A conjecture of Barratt-Jones-Mahowald concerning framed manifolds having Kervaire invariant one

Abstract

To SETTLE the question of the existence or non-existence of a framed manifold having a nontrivial Kervaire invariant (or Arf invariant) is one of the main long-standing problems in algebraic topology. The Kervaire invariant is a Z/Zvalued invariant which may be formulated in many contexts. Originally it occurred as an invariant in framed surgery theory and for this approach the reader may consult Cl23 for example. It is reformulated in [7] in terms of the Adams spectral sequence for the stable homotopy of spheres. In particular the only open cases were reduced to determining whether ht E Exts ‘*+‘(Z/2,2/2) is an infinite cycle, producing a non-trivial element 8, in the 2’[+’ 2 stem of the stable homotopy of spheres. More recently the Kahn-Priddy theorem [8] and the algebraic Kahn-Priddy theorem [lo] have been used to convert it to a problem in the stable homotopy of infinite dimensional real projective space [WP”. The Kahn-Priddy theorem gives a stable map T: [WF’aSO which is a split surjection of stable homotopy groups (localized at the prime 2)