Periodicity Phenomena at the Chromatic Edge, the Chromatic Splitting Conjecture, and the Chromatic Segal Conjecture

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9971850Sadofsky The ``chromatic splitting conjecture'' of Mike Hopkins concerns adecomposition of the localization with respect to the (n-1)stJohnson-Wilson theory of the localization with respect to the nthMorava K-theory (of a finite CW-complex). This is a useful step intrying to understand stable homotopy theory, since the decomposition isexpressed in terms of simpler localizations. In particular, theconjecture would generalize the very beautiful relationships knownbetween the 0th, 1st, and 2nd chromatic levels. Similarly, the``chromatic Segal conjecture'' (due in various forms to combinations ofMike Hopkins, Mark Mahowald, and Doug Ravenel) deals with adecomposition of the Tate cohomology of the localization of a finiteCW-complex with respect to the nth Morava K-theory. Again, thisconjecture intertwines different chromatic levels in a beautiful way;together they give some techniques for inductively determininginformation about higher chromatic levels from lower ones (likerational homotopy theory and K-theory). These conjectures both alsogive data about how the cohomology of the different Morava stabilizergroups are related. The tool used to attack these conjectures is aspectral sequence based on ideas from work of Goerss that computes theBP-homology of various spectra involved in this calculation in termsof the derived functors of inverse limits of comodules over the Hopfalgebroid of BP-operations. Surprisingly, at least in the firstinteresting cases, the higher inverse limits seem to be tractable. This research is concerned with understanding ``homotopy classes offunctions.'' A simple example of the sort of problem studied is givenby considering knot theory. Take a ``knot'' in space to be a loop ofstring embedded in space in some way. One is not really interested in*all* ways of embedding that loop of string; if one can get from oneway of embedding the loop to another just by moving the string alittle bit, one considers those two embeddings to be equivalent.Similarly, if one looks at all ways of putting a loop of string in theplane with the origin removed, one considers two ways the same if theloop of string is wound around the origin the same number of times.As the second example makes clear, it is critical to understand``holes'' in the space to which one is mapping, and how the space fromwhich one is mapping interacts with those holes. There are differentsorts of holes, though; a circle has a hole one can pass something through,while a (hollow) sphere has a different sort of hole. The ``holes''associated to a map (or to a space) can be sorted into differentflavors (usually referred to as different chromatic components).These components do not appear to be independent for spaces and mapsarising from geometric considerations, and the goal is to understandthese dependencies. Solutions to this sort of problem are potentiallyrelated to questions from geometry, but at the same time, surprisingly,to objects arising in number theory, something that gives this work fargreater importance than would either aspect alone.***