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 ...

, a branch of 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 viewe ...

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 invariants that classify topological spaces up to homeomorphism, though usually most classif ...

Jack Morava Jack Johnson Morava is an American homotopy theorist at Johns Hopkins University. Education Of Czech and Appalachian descent, he was raised in Texas' lower Rio Grande valley. An early interest in topology was strongly encouraged by his paren ...

in unpublished preprints in the early 1970s.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 way ...

''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 topol ...

. published the first account of the theories.

Details

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''- ...

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 ea ...

. 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 ass ...

. 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; hence, it can be considered as either analog or generalization of ...

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 eventua ...

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 ...

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 ab ...

''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 color ...

''X'' has the property that ''K''(''n'')_∗(''X'') = 0 if and only if ''n'' is less than a certain number ''N'', called the

type Type may refer to: Science and technology Computing * Typing, producing text via a keyboard, typewriter, etc. * Data type, collection of values used for computations. * File type * TYPE (DOS command), a command to display contents of a file. * Ty ...

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 sp ...

–Smith, every

thick subcategory Thick may refer to: * A bulky or heavyset body shape or overweight * ''Thick'' (album), 1999 fusion jazz album by Tribal Tech * Thick concept, in philosophy, a concept that is both descriptive and evaluative * Thick description, in anthropology, ...

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

Details

See also

References