11.4 K-theory

K-theory is a powerful tool in cohomology theory that studies vector bundles and their invariants. It assigns abelian groups called K-groups to topological spaces or algebraic objects, capturing essential information about vector bundle structures and properties.

K-theory differs from traditional cohomology by focusing specifically on vector bundles. It has wide-ranging applications in algebraic geometry, number theory, and mathematical physics, providing a unified framework for understanding various invariants associated with vector bundles.

Basics of K-theory

• K-theory is a generalization of cohomology theory that studies vector bundles and their associated invariants
• It provides a powerful tool for understanding the topology and geometry of spaces through the lens of vector bundles
• K-theory differs from cohomology theory in that it focuses specifically on vector bundles and their properties, while cohomology theory deals with more general topological invariants

Definition of K-theory

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• K-theory is a cohomology theory that assigns abelian groups, called K-groups, to topological spaces or algebraic objects (rings, schemes, etc.)
• The elements of K-groups represent equivalence classes of vector bundles over the given space or object
• The K-groups capture important information about the structure and properties of vector bundles

Motivation for K-theory

• K-theory was initially developed to study vector bundles and their role in topology and geometry
• It provides a unified framework for understanding various invariants associated with vector bundles, such as Chern classes and characteristic classes
• K-theory has found applications in diverse areas of mathematics, including algebraic geometry, number theory, and mathematical physics

K-theory vs cohomology theory

• While both K-theory and cohomology theory are cohomology theories, they differ in their focus and the objects they study
• Cohomology theory deals with more general topological invariants, such as homology groups and cohomology rings
• K-theory specifically focuses on vector bundles and their associated invariants, providing a more specialized and refined tool for studying topological and geometric properties

Algebraic K-theory

• Algebraic K-theory is a branch of K-theory that studies algebraic objects, such as rings and schemes, using techniques from algebraic topology
• It provides a way to extract topological information from algebraic structures
• Algebraic K-theory has important connections to number theory, algebraic geometry, and the theory of motives

Definition of algebraic K-theory

• Algebraic K-theory assigns K-groups to rings or schemes, which capture information about the structure of projective modules over these objects
• The K-groups are defined using the category of projective modules and the Grothendieck group construction
• Higher K-groups are obtained by considering the homotopy groups of certain classifying spaces associated with the category of projective modules

K-groups of rings

• For a ring $R$, the K-group $[K_0](https://www.fiveableKeyTerm:k_0)(R)$ is defined as the Grothendieck group of the category of finitely generated projective $R$-modules
• Elements of $K_0(R)$ represent stable isomorphism classes of projective modules, with the group operation given by direct sum
• Higher K-groups $[K_n](https://www.fiveableKeyTerm:k_n)(R)$ for $n > 0$ are defined using the homotopy groups of the classifying space of the category of projective $R$-modules

Higher K-groups

• Higher K-groups $K_n(R)$ for $n > 0$ capture more subtle information about the structure of projective modules over a ring $R$
• They are defined using the homotopy groups of the classifying space of the category of projective $R$-modules
• Higher K-groups have important connections to the theory of motives and the study of algebraic cycles

Milnor K-theory

• Milnor K-theory is a variant of algebraic K-theory that is defined for fields
• It is constructed using the tensor algebra of the field and certain relations involving the multiplicative group of the field
• Milnor K-theory has applications in the study of quadratic forms and the cohomology of fields

Quillen's Q-construction

• Quillen's Q-construction is a general method for defining higher K-groups of exact categories, which include categories of projective modules over rings
• It involves constructing a certain topological space, called the Q-construction, from the exact category and taking its homotopy groups
• The Q-construction provides a unified approach to defining higher K-groups and has been influential in the development of algebraic K-theory

Topological K-theory

• Topological K-theory is a branch of K-theory that studies vector bundles over topological spaces
• It assigns K-groups to topological spaces, which capture information about the structure and properties of vector bundles over these spaces
• Topological K-theory has important connections to differential geometry, index theory, and mathematical physics

Definition of topological K-theory

• For a compact Hausdorff space $X$, the K-group $K^0(X)$ is defined as the Grothendieck group of the monoid of isomorphism classes of complex vector bundles over $X$
• Elements of $K^0(X)$ represent stable equivalence classes of vector bundles, with the group operation given by the direct sum of vector bundles
• Higher K-groups $K^{-n}(X)$ for $n > 0$ are defined using the homotopy groups of certain classifying spaces associated with vector bundles over $X$

Vector bundles and K-theory

• Vector bundles are the central objects of study in topological K-theory
• A vector bundle over a topological space $X$ is a family of vector spaces parameterized by points in $X$, satisfying certain local triviality conditions
• K-theory provides a way to classify and study vector bundles up to stable equivalence, capturing important topological and geometric information

Bott periodicity theorem

• The Bott periodicity theorem is a fundamental result in topological K-theory that relates the K-groups of a space to the K-groups of its suspensions
• It states that for a compact Hausdorff space $X$, there are isomorphisms $K^0(X) \cong K^0(\Sigma^2 X)$ and $K^{-1}(X) \cong K^{-1}(\Sigma^2 X)$, where $\Sigma X$ denotes the suspension of $X$
• Bott periodicity has important consequences for the computation of K-groups and the study of vector bundles over spheres and other spaces

Chern character in K-theory

• The Chern character is a homomorphism from the K-theory of a space to its rational cohomology
• It provides a way to relate the K-theoretic invariants of vector bundles to their characteristic classes in cohomology
• The Chern character is an important tool for computing K-groups and understanding the relationship between K-theory and cohomology theory

Atiyah-Hirzebruch spectral sequence

• The Atiyah-Hirzebruch spectral sequence is a computational tool in topological K-theory that relates the K-groups of a space to its cohomology groups
• It provides a systematic way to compute the K-groups of a space using its cohomology and certain differential operators
• The Atiyah-Hirzebruch spectral sequence has been widely used in the computation of K-groups and the study of vector bundles over various spaces

Applications of K-theory

• K-theory has found numerous applications in various branches of mathematics, including algebraic geometry, number theory, topology, and mathematical physics
• It provides a powerful tool for studying geometric and topological properties of spaces and algebraic objects
• The applications of K-theory often involve the interplay between algebraic, geometric, and topological ideas

K-theory in algebraic geometry

• In algebraic geometry, K-theory is used to study the category of coherent sheaves on a scheme
• The K-groups of a scheme capture information about the structure of vector bundles and their moduli spaces
• K-theory has been applied to the study of intersection theory, Riemann-Roch theorems, and the theory of motives in algebraic geometry

K-theory in number theory

• K-theory has important connections to number theory, particularly in the study of algebraic number fields and their rings of integers
• The K-groups of rings of integers provide information about the structure of ideal class groups and the Brauer group
• K-theory has been used to formulate and prove important conjectures in number theory, such as the Quillen-Lichtenbaum conjecture

K-theory in topology

• In topology, K-theory is used to study vector bundles over topological spaces and their associated invariants
• The K-groups of a space provide information about the structure of vector bundles and their moduli spaces
• K-theory has been applied to the study of characteristic classes, index theory, and the topology of manifolds

K-theory in mathematical physics

• K-theory has found applications in various areas of mathematical physics, including string theory, quantum field theory, and condensed matter physics
• In string theory, K-theory is used to classify D-brane charges and study the geometry of spacetime
• In condensed matter physics, K-theory is used to study topological insulators and the quantum Hall effect

Computations and examples

• Computing K-groups is an important aspect of K-theory, as it provides concrete information about the structure of vector bundles and algebraic objects
• Various techniques and tools have been developed for computing K-groups in different settings, including algebraic and topological K-theory
• Examples of K-group computations illustrate the power and applicability of K-theory in various branches of mathematics

Computing K-groups of spaces

• Computing the K-groups of topological spaces often involves the use of spectral sequences, such as the Atiyah-Hirzebruch spectral sequence
• For certain spaces, such as spheres and projective spaces, the K-groups can be computed explicitly using Bott periodicity and the structure of vector bundles over these spaces
• The computation of K-groups of spaces has important applications in topology and geometry, such as the study of characteristic classes and index theory

K-theory of finite fields

• The K-theory of finite fields has important connections to number theory and the study of algebraic number fields
• For a finite field $\mathbb{F}_q$, the K-groups $K_n(\mathbb{F}_q)$ can be computed explicitly using techniques from algebraic K-theory and the properties of the field
• The computation of K-groups of finite fields has applications in the study of zeta functions and the cohomology of algebraic varieties over finite fields

K-theory of group rings

• The K-theory of group rings provides information about the structure of projective modules over these rings and the representation theory of groups
• For certain classes of groups, such as finite groups or free abelian groups, the K-groups of their group rings can be computed using techniques from algebraic K-theory and representation theory
• The computation of K-groups of group rings has applications in the study of group cohomology and the classification of group representations

K-theory of C*-algebras

• The K-theory of C*-algebras is an important tool in noncommutative geometry and the study of quantum spaces
• For certain classes of C*-algebras, such as AF-algebras or Cuntz algebras, the K-groups can be computed explicitly using techniques from operator K-theory and the structure of these algebras
• The computation of K-groups of C*-algebras has applications in the study of noncommutative topology and the classification of C*-algebras

Advanced topics in K-theory

• K-theory is a rich and active area of research, with many advanced topics and ongoing developments
• These advanced topics often involve the interplay between K-theory and other areas of mathematics, such as equivariant topology, noncommutative geometry, and category theory
• The study of advanced topics in K-theory leads to new insights and applications in various branches of mathematics

Equivariant K-theory

• Equivariant K-theory is a generalization of K-theory that takes into account the action of a group on a space or an algebraic object
• It assigns K-groups to spaces or objects equipped with a group action, capturing information about the structure of equivariant vector bundles and representations
• Equivariant K-theory has important applications in the study of transformation groups, orbifolds, and the representation theory of compact Lie groups

Twisted K-theory

• Twisted K-theory is a variant of K-theory that incorporates the notion of twisting by a cohomology class or a gerbe
• It assigns K-groups to spaces or objects equipped with a twisting datum, capturing information about twisted vector bundles and their invariants
• Twisted K-theory has found applications in string theory, where it is used to classify D-brane charges in the presence of a B-field or a H-flux

Bivariant K-theory

• Bivariant K-theory is a generalization of K-theory that assigns K-groups to pairs of C*-algebras or spaces, capturing information about the structure of KK-theory and E-theory
• It provides a unified framework for studying various notions of duality and correspondences in noncommutative geometry and topology
• Bivariant K-theory has important applications in the study of index theory, the Baum-Connes conjecture, and the classification of C*-algebras

Noncommutative K-theory

• Noncommutative K-theory is a generalization of K-theory that studies noncommutative algebras and their modules using techniques from algebraic topology and operator theory
• It assigns K-groups to noncommutative algebras, such as C*-algebras or von Neumann algebras, capturing information about their structure and representations
• Noncommutative K-theory has important applications in the study of noncommutative geometry, quantum groups, and the classification of operator algebras

K-theory of categories

• The K-theory of categories is a generalization of K-theory that assigns K-groups to categories equipped with suitable notions of exact sequences or cofibrations
• It provides a unified framework for studying various notions of K-theory, including algebraic and topological K-theory, as well as their generalizations to more abstract settings
• The K-theory of categories has important applications in the study of motives, higher categories, and the foundations of K-theory itself