What is know about the homotopy groups of $S/3$ where $S/3 = \mathrm{hocofib}(S \xrightarrow{\cdot 3} S)$? Otherwise, is there some reference I can consult for the $BP$ANSS for $S/3$?
1 Answer
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For $3$primary homotopy of $S$ there is early work by
Nakamura, Osamu
Some differentials in the mod 3 Adams spectral sequence.
Bull. Sci. Engrg. Div. Univ. Ryukyus Math. Natur. Sci. No. 19 (1975), 1–25.
and
Tangora, Martin (4OX)
Some homotopy groups mod 3. Conference on homotopy theory (Evanston, Ill., 1974), 227–245,
Notas Mat. Simpos., 1, Soc. Mat. Mexicana, México, 1975.
through degree 103, extended to degree 108 by Ravenel in his green book, but beware of some issues in that extended range. There is ongoing work by Eva Belmont on this that uses modern machine computations.
From these 3local results you can backtrack to extract quite a lot of information on $\pi_*(S/3)$. If you need to go further, maybe try
Arita, Yoshiko (JHROSES); Shimomura, Katsumi (JTOTED)
The chromatic E1term H1M11 at the prime 3.
Hiroshima Math. J. 26 (1996), no. 2, 415–431.
Arita, Yoshiko (JHROSES); Shimomura, Katsumi (JTOTED)
On products of some βelements in the homotopy of the mod 3 Moore spectrum.
Hiroshima Math. J. 27 (1997), no. 3, 477–486.
Shimomura, Katsumi (JKOCHS)
The homotopy groups of the L2localized mod 3 Moore spectrum.
J. Math. Soc. Japan 52 (2000), no. 1, 65–90.
and compare with Nassau's Adams spectral sequence charts at

$\begingroup$ Thank you very much! What a wonderful, detailed response. $\endgroup$ May 16, 2022 at 14:47