most recent 30 from http://mathoverflow.net 2013-06-20T10:06:20Z http://mathoverflow.net/feeds/question/112554 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/112554/do-people-still-use-massey-products-for-computations-in-the-adams-spectral-sequen Joseph Victor 2012-11-16T05:59:29Z 2012-11-16T17:54:20Z

<p>Hey everyone,</p> <p>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.<br> 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?</p> <p>Thanks</p>

http://mathoverflow.net/questions/112554/do-people-still-use-massey-products-for-computations-in-the-adams-spectral-sequen/112600#112600 Peter May 2012-11-16T17:54:20Z 2012-11-16T17:54:20Z

<p>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''). </p>