The algebraic K-theory of ring spectra is intimately related to the geometry of highdimensional manifolds. In a series of papers in the 1970’s and 1980’s, Waldhausen established the deep connection between the K-theory of the sphere spectrum and stable pseudo-isotopy theory. The sphere spectrum has an infinite filtration called the chromatic filtration that forms a tower at each prime p. The layers in the chromatic tower capture periodic phenomena in stable homotopy theory, corresponding to the Morava K-theory “fields”. The chromatic viewpoint has organized the understanding of stable homotopy theory over the last thirty years.
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