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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 $S/3$?

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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 (4-OX)
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 3-local 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 (J-HROSES); Shimomura, Katsumi (J-TOTED)
The chromatic E1-term H1M11 at the prime 3.
Hiroshima Math. J. 26 (1996), no. 2, 415–431.

Arita, Yoshiko (J-HROSES); Shimomura, Katsumi (J-TOTED)
On products of some β-elements in the homotopy of the mod 3 Moore spectrum.
Hiroshima Math. J. 27 (1997), no. 3, 477–486.

Shimomura, Katsumi (J-KOCHS)
The homotopy groups of the L2-localized mod 3 Moore spectrum.
J. Math. Soc. Japan 52 (2000), no. 1, 65–90. 

and compare with Nassau's Adams spectral sequence charts at

http://nullhomotopie.de/charts

http://nullhomotopie.de/charts/m3ext0.pdf

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

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