The Kervaire invariant one element and the double transfer
This is a very nice paper in stable homotopy theory because it shows, by an elementary proof, that the prevailing opinion about a central question in the subject was false. The question is whether the element \(h^ 2_ j\) on the 2-line of the Adams spectral sequence survives to give a class in the stable homotopy groups of spheres. One reason for the importance of this question is that, by early work of Browder, existence of such classes is equivalent to the existence of framed manifolds of Kervaire invariant 1. The classes are known to exist for \(j\leq 5\). The prevailing opinion was that (1) these stable homotopy classes exist for all positive integers \(j\) and (2) they factor through the double transfer map \(P\wedge P\to S^ 0\). Part 1 may still be true, but part 2 is shown here to be false. Indeed it is proved that if there is an element in the stable homotopy groups of spheres which is detected by \(h^ 2_ j\) and factors through the double transfer map, then \(j\leq 4\). In particular, the element detected by \(h^ 2_ 5\) does not factor through the double transfer. The proof proceeds as follows. If there were an element in \(\pi_{2^{j+1}-2}(P \wedge P)\) which maps to the desired element of \(\pi_{2^{j+1}-2} (S^ 0)\), then its Hurewicz image in \(BP_ * (P\wedge P)\) must be annihilated by the Adams operation \(\psi^ 3 -1\). It is not difficult to calculate the order of such a class and to see that, since \(\psi^ 3\) is multiplication by \(3^ n\) on \(BP_{2n} (P\wedge P)\), this implies that we must have \(j\leq 4\).
scientific article; zbMATH DE number 751261