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Sep, 2019
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Chromatic Phenomena in Stable Homotopy Theory

The central object of study in stable homotopy theory is the stable homotopy groups of spheres . They are simpler than unstable counter parts, but still very hard to study.

Thanks to Quillen’s work [ Qui69 ] , we know the complex cobordism MU and its p -local summand BP are useful in the study of stable homotopy theory. Namely, periodicities with respect to the generators v n in the coefficient ring of BP

π*(BP ) ~= ℤ(p)[v1,v2,⋅⋅⋅]

are the key.

This is made explicit by Ravenel [ Rav84 ] . Ravenel’s conjectures, studied mainly by Hopkins and his collaborators, are the driving force for the developement of stable homotopy theory in 90s.

Although stable rational homotopy theory is almost (graded) linear algebra over , there are interesting facts when we allow group actions as is said by Greenlees in [ Gre ] .

References