10.1 K-Theory and characteristic classes
4 min read•july 30, 2024
connects vector bundles to , offering a powerful tool for studying geometric and topological spaces. , like , measure bundle twisting and provide a bridge between K-Theory and cohomology theories.
The , a ring homomorphism from K-Theory to , allows for cohomological computations in K-Theory. This connection, along with spectral sequences like Atiyah-Hirzebruch, helps uncover deep relationships between different mathematical structures.
Characteristic classes in K-Theory
Definition and role of characteristic classes
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- Characteristic classes are cohomology classes associated to vector bundles that measure the twisting or non-triviality of the bundle
- Provide a way to distinguish between different vector bundles over the same base space
- Allow for the construction of maps from K-Theory to various cohomology theories
- Examples of characteristic classes include Chern classes, Pontryagin classes, and
Types of characteristic classes
- Chern classes are associated to complex vector bundles
- Live in the even degree cohomology of the base space with integer coefficients
- Example: The first Chern class of a complex line bundle measures its degree of twisting
- Pontryagin classes are associated to real vector bundles
- Live in the cohomology of the base space with integer coefficients in degrees divisible by 4
- Example: The first Pontryagin class of a real is related to its second Stiefel-Whitney class
- Stiefel-Whitney classes are associated to real vector bundles
- Live in the cohomology of the base space with Z/2Z coefficients
- Example: The first Stiefel-Whitney class of a real vector bundle determines its orientability
Chern character and K-Theory
Definition and properties of the Chern character
- The Chern character is a ring homomorphism from the K-Theory of a space X to the rational cohomology of X
- Defined using the Chern classes of complex vector bundles
- For a line bundle L, the Chern character is given by ch(L) = exp(c_1(L)), where c_1(L) is the first Chern class of L
- For a vector bundle E, the Chern character is defined as ch(E) = (E) + c_1(E) + (1/2)c_1(E)^2 - c_2(E) + ..., where c_i(E) are the Chern classes of E
Relation to K-Theory
- The Chern character satisfies the properties of a ring homomorphism:
- ch(E + F) = ch(E) + ch(F), where + denotes the sum of vector bundles
- ch(E * F) = ch(E) * ch(F), where * denotes the tensor product of vector bundles
- Induces an isomorphism between the rational K-Theory of a space and its rational cohomology
- Allows for the computation of K-Theory groups using cohomological methods
- Example: The Chern character of a sum of line bundles is the sum of their exponentials of first Chern classes
K-Theory vs cohomology theories
K-Theory as a generalized cohomology theory
- K-Theory satisfies the Eilenberg-Steenrod axioms except for the dimension axiom
- Allows for the construction of K-Theory groups in negative degrees
- Example: The K-Theory of a point is Z in even degrees and 0 in odd degrees
Relation to ordinary cohomology
- The Chern character provides a link between K-Theory and theories
- Induces a rational isomorphism between K-Theory and rational cohomology
- The relates the K-Theory of a space to its ordinary cohomology
- Example: The K-Theory of the complex projective space CP^n is isomorphic to a truncated polynomial ring over Z
Additional structures in K-Theory
- The Adams operations in K-Theory are analogous to the Steenrod operations in cohomology
- Provide additional structure on K-Theory groups
- Can be used to study the torsion in K-Theory
- Example: The Adams operations can detect the non-triviality of certain vector bundles over spheres
Atiyah-Hirzebruch spectral sequence for K-Theory
Definition and construction
- The Atiyah-Hirzebruch spectral sequence (AHSS) is a spectral sequence that converges to the K-Theory groups of a space X
- Relates the K-Theory of X to its ordinary cohomology
- The E_2 page of the AHSS is given by E_2^{p,q} = H^p(X; K^q(point)), where H^p denotes ordinary cohomology and K^q(point) is the K-Theory of a point in degree q
Differentials and torsion
- The differentials in the AHSS are related to the Steenrod operations in cohomology
- Provide a way to determine the torsion in the K-Theory groups
- Example: The differentials in the AHSS can detect the torsion in the K-Theory of lens spaces
Computation of K-Theory groups
- To calculate the K-Theory groups using the AHSS:
- Compute the E_2 page using the cohomology of X and the known values of K^q(point)
- Determine the differentials and compute the subsequent pages of the spectral sequence
- The K-Theory groups are given by the limit of the spectral sequence, i.e., K^n(X) = lim_{p+q=n} E_\infty^{p,q}
- The AHSS is particularly useful for computing the K-Theory of spaces with known cohomology (spheres, projective spaces, certain homogeneous spaces)
- Example: The AHSS can be used to compute the K-Theory of the complex Grassmannian, which is a quotient of the unitary group
Key Terms to Review (16)
Atiyah-Hirzebruch Spectral Sequence: The Atiyah-Hirzebruch spectral sequence is a powerful tool in algebraic topology that provides a way to compute the K-theory of a space from its cohomology. It connects the geometry of vector bundles to topological invariants, allowing for the classification of vector bundles through the lens of K-theory and characteristic classes.
Characteristic classes: Characteristic classes are a way to associate algebraic invariants to vector bundles, which help in understanding their geometric and topological properties. They are crucial in many areas of mathematics, such as differential geometry and topology, providing insights into the structure of bundles and leading to significant results like the Atiyah-Singer index theorem.
Chern character: The Chern character is an important topological invariant associated with complex vector bundles, which provides a connection between K-theory and cohomology. It captures information about the curvature of the vector bundle and its underlying geometric structure, serving as a bridge in various applications, from fixed point theorems to differential geometry.
Chern classes: Chern classes are a set of characteristic classes associated with complex vector bundles, providing vital topological invariants that help classify vector bundles over a manifold. They connect deeply with various fields such as geometry, topology, and algebraic geometry, allowing us to analyze vector bundles through their topological properties.
Cohomology: Cohomology is a mathematical concept used to study topological spaces through algebraic invariants, providing a way to classify and analyze their structures. It connects various areas of mathematics, including geometry and algebra, and plays a crucial role in understanding vector bundles, characteristic classes, and K-theory. Cohomology theories allow for computations that lead to insights into the properties of spaces and the nature of continuous mappings between them.
Commutative algebra: Commutative algebra is a branch of mathematics that studies commutative rings and their ideals, focusing on the properties of algebraic structures where the multiplication operation is commutative. This field lays the groundwork for various mathematical concepts, including algebraic geometry and number theory, connecting the behavior of rings to geometric properties of shapes defined by polynomial equations.
Homotopy: Homotopy is a concept in topology that describes a continuous deformation between two continuous functions or shapes, showing how one can be transformed into the other without tearing or gluing. This idea is crucial for understanding equivalence in spaces and mappings, allowing mathematicians to classify spaces based on their topological properties, which connects deeply with the construction of groups and characteristic classes.
Index Theory: Index theory is a branch of mathematics that studies the relationship between the analytical properties of differential operators and topological invariants of manifolds. It provides a powerful tool for understanding how various geometric and topological aspects influence the behavior of solutions to differential equations, linking analysis, topology, and geometry.
K-groups: K-groups are algebraic invariants in K-Theory that categorize vector bundles over a topological space or scheme. They provide a way to study and classify these bundles, revealing deep connections between geometry and algebra through various mathematical contexts.
K-Theory: K-Theory is a branch of mathematics that studies vector bundles and their generalizations through the construction of K-groups, which provide a way to classify and understand vector bundles up to isomorphism. It connects various areas of mathematics, including topology, algebra, and geometry, offering insights into fixed point theorems, quantum field theory, and even string theory.
Ordinary cohomology: Ordinary cohomology is a mathematical tool that assigns algebraic invariants to topological spaces, allowing for the classification and comparison of these spaces. It helps in understanding the properties of manifolds and can be seen as a way to 'measure' the shape and structure of spaces by capturing information about their holes at different dimensions. This concept plays a significant role in various advanced topics such as spectral sequences and connections with characteristic classes.
Rank: Rank is a fundamental concept in the study of vector bundles, referring to the dimension of the fibers associated with a vector bundle at each point in the base space. It captures important information about how the structure of the vector bundle varies across different points, and plays a crucial role in various operations that can be performed on vector bundles, as well as in understanding their properties through K-Theory and characteristic classes.
Rational cohomology: Rational cohomology is a type of cohomology theory that studies topological spaces using rational numbers as coefficients, simplifying many properties and making calculations more manageable. This approach allows for the computation of invariants that reveal structural information about spaces, particularly in relation to algebraic topology and K-Theory, where understanding the relationships between different cohomological tools is crucial.
Stiefel-Whitney Classes: Stiefel-Whitney classes are characteristic classes associated with real vector bundles, serving as a topological invariant that provides insight into the geometry of the bundle. These classes help distinguish non-isomorphic vector bundles and are crucial in applications such as vector bundle classification, cobordism theory, and K-Theory. They offer a way to understand how the topology of a manifold interacts with its vector bundles, contributing to deeper insights in algebraic topology and differential geometry.
Topological Ring: A topological ring is a mathematical structure that combines the properties of both a ring and a topological space. In this setup, the ring operations (addition and multiplication) are continuous with respect to the topology, meaning that small changes in input lead to small changes in output. This concept allows for the study of algebraic structures in a more flexible way, incorporating notions of convergence and continuity that are vital for understanding various areas in mathematics, including K-theory and characteristic classes.
Vector Bundle: A vector bundle is a mathematical structure that consists of a base space and a collection of vector spaces associated with each point in the base space. This concept allows for the study of smooth manifolds and serves as a fundamental tool in various areas of mathematics, connecting topology, geometry, and algebra through concepts like classification and characteristic classes.