Morava K-theory

Short description: Cohomology theory

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Morava K-theory is one of a collection of cohomology theories introduced in algebraic topology by Jack Morava in unpublished preprints in the early 1970s.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) 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

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

• For every prime number $p$ and every natural number $n\in \mathbb{\mathbb{N}}$, 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 $\mathbb{𝕊}$ (with ${\mathbb{𝕊}}_n=S^n$), its Bousfield localizations $L_{K(n)}\mathbb{𝕊}$ are also called $K(n)$-local spheres.

See also

• Chromatic homotopy theory
• Morava E-theory

References

• "BP operations and Morava's extraordinary K-theories.", Math. Z. 144 (1): 55−75, 1975, doi:10.1007/BF01214408 
• Hovey-Strickland, "Morava K-theory and localisation"
• Lurie, Jacob (2010-04-27). "Morava E-Theory and Morava K-Theory (Lecture 22)". Harvard University. https://people.math.harvard.edu/~lurie/252xnotes/Lecture22.pdf. 
• Lurie, Jacob (2010-04-27). "The Bousfield Classes of E(n) and K(n) (Lecture 23)". Harvard University. https://people.math.harvard.edu/~lurie/252xnotes/Lecture23.pdf. 
• Lurie, Jacob (2010-04-27). "Uniqueness of Morava K-Theory (Lecture 24)". Harvard University. https://people.math.harvard.edu/~lurie/252xnotes/Lecture24.pdf. 
• Nilpotence and periodicity in stable homotopy theory, Annals of Mathematics Studies, 128, Princeton University Press, 1992 
• Würgler, Urs (1991), "Morava K-theories: a survey", Algebraic topology Poznan 1989, Lecture Notes in Math., 1474, Berlin: Springer, pp. 111–138, doi:10.1007/BFb0084741, ISBN 978-3-540-54098-4 

External links

• Morava K-theory on the nLab