K theory

The mathematical sub-area of K- theory deals with the study of vector bundles on topological spaces ( topological K-theory ) or of rings or schemes ( algebraic K-theory ). The name K-Theory was created by Alexander Grothendieck ; the K stands for "class" in a very general sense.

history

In order to generalize his work on the Riemann-Roch theorem, Grothendieck developed a new functor on the category of smooth algebraic varieties . The elements of were classes of algebraic vector bundles above . This theory had similar properties to classical cohomology theories . Characteristic classes , particularly the Chern character , define morphisms of in cohomology theories. ${\ displaystyle K (X)}$ ${\ displaystyle X}$ ${\ displaystyle K (X)}$ ${\ displaystyle X}$ ${\ displaystyle K (X)}$

Immediately after Grothendieck, Atiyah and Hirzebruch considered an analog construction for any compact spaces , the topological K-theory , today mostly referred to as . This topological K-theory is easier to calculate than Grothendieck's K-groups, for example the Chern character gives an isomorphism and one has Bott periodicity . ${\ displaystyle X}$ ${\ displaystyle K ^ {top} (X)}$ ${\ displaystyle K (X)}$ ${\ displaystyle K ^ {top} (X) \ otimes \ mathbb {Q} \ simeq H ^ {*} (X; \ mathbb {Q})}$

Topological K-theory has cohomology operations which are defined by means of outer products of vector bundles (so-called Adams operations) and thus have a more geometric nature than the Steenrod operations in singular cohomology. These operations had spectacular uses in the 1960s. For example, Frank Adams used it to calculate the maximum number of linearly independent vector fields on spheres of any dimension. Other applications have found in global analysis (one of the proofs of the Atiyah-Singer index theorem used topological K-theory) and the theory of C ^* algebras.

The generalization of topological K-theory in non-commutative geometry led to the K-theory of Banach algebras .

The algebraic K-groups were defined by Bass , they had applications in solutions of the "congruence subgroup problem" and in the s-cobordism theorem . ${\ displaystyle K_ {1}}$

Next, Milnor gave a definition of the algebraic K-groups . Their computation for solids ( Matsumoto's theorem ) was the basis for applications of algebra and number theory, in connection with the Brauer group and Galois cohomology . ${\ displaystyle K_ {2}}$ ${\ displaystyle K_ {2}}$

There were then different approaches to defining higher K groups. The definition generally used today was proposed in 1974 by Daniel Quillen at the International Congress of Mathematicians.

Topological K theory

It is a firm, compact Hausdorff space . Then the quotient of the free Abelian group on the isomorphism classes of complex vector bundles is over after the subgroup that of elements of the form ${\ displaystyle X}$ ${\ displaystyle K (X)}$ ${\ displaystyle X}$

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

for vector bundles is generated. This construction, which is based on the construction of whole numbers from natural numbers , is called the Grothendieck group (after Alexander Grothendieck ). ${\ displaystyle E, F}$

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

K is a contravariant functor on the category of compact Hausdorff spaces. It fulfills Bott periodicity with period 2.

If one carries out the analog constructions with real vector bundles, one obtains the real K-theory . For this, Bott periodicity with period applies , i.e. H. . ${\ displaystyle KO (X)}$ ${\ displaystyle 8}$ ${\ displaystyle {\ tilde {K}} O ^ {n + 8} (X) = {\ tilde {K}} O ^ {n} (X)}$

Algebraic K theory

Let be a unitary ring , the group of invertible matrices above and the classifying space of , that is, an aspherical space with a fundamental group . Because the group of elementary matrices is perfect and a normal divisor, the plus construction can be used. The algebraic K-theory of the ring is defined as ${\ displaystyle R}$ ${\ displaystyle \ textstyle GL (R) = \ bigcup _ {n \ geq 0} GL (n, R)}$ ${\ displaystyle R}$ ${\ displaystyle BGL (R)}$ ${\ displaystyle GL (R)}$ ${\ displaystyle GL (R)}$ ${\ displaystyle E (R) = \ left [GL (R), GL (R) \ right]}$ ${\ displaystyle R}$

{\ displaystyle K_ {i} (R): = \ pi _ {i} (BGL ^ {+} (R))}

for . ${\ displaystyle i \ geq 1}$

A variant of the algebraic K-theory (which is not isomorphic to the one defined above) is Milnor's K-theory . Their connection with etal cohomology is the subject of the Milnor conjecture , for the proof of which Vladimir Wojewodski was awarded the Fields medal at the 2002 international mathematicians ' congress . The proof is based on the homotopy theory of algebraic varieties developed by Wojewodski and the motivic cohomology developed by Beilinson and Lichtenbaum . ${\ displaystyle i> 2}$

The most comprehensive definition of an algebraic theory was given by D. Quillen and uses the Q construction . ${\ displaystyle K}$

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. ${\ displaystyle X}$ ${\ displaystyle C (X)}$ ${\ displaystyle X \ rightarrow \ mathbb {C}}$ ${\ displaystyle X \ mapsto C (X)}$

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.

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
Jacek Brodzki: An Introduction to K-theory and Cyclic Cohomology. arxiv : funct-an / 9606001 .
Allen Hatcher: Vector bundles and K-theory ( math.cornell.edu ).
Daniel Quillen: Higher algebraic K-theory: I. In: H. Bass (Ed.): Higher K-Theories. Lecture Notes in Mathematics, Volume 341. Springer-Verlag, Berlin 1973, ISBN 3-540-06434-6 .
Charles Weibel: An introduction to algebraic K-theory, ( math.rutgers.edu ).
Bruce Blackadar: K-Theory for Operator Algebras. Springer Verlag, 1986, ISBN 3-540-96391-X .
Karlheinz Knapp: vector bundle. ( link.springer.com ).

Web links

Max Karoubi: Lectures on K-theory. (PDF).
Christian Voigt: K-theory of operator algebras. (PDF).
Daniel Grayson: On the K-theory of fields. (PDF).