K-theory

English

Etymology

From circa 1960. The K stands for German Klasse (“class”).

The theory developed out of algebraic geometry after the 1957 publication of work by German-born French mathematician Alexander Grothendieck.

Noun

K-theory (uncountable)

1. (uncountable, algebra, algebraic geometry, algebraic topology) The study of rings R generated by the set of vector bundles over some topological space or scheme;
   (dated, obsolete) that part of algebraic topology comprising what is now called topological K-theory.
   • 1994, Jonathan Rosenberg, Algebraic K-Theory and Its Applications, Springer, page 1:

     K-theory as an independent discipline is a fairly new subject, only about 35 years old.

   • 2000, M. Rørdam, F. Larsen, Flemming Larsen, N. Laustsen, An Introduction to K-Theory for C*-Algebras, Cambridge University Press, page ix:

     K-theory was developed by Atiyah and Hirzebruch in the 1960s based on work of Grothendieck in algebraic geometry. It was introduced as a tool in C^*-algebra theory in the early 1970s through some specific applications described below. Very briefly, K-theory (for C^*-algebras) is a pair of functors, called K₀ and K₁, that to each C^*-algebra A associate two Abelian groups K₀(A) and K₁(A).

2. (countable) The cohomology generated by the set of vector bundles over some topological space or scheme.
   • 1978, Michiel Hazewinkel, Formal Groups and Applications‎^[1], Harcourt Brace Jovanovich (Academic Press), page xi:

     The theory of formal groups has found a number of rather spectacular applications in recent years in number theory, arithmetical algebraic geometry, algebraic geometry, and algebraic topology, ranging from congruences for the coefficients of modular forms and local class field theory to extraordinary K-theories and (indirectly) results on the homotopy groups of spheres.

   • 2007, Mathematical Reviews, American Mathematical Society, page 4338:

     This has changed in recent years: on the one hand, bivariant K-theories were defined by the author for other categories of algebras [Doc. Math. 2 (1997), 139–182 (electronic); MR1456322 (98h: 19006)]; on the other hand, the local cyclic homology theory by M. Puschnigg works for small algebras and C^*-algebras alike […] .

   • 2014, Stephan Stolz, Topology and Field Theories, American Mathematical Society, page 102:

     In particular, the Morava K-theory $K(1)$ appears as a direct summand of 'complex K-theory modulo $p$'. However, the higher Morava K-theories $\left\{K(n)\right\}_{n\geq 2}$ are much more mysterious from a geometric point of view.
     REMARK 5.8. For a fixed prime number $p$, the Morava K-theories $K(n)$ are defined for $0<n<\infty$.

Usage notes

• In mathematics:
  • Following Alexander Grothendieck's 1957 publication of a proof of the Grothendieck–Riemann–Roch theorem, K-theory was developed as a subfield of algebraic topology (notably in relation to C^*-algebra theory). This subfield is now known as topological K-theory.
  • Subsequently, K-theory was generalised as a subfield of algebraic geometry, now often called algebraic K-theory.
    • This development allows a revisionist definition of topological K-theory as "the application of tools from K-theory to problems in algebraic topology."
  • In its current state, K-theory (in the area of study sense) comprises algebraic K-theory and topological K-theory.
  • The term K-theory also refers (in its countable sense) to the cohomology (a ring whose multiplication operation is tensor product) that is the central subject of K-theory.
  • Applications have also been found in:
    • The study of operator algebras (within functional analysis), where K-theory is a fundamental tool.
    • The study of certain kinds of invariant of large matrices.
• In physics:
  • In condensed matter physics, K-theory is a tool used to classify certain materials of importance.
  • In string theory, a variation called twisted K-theory (aka K-theory with local coefficients) is the basis of an application called K-theory classification, by which certain (hypothetical) objects are classified.

Hyponyms

• (study of rings generated by vector bundles): algebraic K-theory, topological K-theory

Derived terms

• algebraic K-theory
• complex K-theory
• K-theory classification
• Morava K-theory
• real K-theory
• topological K-theory
• twisted K-theory

Translations

study of rings generated by vector bundles

cohomology generated by vector bundles

Further reading

• Algebraic K-theory on Wikipedia.Wikipedia
• Topological K-theory on Wikipedia.Wikipedia
• Twisted K-theory on Wikipedia.Wikipedia
• Cohomology on Wikipedia.Wikipedia
• Grothendieck–Riemann–Roch theorem on Wikipedia.Wikipedia
• K-theory (physics) on Wikipedia.Wikipedia
• Algebraic K-theory on Encyclopedia of Mathematics
• K-theory on Encyclopedia of Mathematics