Question

Hey all,

I know that in some versions of the Adams Spectral Sequence you can easily identify the image of $J$, and I was wondering if there was a way to identify the image of $J$ in the $E_2$ page of the classical mod-2 version, especially for $t-s=3$ (mod 4). Since the order of this image is known and it is known that the image is a direct summand, it isn't so hard to find it in $E_\infty$. Of course, if you can identify $Im(J)$ in an earlier page, then you learn a huge amount about the differentials in that column. This might imply that identifying the image of $J$ is almost as hard as calculating the differentials, so maybe this is too much to hope for, but maybe just maybe there's a trick.

Thanks

Answer 1

The image of $J$ is pretty easy to see in the Adams $E_2$ term: it consists of the elements along the vanishing line, plus, in dimensions 8k-1, of the towers that end near the vanishing line.

This identification is due to Mahowald, see The order of the image of the J-homomorphism for the announcement ( http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.bams/1183532412 ) and On bo-resolutions ( http://projecteuclid.org/DPubS?verb=Display&version=1.0&service=UI&handle=euclid.pjm/1102736799&page=record ) for details.