What is so 'coloured' on Chromatic Homotopy Theory

Question

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.

Answer 1

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, ...

Answer 2

The metaphor is very apt musically, and quite precise at $p=2$, where the $2^3$ - periodicity (or `diapason') defines the classical octave in which pitches reproduce themselves. Perhaps this is a reasonable place to suggest that

$K(n)_*$ is most naturally regarded as a $Z_2$-graded (pro-)'etale abelian groupscheme over $F_p$, with a natural Gal$(F_q/F_p)$-action on its group of $F_q$-valued points $K(n)^*(-;F_q)$, thereby defining a further $(q-1)$-cyclic refinement of the grading, with ($q = p^n$ as in Atiyah-Tall).

In other words, the $K(n)$s are naturally indexed by the finite fields $F_q, q = p^n$.

Answer 3

After introducing chromatic filtrations on stable homotopy groups in his orange book (p. 24), Ravenel writes

We use the word ‘chromatic’ here for the following reason. The $n$-th subquotients in the chromatic filtration consists of $v_n$-periodic elements. As illustrated in 2.4.2, these elements fall into periodic families. The chromatic filtration is thus like a spectrum in the astronomical sense in that it resolves the stable homotopy groups of a finite complex into periodic families of various periods. Comparing these to the colors of the rainbow led us to the word ‘chromatic.’