the purpose of this paper is to disprove such a belief: If the Kervaire invariant one element 0j E rc$,+ I _ 2(S”) exists andfactors through the double transfer PA P -+ So, then j I 4 (Theorem 3.1). We will prove this result as follows: In section 1, we show any such a double transfer lift has a BP-Hurewicz image with a gigantic order. In section 2, we study the BP-Adams operation on BP,,,,(P A P), and show that gigantic order elements in BP,,,,(P A P) cannot be in the BP-Hurewicz image. And, in section 3, these results are combined to prove Theorem 3.1.