stable homotopy theory In mathematics, stable homotopy theory is the part of homotopy theory (and thus algebraic topology) concerned with all structure and phenomena that remain after sufficiently many applications of the suspension functor. A founding result was the F ...

, a branch of

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Morava K-theory is one of a collection of

cohomology theories In mathematics, specifically in homology theory and algebraic topology, cohomology is a general term for a sequence of abelian groups, usually one associated with a topological space, often defined from a cochain complex. Cohomology can be viewed ...

introduced in

algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...

by Jack Morava in unpublished preprints in the early 1970s. For every

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

''p'' (which is suppressed in the notation), it consists of theories ''K''(''n'') for each nonnegative integer ''n'', each a

ring spectrum In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy a ...

in the sense of

homotopy theory In mathematics, homotopy theory is a systematic study of situations in which maps can come with homotopies between them. It originated as a topic in algebraic topology but nowadays is studied as an independent discipline. Besides algebraic topolog ...

. published the first account of the theories.

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The theory ''K''(0) agrees with

singular homology In algebraic topology, singular homology refers to the study of a certain set of algebraic invariants of a topological space ''X'', the so-called homology groups H_n(X). Intuitively, singular homology counts, for each dimension ''n'', the ''n''-d ...

with rational coefficients, whereas ''K''(1) is a summand of mod-''p''

complex K-theory In mathematics, topological -theory is a branch of algebraic topology. It was founded to study vector bundles on topological spaces, by means of ideas now recognised as (general) K-theory that were introduced by Alexander Grothendieck. The early ...

. The theory ''K''(''n'') has coefficient ring :F_''p'' 'v''_''n'',''v''_''n''⁻¹ where ''v''_''n'' has degree 2(''p''^''n'' − 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 \times Y) \cong K(n)_*(X) \otimes_ K(n)_*(Y).

* They are "fields" in the

category Category, plural categories, may refer to: Philosophy and general uses * Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) *Categories (Peirce) * ...

ring spectra In stable homotopy theory, a ring spectrum is a spectrum ''E'' together with a multiplication map :''μ'': ''E'' ∧ ''E'' → ''E'' and a unit map : ''η'': ''S'' → ''E'', where ''S'' is the sphere spectrum. These maps have to satisfy a ...

. In other words every

module spectrum In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra. The ∞-category of (say right) module spectra is stable A stable is a building in which livestock, especially horses ...

over ''K''(''n'') is free, i.e. a

wedge A wedge is a triangular shaped tool, and is a portable inclined plane, and one of the six simple machines. It can be used to separate two objects or portions of an object, lift up an object, or hold an object in place. It functions by converti ...

suspensions In chemistry, a suspension is a heterogeneous mixture of a fluid that contains solid particles sufficiently large for sedimentation. The particles may be visible to the naked eye, usually must be larger than one micrometer, and will eventually ...

of ''K''(''n''). * They are complex oriented (at least after being periodified by taking the wedge sum of (''p''^''n'' − 1) shifted copies), and the

formal group In mathematics, a formal group law is (roughly speaking) a formal power series behaving as if it were the product of a Lie group. They were introduced by . The term formal group sometimes means the same as formal group law, and sometimes means one o ...

they define has

height Height is measure of vertical distance, either vertical extent (how "tall" something or someone is) or vertical position (how "high" a point is). For example, "The height of that building is 50 m" or "The height of an airplane in-flight is abou ...

''n''. * Every finite ''p''-local

spectrum A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors i ...

''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 Hopkins is an English, Welsh and Irish patronymic surname. The English name means "son of Hob". ''Hob'' was a diminutive of ''Robert'', itself deriving from the Germanic warrior name ''Hrod-berht'', translated as "renowned-fame". The Robert spell ...

–Smith, every thick subcategory of the

of finite ''p''-local spectra is the subcategory of type-''n'' spectra for some ''n''.

References

* *Hovey-Strickland,
Morava K-theory and localisation
* *{{citation, mr=1133896 , last=Würgler, first= Urs , chapter=Morava K-theories: a survey, title= Algebraic topology Poznan 1989, pages= 111–138 , series=Lecture Notes in Math., volume= 1474, publisher= Springer, location= Berlin, year= 1991 , doi=10.1007/BFb0084741, isbn=978-3-540-54098-4 Algebraic topology Cohomology theories

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See also

References