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Morava K theory

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In stable homotopy theory, a branch of mathematics, Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s. For every prime number p (which is suppressed in the notation), it consists of theories K(n) for each nonnegative integer n, each a ring spectrum in the sense of homotopy theory. Johnson & Wilson (1975) published the first account of the theories.

Details

The theory K(0) agrees with singular homology with rational coefficients, whereas K(1) is a summand of mod-p complex K-theory. The theory K(n) has coefficient ring

Fp[vn,vn−1]

where vn has degree 2(pn − 1). In particular, Morava K-theory is periodic with this period, in much the same way that complex K-theory has period 2.

These theories have several remarkable properties.

  • They have Künneth isomorphisms for arbitrary pairs of spaces: that is, for X and Y CW complexes, we have
    • K ( n ) ∗ ( X × Y ) ≅ K ( n ) ∗ ( X ) ⊗ K ( n ) ∗ K ( n ) ∗ ( Y ) .
  • They are "fields" in the category of ring spectra. In other words every module spectrum over K(n) is free, i.e. a wedge of suspensions of K(n).
  • They are complex oriented (at least after being periodified by taking the wedge sum of (pn − 1) shifted copies), and the formal group they define has height n.
  • Every finite p-local spectrum X has the property that K(n)∗(X) = 0 if and only if n is less than a certain number N, called the type of the spectrum X. By a theorem of Devinatz–Hopkins–Smith, every thick subcategory of the category of finite p-local spectra is the subcategory of type-n spectra for some n.

    • References

    Morava K-theory Wikipedia
    (Text) CC BY-SA