Topological K-theory

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 . ${\displaystyle X}$

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 ${\displaystyle K(X)}$ ${\displaystyle X}$

${\displaystyle [E\oplus F]-[E]-[F]}$

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. ${\displaystyle E,F}$ ${\displaystyle X}$ ${\displaystyle \oplus }$ ${\displaystyle K(X)}$

If one considers real vector bundles instead, one obtains the real theory ${\displaystyle K}$ . For a better definition, the K-theory of complex vector bundles is also called complex K-theory. ${\displaystyle KO(X)}$

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 ${\displaystyle E}$ ${\displaystyle F}$ ${\displaystyle X}$ ${\displaystyle K(X)}$ ${\displaystyle G}$

${\displaystyle E\oplus G\cong F\oplus G}$

The tensor product of vector bundles becomes a commutative ring with one element. ${\displaystyle K(X)}$

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 . ${\displaystyle K}$ ${\displaystyle {\tilde {K}}(X)}$ ${\displaystyle {\tilde {K}}^{n}(X)={\tilde {K}}(S^{n}X)}$ ${\displaystyle S}$

properties

• ${\displaystyle K}$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 . ${\displaystyle BU}$ ${\displaystyle K(X)}$ ${\displaystyle X\to BU}$
• There is a natural ring homomorphism called the Chern character . ${\displaystyle K(X)\to H^{*}(X,Q)}$

Bott periodicity

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

• ${\displaystyle K(X\times S^{2})=K(X)\otimes K(S^{2}),}$and here is the class of the tautological bundle over . ${\displaystyle K(S^{2})=\mathbb {Z} [H]/(H-1)^{2};}$ ${\displaystyle H}$ ${\displaystyle S^{2}=\mathbb {C} P^{1}}$
• ${\displaystyle {\tilde {K}}^{n+2}(X)={\tilde {K}}^{n}(X)}$
• ${\displaystyle \Omega ^{2}\mathrm {BU} \simeq \mathrm {BU} \times \mathbf {Z} }$.

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

→ Main article : K-theory of 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. ${\displaystyle X}$ ${\displaystyle C(X)}$ ${\displaystyle X\rightarrow \mathbb {C} }$ ${\displaystyle X\mapsto C(X)}$

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

• Max Karoubi: Lectures on K-theory. (PDF).

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 .