Hey everyone,

It seems to me like in the literature of the Adams Spectral Sequence, older publications (Toda, May, Tengora+Mahowald) make heavy and explicit use of Massey Products for computations.
More "recent" sources (Kahn, Milgram, Ravenel, May (again), Bruner) seem to like to make references to the Massey products, even just for the sake of naming, but use Steenrod Squares for actual computations. The few computations I know of using Massey products can be more easily done with Steenrod Squares, but this may just be my ignorance about Massey products. My question then is do these things still play an important computational role?


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    $\begingroup$ But how many people are really doing ASS calculations? If you look at, say, the calculation of the homotopy of tmf at $p=2,3$ then it makes heavy use of Massey products. $\endgroup$
    – Drew Heard
    Nov 16, 2012 at 9:41
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    $\begingroup$ Also, if you check out the work of Shimomura and colleagues, who use the chromatic spectral sequence in a fairly heavy way, I'm sure you'll find Massey products a plenty! $\endgroup$
    – Drew Heard
    Nov 16, 2012 at 9:49
  • $\begingroup$ Drew, I want to be doing ASS calculations!!!!! But I'm just a little undergrad and there is soooo muuuuuch literature. Thanks for the comment though. $\endgroup$ Nov 16, 2012 at 22:14
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    $\begingroup$ If you specifically want to see these in action, 'Bordism, Stable Homotopy, and Adams Spectral Sequences' by Kochman, does some serious $p=2$ calculations of the stable homotopy groups of spheres $\endgroup$
    – Drew Heard
    Nov 17, 2012 at 0:00

1 Answer 1


Why would anything computationally useful be obsolete? Massey products and Toda brackets are intrinsic to stable homotopy theory. It is guaranteed in advance that every element of $E_2$ of the classical Adams spectral sequence (for the homotopy groups of spheres, a similar statement holds more generally for the ASS computing maps between spectra) above the $s=1$ line is decomposable in terms of matric Massey products. Similarly, every element of the stable homotopy groups of spheres is decomposable in terms of matric Toda brackets built from the Hopf invariant one elements. Drew's comments are on the mark: few people nowadays do these kinds of calculations, or know how to do them, which is a pity. We still know relatively little about concrete calculations of stable homotopy groups. These operations are complementary to, not in competition with, Steenrod operations and their related homotopy operations (see e.g. Bruner's contributions to $H_{\infty}$ ring spectra and their applications'').

  • $\begingroup$ Thank you for the response. I apologize because I am a still learning the literature and am probably missing some important point. In H_\infty ring spectra and applications, Bruner does not seem to use Massey Products at all, unless it is in some subtle way I cannot pick up on. He writes down a lot of Massey products, even citing a theorem about their interactions with Steenrod Squares, but does not seem to actually use them for computations. The same holds for Milgram's big paper on the topic, and this is what confuses me. $\endgroup$ Nov 16, 2012 at 21:41
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    $\begingroup$ I sympathize with "there is sooo muuuch literature''. The people who do the most calculational of calculational work really do use Massey products and Toda brackets (I have one student working that way right now). Look at papers of Mahowald and collaborators to see more such things in action, for example papers on the Kervaire invariant one problem in dimensions 30 and 62 (but they may be hard reading). $\endgroup$
    – Peter May
    Nov 17, 2012 at 0:31

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