Morava K-theory

Template:Short description 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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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. Template:Harvtxt published the first account of the theories.

Overview

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

F_p[v_n,v_n⁻¹]

where v_n 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.

Properties

• They have Künneth isomorphisms for arbitrary pairs of spaces: that is, for X and Y CW complexes, we have

$K(n{)}_{\ast }(X\times Y)\cong K(n{)}_{\ast }(X){\otimes }_{K(n{)}_{\ast }}K(n{)}_{\ast }(Y).$

• For every prime number $p$ and every natural number $n\in \mathbb{N}$, Morava K-theory $K(n)$ is a "field" in the category of ring spectra, meaning the graded ring ${\pi }_{0}K(n)$ is even a graded field, meaning every nonzero homogeneous element in it is invertible. Therefore every module spectrum over $K(n)$ is free, meaning it's 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.
• A connected closed spin 10-manifold is generalized oriented in second integral Morava K-theory $\tilde{K}(2)$ (for $p=2$) if and only if its seventh integral Stiefel–Whitney class $W_7$ vanishes.

K(n)-local spheres

For the sphere spectrum $𝕊$ (with ${𝕊}_n=S^n$), its Bousfield localizations $L_{K(n)}𝕊$ are also called $K(n)$-local spheres.

See also

• Chromatic homotopy theory
• Morava E-theory

References

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• Hovey-Strickland, "Morava K-theory and localisation"
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External links

• Morava K-theory on the nLab