extends classical K-theory to rings and schemes, providing powerful tools for studying algebraic and geometric properties. It connects deeply to topology, algebraic geometry, and number theory, offering insights into various mathematical structures.
The main constructions are and . These approaches define K-groups differently but yield equivalent results for suitable categories. Understanding their similarities and differences is crucial for applying K-theory effectively.
Algebraic K-theory
Algebraic K-theory is a generalization of classical K-theory that assigns K-groups to rings and schemes, providing a powerful tool for studying their algebraic and geometric properties
It has deep connections to various areas of mathematics, including topology, algebraic geometry, and number theory
The two main constructions of algebraic K-theory are Quillen's Q-construction and Waldhausen's S-construction, which offer different approaches to defining the K-groups
Quillen's Q-construction
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Quillen's Q-construction assigns a topological space Q(C) to a category C with exact sequences, such as the category of projective modules over a ring
The K-groups are then defined as the homotopy groups of Q(C): Ki(C)=πi(Q(C))
The Q-construction has the advantage of being functorial and compatible with sequences (Quillen's localization theorem)
Examples: K0(R) recovers the Grothendieck group of a ring R, while K1(R) is related to the group of units GL(R)
Waldhausen's S-construction
Waldhausen's S-construction assigns a topological space S(C) to a category C with cofibrations and weak equivalences, such as the category of chain complexes over a ring
The K-groups are defined as the homotopy groups of S(C): Ki(C)=πi(S(C))
The S-construction is more general than the Q-construction and can be applied to a wider range of categories (Waldhausen categories)
Examples: The S-construction can be used to define the K-theory of spaces, spectra, and exact categories
Comparison of constructions
Both the Q-construction and S-construction yield the same K-groups for suitable categories, such as the category of projective modules over a ring
The comparison is often established via the additivity theorem, which relates the K-theory of a category to the K-theory of its subcategories
The choice of construction depends on the specific context and the properties one wishes to exploit (functoriality, localization, etc.)
Example: The Q-construction is more suitable for studying the K-theory of schemes, while the S-construction is more amenable to the study of the K-theory of spaces and spectra
Topological K-theory
assigns K-groups to topological spaces, which encode information about the vector bundles over the space
It has applications in various areas of mathematics, including topology, geometry, and mathematical physics
The two main results in topological K-theory are and the , which provide powerful tools for computing K-groups
Bott periodicity
Bott periodicity states that the K-groups of a space X are periodic with period 2: Ki(X)≅Ki+2(X)
The periodicity is realized by the Bott map β:Ki(X)→Ki+2(X), which is an isomorphism
Bott periodicity has important consequences, such as the computation of the K-groups of spheres and the classification of stable vector bundles
Example: The K-groups of the sphere Sn are given by K0(Sn)≅Z for even n and K1(Sn)≅Z for odd n
Atiyah-Hirzebruch spectral sequence
The Atiyah-Hirzebruch spectral sequence (AHSS) relates the K-theory of a space X to its ordinary cohomology H∗(X;Z)
The AHSS is a spectral sequence with E2-page given by E2p,q=Hp(X;Kq(pt)) and converging to the K-theory of X
The differentials in the AHSS are related to the Steenrod operations in cohomology and provide a means of computing K-groups from cohomology
Example: For a CW-complex X with only even-dimensional cells, the AHSS collapses at the E2-page, yielding an isomorphism K0(X)≅⨁iH2i(X;Z)
Conner-Floyd Chern character
The is a natural transformation ch:K∗(X)→H∗(X;Q) from K-theory to rational cohomology
It is a refinement of the classical Chern character for vector bundles and is compatible with the Bott periodicity and the AHSS
The Conner-Floyd Chern character provides a link between K-theory and cohomology and is an important tool in the study of the K-theory of manifolds
Example: For a compact oriented manifold X, the Conner-Floyd Chern character induces an isomorphism K∗(X)⊗Q≅H∗(X;Q)
Operator K-theory
Operator K-theory is a generalization of topological K-theory that assigns K-groups to C∗-algebras, which are the noncommutative analogs of topological spaces
It has applications in various areas of mathematics, including noncommutative geometry, , and representation theory
The two main approaches to operator K-theory are and Kasparov's KK-theory, which provide powerful tools for studying the K-theory of C∗-algebras
Fredholm modules
A Fredholm module over a C∗-algebra A is a pair (H,F), where H is a Hilbert space and F is a bounded operator on H satisfying certain conditions
Fredholm modules can be used to define the groups Ki(A), which are the dual of the K-theory groups Ki(A)
The index pairing between K-theory and K-homology, given by Ki(A)×Ki(A)→Z, is a fundamental tool in operator K-theory
Example: The Dirac operator on a compact spin manifold M defines a Fredholm module over the C∗-algebra C(M) of continuous functions on M
KK-theory of Kasparov
Kasparov's KK-theory is a bivariant version of operator K-theory that assigns abelian groups KK(A,B) to pairs of C∗-algebras A and B
The KK-groups are defined using Kasparov modules, which are a generalization of Fredholm modules
KK-theory satisfies a powerful set of axioms (homotopy invariance, stability, split exactness) that make it a flexible tool for studying the K-theory of C∗-algebras
Example: The KK-theory of the C∗-algebra of a discrete group G is closely related to the for G
Baum-Connes conjecture
The Baum-Connes conjecture is a far-reaching conjecture in operator K-theory that relates the K-theory of the reduced C∗-algebra of a group G to the equivariant K-homology of the classifying space for proper actions of G
It can be formulated using an assembly map μ:K∗G(EG)→K∗(Cr∗(G)), which is conjectured to be an isomorphism
The Baum-Connes conjecture has important consequences in topology, geometry, and representation theory, such as the Novikov conjecture and the Kadison-Kaplansky conjecture
Example: The Baum-Connes conjecture holds for amenable groups, hyperbolic groups, and discrete subgroups of Lie groups
Cyclic homology
is a homology theory for algebras that is closely related to algebraic K-theory and provides a bridge between K-theory and differential geometry
It has applications in various areas of mathematics, including noncommutative geometry, Lie theory, and mathematical physics
The two main ingredients in cyclic homology are and Connes' periodicity operator, which give rise to a rich algebraic structure
Cyclic objects and complexes
A cyclic object in a category C is a functor A:Λop→C, where Λ is the cyclic category
The cyclic category Λ encodes the combinatorics of cyclic symmetry and has objects [n] and morphisms generated by face maps, degeneracy maps, and cyclic permutations
A cyclic complex is a chain complex equipped with an action of the cyclic category, and the cyclic homology of a cyclic complex is defined as the homology of its total complex
Example: The Hochschild complex of an algebra A is a cyclic complex, whose cyclic homology is called the cyclic homology of A
Connes' periodicity operator
Connes' periodicity operator S:HCn(A)→HCn−2(A) is a map between the cyclic homology groups of an algebra A that plays a crucial role in the structure of cyclic homology
The periodicity operator is defined using the cyclic structure of the Hochschild complex and satisfies S2=0
The periodicity operator gives rise to a long exact sequence (Connes' SBI sequence) relating Hochschild homology, cyclic homology, and negative cyclic homology
Example: For the algebra of smooth functions on a manifold M, the periodicity operator corresponds to the de Rham differential on differential forms
Cyclic vs Hochschild homology
Hochschild homology HH∗(A) and cyclic homology HC∗(A) are two closely related homology theories for algebras
There is a natural map I:HH∗(A)→HC∗(A), called the antisymmetrization map, which is induced by the inclusion of the Hochschild complex into the cyclic complex
The cyclic homology of an algebra contains more information than its Hochschild homology, as it takes into account the additional cyclic symmetry
Example: For the algebra of functions on a manifold, Hochschild homology corresponds to differential forms, while cyclic homology corresponds to de Rham cohomology
Algebraic vs topological K-theory
Algebraic K-theory and topological K-theory are two distinct but related theories that assign K-groups to different types of objects (rings/schemes vs topological spaces)
Despite their differences, there are deep connections between algebraic and topological K-theory, which are often expressed via the Chern character
Understanding the similarities and differences between these two theories is crucial for applications in areas such as algebraic geometry and topology
Comparison via Chern character
The Chern character is a natural transformation ch:K∗(X)→H∗(X;Q) from algebraic or topological K-theory to rational cohomology
In the algebraic setting, the Chern character maps the algebraic K-theory of a scheme X to its Chow groups (algebraic cycles modulo rational equivalence)
In the topological setting, the Chern character maps the topological K-theory of a space X to its rational cohomology
The Chern character is compatible with various operations in K-theory and cohomology (products, pullbacks, etc.) and provides a powerful tool for comparing K-theory and cohomology
Example: For a smooth projective variety X over C, the Chern character induces an isomorphism K∗(X)⊗Q≅⨁iH2i(X;Q) between algebraic K-theory and singular cohomology
Advantages and limitations
Algebraic K-theory has the advantage of being defined for a wide range of objects (rings, schemes, categories) and capturing subtle arithmetic information
Topological K-theory, on the other hand, is more computable and has a rich geometric interpretation in terms of vector bundles
The Chern character provides a link between the two theories but is only an isomorphism after tensoring with Q, losing torsion information
In some cases, the algebraic and topological K-theory of an object can be quite different, reflecting the distinct nature of the two theories
Example: For the integers Z, the algebraic K-groups Ki(Z) contain deep arithmetic information (e.g., about the Riemann zeta function), while the topological K-groups Ki(S1) are simply Z in even degrees and 0 in odd degrees
Applications of higher K-theory
Higher K-theory, including both algebraic and topological K-theory, has numerous applications in various areas of mathematics
Some of the most prominent applications include index theory, the Novikov conjecture, and assembly maps in the Baum-Connes conjecture
These applications demonstrate the power of K-theory as a unifying language for studying problems in geometry, topology, and analysis
Index theory and geometry
Index theory studies the relationship between the analytical properties of differential operators (e.g., the Dirac operator) and the geometry of the underlying spaces
K-theory plays a crucial role in index theory, as the index of an elliptic operator can be interpreted as a K-homology class, while the symbol of the operator defines a K-theory class
The Atiyah-Singer index theorem expresses the index of an elliptic operator in terms of (characteristic classes) of the underlying manifold and the operator
Example: The Dirac operator on a compact spin manifold has an index that can be computed using the A^-genus of the manifold, which is a characteristic class defined in terms of the Pontryagin classes
Novikov conjecture
The Novikov conjecture is a famous conjecture in topology that asserts the homotopy invariance of certain higher signatures of manifolds
It can be formulated using the K-theory of group C∗-algebras and assembly maps, which relate the K-homology of the classifying space of a group to the K-theory of its group C∗-algebra
The Novikov conjecture has important consequences in geometry and topology, such as the rigidity of certain classes of manifolds and the existence of non-triangulable manifolds
Example: The Novikov conjecture holds for hyperbolic groups, implying that the higher signatures of hyperbolic manifolds are homotopy invariants
Assembly maps and isomorphism conjectures
Assembly maps are natural maps between the K-homology of the classifying space of a group and the K-theory of its group C∗-algebra
The Baum-Connes conjecture asserts that the assembly map for the K-theory of the reduced group C∗-algebra is an isomorphism
There are various other isomorphism conjectures in K-theory, such as the Farrell-Jones conjecture for the algebraic K-theory of group rings and the Bost conjecture for the K-theory of number fields
These conjectures have important implications in geometry, topology, and number theory, and their study has led to the development of powerful new techniques in K-theory
Example: The Baum-Connes conjecture for a discrete group G implies the Novikov conjecture for G and the Kadison-Kaplansky conjecture on the existence of idempotents in the reduced group C∗-algebra of G
Key Terms to Review (28)
Alain Connes: Alain Connes is a French mathematician known for his foundational work in noncommutative geometry, a field that extends classical geometry to accommodate the behavior of spaces where commutativity fails. His contributions have led to new understandings of various mathematical structures and their applications, bridging concepts from algebra, topology, and physics.
Atiyah-Hirzebruch Spectral Sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a systematic way to compute generalized cohomology theories, particularly K-theory. It arises from the study of vector bundles and their associated characteristic classes, leading to a connection between topology and algebraic structures. This spectral sequence is especially useful in computing higher K-theories, revealing deep insights into the structure of vector bundles over topological spaces.
Baum-Connes Conjecture: The Baum-Connes Conjecture is a fundamental hypothesis in noncommutative geometry that relates the K-theory of a space to its topology, particularly through the use of group C*-algebras. It posits that the K-homology of a space can be computed using the K-theory of its associated C*-algebra, linking topological properties with algebraic structures in a profound way. This conjecture has important implications for understanding index theory and the topology of manifolds.
Bott Periodicity: Bott periodicity refers to a fundamental result in topology and K-theory, stating that the K-theory groups of the complex projective space exhibit periodic behavior, specifically that $$K^n(X) \cong K^{n+2}(X)$$ for any space X. This periodicity is crucial in understanding how vector bundles behave over different spaces, influencing areas like higher K-theory, K-homology, and the relationship between topological spaces and C*-algebras.
C*-algebras: A c*-algebra is a type of algebra that consists of a set of continuous linear operators on a Hilbert space, equipped with a norm and an involution that satisfy specific algebraic and topological properties. These structures play a crucial role in various mathematical areas, including functional analysis and quantum mechanics, and provide a framework for understanding continuous functions, projective modules, higher K-theory, Bott periodicity, and the Seiberg-Witten map.
Conner-Floyd Chern Character: The Conner-Floyd Chern character is a mathematical construct used in noncommutative geometry and higher K-theory, representing a way to associate topological invariants with the geometry of a manifold. It generalizes classical Chern characters from complex geometry, allowing for an extension into the realm of noncommutative spaces, connecting deeply with the understanding of vector bundles and their transformations in higher dimensions.
Cyclic homology: Cyclic homology is a branch of algebraic topology and homological algebra that extends classical homology theories to incorporate a cyclic symmetry. It provides tools for studying algebras, particularly noncommutative algebras, by capturing their structure in a way that reflects both algebraic and geometric properties. This concept is deeply linked to higher K-theory, cohomological theories, and the study of invariants associated with algebraic structures.
Cyclic objects and complexes: Cyclic objects and complexes are structures in algebraic topology and noncommutative geometry that capture the essence of periodicity in homological algebra. They allow for the study of modules or complexes that exhibit a cyclic symmetry, which can be critical in defining invariants in higher K-theory. By understanding these objects, one can explore deeper relationships between topology, algebra, and geometry.
Formal Groups: Formal groups are mathematical structures that generalize the concept of groups in a way that allows for the study of algebraic and geometric properties through a more flexible lens. These groups can be seen as a tool to understand various phenomena in higher algebra and topology, particularly in the context of K-theory where they help capture the information about the structure of rings and schemes.
Fredholm modules: Fredholm modules are mathematical structures that generalize the concept of a Dirac operator acting on sections of a vector bundle over a noncommutative space. They play a significant role in noncommutative geometry, allowing one to define K-theory and compute index invariants. These modules provide a framework for understanding the relationship between geometry and analysis in settings where traditional methods may fail, connecting deep ideas in topology, functional analysis, and operator algebras.
Gysin sequence: The gysin sequence is a fundamental tool in algebraic topology that relates K-theory and homology for a fibration, allowing the computation of higher K-theory groups. This sequence connects various topological spaces through their associated vector bundles, providing insights into the relationships between their K-theory classes. It plays a crucial role in understanding the structure and properties of spaces, especially when considering fibrations and their associated spectral sequences.
Higher K-theory: Higher K-theory is an advanced concept in algebraic topology and noncommutative geometry that generalizes classical K-theory to higher dimensions, providing tools to study vector bundles and other geometric structures on topological spaces. It extends the idea of counting vector bundles to include more intricate information about the space and its algebraic properties, such as noncommutative algebras and their representations.
Index Theory: Index theory is a mathematical framework that relates analytical properties of differential operators to topological invariants of the underlying space. This theory is fundamental in understanding the relationship between geometry, analysis, and topology, especially in noncommutative settings where traditional geometric concepts are generalized.
K-cohomology: K-cohomology is a type of cohomology theory that provides invariants for topological spaces or algebraic varieties, connecting geometric properties to algebraic data. It extends the classical notion of cohomology to incorporate higher-dimensional features and captures information related to vector bundles and their classifications in a way that is particularly relevant in the study of noncommutative geometry.
K-homology: k-homology is an important concept in noncommutative geometry that extends the classical notion of homology to spaces equipped with a Dirac operator. It connects topology with analysis, allowing for a way to study spaces by understanding their geometric and spectral properties through the lens of Dirac operators. This concept also plays a crucial role in understanding how these spaces can be related to K-theory and provides insights into the structure of noncommutative spaces.
Kk-theory of Kasparov: The kk-theory of Kasparov is a framework in noncommutative geometry that extends the classical K-theory to the setting of operator algebras and is particularly useful in studying the topology of spaces through the lens of C*-algebras. It connects K-theory with the representation theory of groups and provides tools for analyzing the structure of noncommutative spaces, offering a way to classify and relate different operator algebras.
Localization: Localization is a mathematical process that focuses on studying objects in a neighborhood around a point or in relation to a specific subset. This approach allows for the examination of properties and structures that may behave differently in local contexts compared to global ones. In various branches of mathematics, particularly in K-theory and Hochschild cohomology, localization plays a vital role in understanding how certain structures can be simplified or understood more deeply by focusing on these specific regions.
Mayer-Vietoris Sequence: The Mayer-Vietoris Sequence is a powerful tool in algebraic topology that provides a way to compute the homology or K-theory of a topological space by breaking it down into simpler pieces. It essentially states that if a space can be decomposed into two open subsets whose intersection has certain properties, then there exists a long exact sequence relating the homology (or K-theory) of the individual pieces and their intersection. This sequence plays a crucial role in connecting various concepts in noncommutative geometry, particularly when examining K0 and K1 groups, higher K-theory, and KK-theory.
Michael Atiyah: Michael Atiyah was a renowned British mathematician known for his significant contributions to geometry, topology, and the development of noncommutative geometry. He played a key role in connecting the fields of mathematics and theoretical physics, particularly through his work on the index theorem, which relates analysis, geometry, and topology.
Quantum Groups: Quantum groups are algebraic structures that generalize the concept of groups and are essential in the study of noncommutative geometry and mathematical physics. They play a pivotal role in the representation theory of noncommutative spaces and provide a framework for understanding symmetries in quantum mechanics, connecting seamlessly to various concepts in geometry and algebra.
Quillen's Q-construction: Quillen's Q-construction is a method in algebraic K-theory that constructs a space from a category, which allows for the definition of higher K-groups. This construction helps to relate algebraic topology with homological algebra by providing a way to study the stable homotopy types of categories. It is instrumental in the development of higher K-theory, allowing mathematicians to analyze vector bundles and other algebraic structures in a deeper way.
Spectral Sequences: Spectral sequences are powerful computational tools used in algebraic topology and homological algebra to compute homology and cohomology groups. They allow for the systematic organization of information from a complex into a series of pages, making it easier to extract the desired algebraic invariants and understand the relationships between them.
Spectral Triples: Spectral triples are mathematical structures used in noncommutative geometry that generalize the notion of a geometric space by combining algebraic and analytic data. They consist of an algebra, a Hilbert space, and a self-adjoint operator, which together capture the essence of both classical geometry and quantum mechanics, making them a powerful tool for studying various mathematical and physical concepts.
Stable Equivalence: Stable equivalence is a concept in algebraic K-theory that describes a form of isomorphism between vector bundles or more generally, between modules over a ring after adding trivial bundles or modules. It provides a way to compare objects in a stable manner, allowing for a broader classification of structures by ignoring certain dimensions and focusing on their essential properties.
Topological invariants: Topological invariants are properties of a topological space that remain unchanged under homeomorphisms, essentially capturing the intrinsic structure of the space. These invariants allow mathematicians to classify spaces and understand their essential features, providing crucial insights into geometry and topology. They play an important role in various mathematical theories, including the study of noncommutative geometry, where spaces may not have a traditional geometric interpretation but still possess invariant properties.
Topological K-theory: Topological K-theory is a branch of mathematics that studies vector bundles over topological spaces, primarily through the lens of algebraic topology. It provides a way to classify vector bundles using homotopy theory and offers insight into their properties, which connects to various mathematical fields such as geometry, analysis, and even physics. This concept underpins the construction of K0 and K1 groups, lays the groundwork for higher K-theory, and intertwines with Bott periodicity and cyclic cohomology.
Von Neumann algebras: Von Neumann algebras are a special class of *-algebras that are closed under the weak operator topology and include all bounded linear operators on a Hilbert space. They provide a framework to study various aspects of quantum mechanics and are essential in the development of noncommutative geometry, particularly in understanding projective modules and higher K-theory.
Waldhausen's S-construction: Waldhausen's S-construction is a method used in algebraic K-theory to construct spaces whose K-theory is easier to compute. It provides a way to build a classifying space for a category of spaces, often applied in the context of higher K-theory. This construction is particularly significant because it relates to the stabilization process and leads to spectral sequences that help compute K-groups, bridging topology and algebra.