# Chromatic Filtration of Burnside Ring

I simply participated in a workshop on the colorful purification of the Burnside ring. I recognized it reasonably well, yet at no factor did any person offer a specific definition of what a colorful purification in fact is, and also I have had some problem locating one. Thinking I have the essential specific history, i.e. expertise of purifications, etc can any person offer me a description of what makes a purification "chromatic"?

There are joint documents by Benedict Gross (a number philosopher) and also Mike Hopkins (a topologist) on official teams and also the colorful tower, that might be extra obtainable as a fast intro to the principles. They attempt to clarify several of this product to non - topologists, and also their presentation from virtually 20 years earlier does not rely upon the equipment of heaps, "brave new algebra" or infinity - groups. The recap paper in the AMS Bulletin has a thesaurus connecting secure homotopy and also quasicoherent sheaves, specifically they clarify that the colorful purification is similar to the Cousin facility. See :

http://www.ams.org/journals/bull/1994-30-01/S0273-0979-1994-00438-0/

It seems non - technological, and also about equal to "periodic".

A description is offered by Ravenel.

The "original paper" seems :

Miller, Haynes R. ; Ravenel, Douglas C. ; Wilson, W. Stephen. "Periodic sensations in the Adams - Novikov spooky sequence." Ann. Mathematics. (2) 106 (1977 ), no. 3, 469-- 516. MR458423 JSTOR

yet the term is not made use of there. These type of points were for homotopy of rounds, a significant subject in algebraic topology.

Probably the purification reviewed for the Burnside ring had a comparable feel.

Expanding on Jack is solution, there is an entire amazing area called "chromatic homotopy theory". I do not recognize much concerning it, I'll claim a couple of mottos given that I saw you additionally asked the inquiry "geometry or topology". With any luck none are incorrect. (If so, someone please remedy me!)

~~Colorful homotopy theory in some way cuts the Adams-Novikov spectral sequence based upon degrees of periodicity (therefore the name "chromatic" - - it is intended to create the image of light refracting via a prism or something). ~~ [cf. Sean is solution for a better educated account. ]

- Degree 0 is average cohomology with $\mathbb{Q}$ coefficients. There, the communication of geometry and also homotopy theory is offered by assimilation.
- Degree 1 is K - concept. There, the communication of geometry and also homotopy theory is offered by the Atiyah - Singer index theory.
- Degree 2 is elliptic cohomology (and also something called topological modular kinds, which in some way freely is the "universal elliptic cohomology theory"). It is unidentified just how to link the void with geometry, yet this is something individuals are working with.
- All the various other limited degrees are unidentified. A couple of individuals are attempting to identify what degree 3 needs to be.
- Degree $\infty$ is intricate cobordism.
- All the degrees (until now?) are complex-oriented. The facility cobordism range $MU$ lugs the global official team regulation, and also intricate alignments of a range $E$ are gotten as multiplicative maps of ranges $MU\rightarrow E$. (In reality, $(MU_*,MU_*MU)$ corepresents the moduli pile of official
*teams*!)

If you intend to in fact find out some features of this, attempt :

There is no, regarding i recognize, basic colorful purification for object of type X. There is a colorful purification of the secure homotopy teams of rounds. You can attempt to understand some guided system that is "chromatic" linked to a space/spectrum by considering the maps in between its localizations relative to Morava K - concept (or Morava E - concept or Johnson - Wilson, or what have you) and also just how they mesh, yet i assume this is basically the colorful merging thm, (there are limited type and also p - neighborhood theories).

A vital active ingredient to note is Carlsson is Thm/the Segal opinion, which mentions, i think, that secure cohomotopy is the finished Burnside ring. Possibly you can feed several of the colorful suggestions via all the effort that entered into the Segal Conjecture and also get something in this way.

Yet possibly that is not it in all. A note concerning Aarons solution, the colorful purification does not originate from the ANSS, although it does separate the means you claim however. Ravenel invests time in Nilpotence and also Periodicity speaking about this things. I assume it concerns self maps generating reproduction by $v_n$ in $K(n)_*(X)$, and also you filter the self maps of X in this way. The $v_n$ are linked to the concept of official team regulations thoroughly. The means I consider the colorful purification, and also this can be off, is by what degree of (intricate oriented) cohomology concept i require to identify that a self map is not unimportant, where by degree i suggest elevation of the official team regulation representing the certain facility oriented concept. The elevation of the official team regulations control the purification and also just how it acts. (This is just how I consider it, yet taking into account the below, it is type of scrap, I will certainly leave it due to the fact that it is much less technological.)

I think the paper Jack connected to is where they create the colorful spooky series which merges to the $E_2$ regard to ANSS which is insane tough to calculate (i do not assume any person recognizes all the 2 line, and also if they do after that no person recognizes every one of the 3 line, and also by all i suggest bent on $t-s=50$) So there is a purification on the $E_2$ regard to ANSS that generates this spooky series, and also I assume this is the colorful purification.

(Also, it is a technological term, and also it does not simply suggest routine.)

**MODIFY : **
I simply located the list below sentence while reviewing the testimonial of "$v_n$ routine components in ring ranges and also applications to bordism theory" by Hovey (the testimonial is by stong) :
"Let $R$ be a ring range. If $v \in \pi_k R$, $v$ is called a $v_n$ component if $K(n)_*(v) \in K(n)_*(R)$ is a device and also $K(i)_*(v) \in K(i)_*(R)$ is nilpotent for all $i \neq n$. "

I will certainly be back in a little bit to examine it versus an analysis of Ravenel is orange publication.

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