Topological K-theory

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In mathematics , especially in algebraic topology , the topological K theory deals with the study of vector bundles on topological spaces . The name K-Theory was created by Alexander Grothendieck ; the K stands for "class" in a very general sense.

Definitions

It is a firm, compact Hausdorff space .

Then the quotient of the free Abelian group on the isomorphism classes of the stable equivalent complex vector bundles is over after the subgroup that of elements of the form

for any complex vector bundle is generated via . The Whitney sum denotes the vector bundles. This construction, which is based on the construction of whole numbers from natural numbers , is called the Grothendieck group (after Alexander Grothendieck ). One can think of elements of as formal sums and differences of (isomorphism classes of) complex vector bundles.

If one considers real vector bundles instead, one obtains the real theory . For a better definition, the K-theory of complex vector bundles is also called complex K-theory.

Two vector bundles and on define the same element in if and only if they are stable equivalent , i.e. H. if there is a trivial bundle of vectors such that

The tensor product of vector bundles becomes a commutative ring with one element.

The concept of the rank of a vector bundle is carried over to elements of the theory. The reduced K-theory is the subgroup of the elements of rank 0. The designation is also introduced; denotes the reduced hanging .

properties

  • is a contravariant functor on the category of compact Hausdorff spaces.
  • There is a topological space so that elements from the homotopy classes correspond to maps .
  • There is a natural ring homomorphism called the Chern character .

Bott periodicity

This periodicity phenomenon, named after Raoul Bott , can be formulated in the following equivalent ways:

  • and here is the class of the tautological bundle over .
  • .

In real K theory there is a similar periodicity with period 8.

calculation

The (complex or real) topological K-theory is a generalized cohomology theory and can often be calculated with the help of the Atiyah-Hirzebruch spectral sequence.

K-theory for Banach algebras

The topological K-theory can be extended to general Banach algebras, where the C * algebras play an important role. The topological K-theory of compact spaces can be reformulated as the K-theory of the Banach algebras of continuous functions and then transferred to any Banach algebras, even the one element of the algebras can be dispensed with. Since the assignment is a contravariant functor from the category of compact Hausdorff spaces to the category of Banach algebras and since the topological K-theory is also contravariant, we get a covariant functor from the category of Banach algebras to the category of Abelian groups.

Since non-commutative algebras can also occur here, one speaks of non-commutative topology. The K-theory is an important subject of investigation in the theory of C * algebras.

See also

Algebraic K theory

literature

  • Michael Atiyah: K -theory. Notes by DW Anderson. Second edition. Advanced Book Classics. Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. ISBN 0-201-09394-4
  • Allen Hatcher: Vector bundles and K-theory ( math.cornell.edu ).
  • Karlheinz Knapp: vector bundle. ( link.springer.com ).

Web links

swell

  1. Atiyah , Hirzebruch : Vector bundles and homogeneous spaces. In: Proc. Sympos. Pure Math. Volume III. American Mathematical Society, Providence, RI 1961, pp. 7-38.
  2. Blackadar : K-Theory for Operator Algebras. Springer Verlag, 1986, ISBN 3-540-96391-X .