As the title suggest, I would like know the motivation/ historical background why chromatic homotopy theory is called 'chromatic'. Literally, what analogy to colors it might have. Accordings to wikipedia the origin of such homotopy theories bases on Quillen's work on cohomology theories of formal groups. This allows due to wikipedia to study that studies complex oriented cohomology theories from the 'chromatic' point of view.

This involves classification of these theories by so called 'chromatic levels'.

Has this 'visualization' regarding the theories as 'lying on certain levels' some funny analogy to colors regarded from physical viewpoint as a part of electromagnetic spectrum? Sound probably silly but I am quite curious what was the background for the choice of the name 'chromatic' in this context.

  • 7
    $\begingroup$ From here: "[...] We now know that the stable homotopy groups of spheres arise in periodic families, and thus admit a chromatic decomposition by their periods, in a manner similar to how light waves have different colors, determined by their wave-lengths." I will let someone who actually understands the topic and can explain the passage elaborate in an answer though. $\endgroup$ – Wojowu Jan 20 at 1:20

This term is surely due to Doug Ravenel. In the mid 1970's, he and collaborators Steve Wilson and Haynes Miller constructed and exploited a "chromatic spectral sequence" for computing Ext groups in the Adams-Novikov spectral sequence. This is featured in the Doug's book titled Complex cobordism and stable homotopy groups of spheres published in the early 1980s. As he relates in the intro to that book, he wanted to call it The music of the spheres, but the publisher persuaded him not to. So chromatic might be more musical than luminary! At any rate, the term refers to the different sorts of periodicity one finds in stable homotopy: discovered first in the algebra, and then later in actual maps.

The relationship to complex oriented theories is critical. Quillen wrote his famous paper noting the connection with formal groups. Jack Morava realized that classification theorems for these proved by number theorists were a guide to cohomology theories that should exist - now called the Morava K-theories - periodic with different periodicies - and also a guide of how to organize our thinking about stable homotopy. Miller-Ravenel-Wilson made concrete use of this stuff and Doug was inspired to make his famous conjectures, mainly proved in the mid 1980s by the next generation to come along, ...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.