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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

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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 ( Link ) and On bo-resolutions ( Link ) for details.

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  • $\begingroup$ That seems too good to be true! I will now read the paper. $\endgroup$
    – Joseph Victor
    Commented Sep 3, 2012 at 20:11
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    $\begingroup$ A later paper by Donald M Davis and Mark Mahowald: The image of the stable J-homomorphism, Topology 28 (1989), no. 1, 39–58, is even more directly about this question. Its Theorem 1.1 gives the Adams filtration and sphere of origin of each element of the image of J. $\endgroup$
    – John Rognes
    Commented May 13, 2015 at 19:53

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