🔢Algebraic K-Theory Unit 8 – Chern Character and Adams Operations

Definition and Basics

• Chern character is a ring homomorphism from the topological K-theory of a space $X$ to the rational cohomology of $X$, denoted as $ch: K(X) \to H^*(X;\mathbb{Q})$
• Provides a connection between the algebraic world of vector bundles and the topological world of cohomology
• For a complex vector bundle $E$ over $X$, the Chern character is defined as $ch(E) = \sum_{i=0}^{\infty} \frac{c_i(E)}{i!}$, where $c_i(E)$ denotes the $i$-th Chern class of $E$
  • Chern classes are characteristic classes associated with complex vector bundles, measuring the twisting of the bundle
  • First Chern class $c_1(E)$ represents the obstruction to the existence of a nowhere-zero section of $E$
• Chern character is a rational isomorphism, meaning it becomes an isomorphism after tensoring with the rational numbers $\mathbb{Q}$
• Preserves the ring structure, i.e., $ch(E \oplus F) = ch(E) + ch(F)$ and $ch(E \otimes F) = ch(E) \cdot ch(F)$ for vector bundles $E$ and $F$
• Generalizes to higher K-theory groups, such as $K_1(X)$, by considering the Chern character of the associated suspension bundle

Historical Context

• Chern character was introduced by Shiing-Shen Chern in the 1940s as a tool to study the relationship between vector bundles and cohomology
• Motivated by the Grothendieck-Riemann-Roch theorem, which relates the Euler characteristic of coherent sheaves to their Chern characters
• Became a fundamental tool in the development of K-theory, a generalized cohomology theory that studies vector bundles and their stable equivalence classes
• Played a crucial role in the proof of the Atiyah-Singer index theorem, which connects the analytical index of an elliptic operator to the topological index given by the Chern character
• Further generalized to equivariant K-theory and other variants of K-theory, such as algebraic K-theory and topological K-theory of C*-algebras
• Continues to be an active area of research, with applications in various fields like algebraic geometry, differential geometry, and mathematical physics

Chern Character Construction

• Construction of the Chern character relies on the splitting principle, which allows the reduction of computations to the case of line bundles
• For a complex vector bundle $E$ of rank $n$ over a space $X$, the splitting principle states that there exists a space $F(E)$ and a map $f: F(E) \to X$ such that the pullback bundle $f^*(E)$ splits as a direct sum of line bundles
  • Formally, $f^*(E) \cong L_1 \oplus L_2 \oplus \cdots \oplus L_n$, where each $L_i$ is a line bundle over $F(E)$
  • The map $f$ is called the flag bundle associated with $E$, and it induces an injective ring homomorphism $f^*: H^*(X;\mathbb{Q}) \to H^*(F(E);\mathbb{Q})$
• Using the splitting principle, the Chern character of $E$ can be defined as $ch(E) = f_*(ch(f^*(E)))$, where $f_*$ denotes the pushforward in cohomology
  • For a line bundle $L$ with first Chern class $x = c_1(L)$, the Chern character is given by the formal expansion $ch(L) = e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots$
  • The Chern character of the direct sum of line bundles is the sum of their individual Chern characters
• The construction is independent of the choice of the flag bundle $f: F(E) \to X$, making the Chern character well-defined
• Chern character is natural with respect to pullbacks, i.e., for a map $g: Y \to X$, we have $ch(g^*(E)) = g^*(ch(E))$

Properties of Chern Character

• Chern character is a ring homomorphism, preserving the additive and multiplicative structure of K-theory
  • For vector bundles $E$ and $F$, $ch(E \oplus F) = ch(E) + ch(F)$ (additivity)
  • $ch(E \otimes F) = ch(E) \cdot ch(F)$ (multiplicativity)
• Satisfies the normalization property: for a trivial bundle $\varepsilon^n$ of rank $n$, $ch(\varepsilon^n) = n$
• Commutes with the exterior product: for spaces $X$ and $Y$, and bundles $E$ over $X$ and $F$ over $Y$, $ch(E \boxtimes F) = ch(E) \times ch(F)$, where $\boxtimes$ denotes the external tensor product and $\times$ the cross product in cohomology
• Relates the Chern character of a tensor product to the Chern characters of the factors via the splitting principle: $ch(E \otimes F) = ch(E) \cdot ch(F)$
  • Consequence of the Whitney sum formula for Chern classes and the multiplicativity of the exponential function
• Behaves well under the Adams operations $\psi^k$, which are ring homomorphisms on K-theory: $ch(\psi^k(E)) = k^{-n} \cdot ch(E)$ for a vector bundle $E$ of rank $n$
• Chern character of a virtual bundle (formal difference of vector bundles) can be defined using the additivity property: $ch([E] - [F]) = ch(E) - ch(F)$
• Provides a connection between the K-theory of a space and its rational cohomology, enabling the use of cohomological methods in the study of vector bundles

Adams Operations: Introduction

• Adams operations, denoted as $\psi^k$ for $k \in \mathbb{Z}$, are a family of ring homomorphisms on K-theory that generalize the exterior power operations
• Introduced by Frank Adams in the 1960s as a tool to study vector bundles and their stable equivalence classes
• For a complex vector bundle $E$ over a space $X$, the $k$-th Adams operation $\psi^k(E)$ is a virtual vector bundle in $K(X)$ that shares the same Chern character as the $k$-th exterior power of $E$
  • Exterior power $\Lambda^k(E)$ is the $k$-th antisymmetric tensor product of $E$ with itself, representing $k$-forms on the fibers of $E$
  • Adams operations provide a more algebraic and computable approach to studying exterior powers
• Adams operations satisfy several important properties:
  • Ring homomorphisms: $\psi^k(E \oplus F) = \psi^k(E) \oplus \psi^k(F)$ and $\psi^k(E \otimes F) = \psi^k(E) \otimes \psi^k(F)$
  • Composition rule: $\psi^k \circ \psi^l = \psi^{kl}$ for all $k, l \in \mathbb{Z}$
  • Normalization: $\psi^1 = \mathrm{id}$ (identity) and $\psi^0 = \varepsilon$ (augmentation)
• Closely related to the Grothendieck group of vector bundles and the lambda operations $\lambda^k$, which represent the exterior powers
• Play a fundamental role in the study of K-theory and its connections to other areas of mathematics, such as representation theory and number theory

Computation of Adams Operations

• Computation of Adams operations relies on their relation to the Chern character and the lambda operations
• For a line bundle $L$ with first Chern class $x = c_1(L)$, the $k$-th Adams operation is given by $\psi^k(L) = L^{\otimes k}$, the $k$-th tensor power of $L$
  • Chern character of $\psi^k(L)$ is $ch(\psi^k(L)) = e^{kx} = 1 + kx + \frac{(kx)^2}{2!} + \frac{(kx)^3}{3!} + \cdots$
• Using the splitting principle, the Adams operations on a vector bundle $E$ of rank $n$ can be computed by splitting $E$ into a sum of line bundles $E \cong L_1 \oplus L_2 \oplus \cdots \oplus L_n$
  • $\psi^k(E) = \psi^k(L_1) \oplus \psi^k(L_2) \oplus \cdots \oplus \psi^k(L_n)$
  • Chern character of $\psi^k(E)$ is the sum of the Chern characters of the individual line bundles
• Adams operations can also be computed using the Newton-Girard formulas, which express the Adams operations in terms of the lambda operations
  • $\psi^k(E) = \sum_{i=1}^n (-1)^{i-1} k^{n-i} \lambda^i(E)$, where $\lambda^i(E)$ denotes the $i$-th exterior power of $E$
  • Allows for the computation of Adams operations without explicitly splitting the vector bundle
• In the case of a virtual bundle $[E] - [F]$, the Adams operations are computed using the ring homomorphism property: $\psi^k([E] - [F]) = \psi^k(E) - \psi^k(F)$
• Computation of Adams operations in higher K-theory groups, such as $K_1(X)$, involves the use of the relative Chern character and the boundary homomorphism in K-theory

Applications in K-Theory

• Adams operations provide a powerful tool for studying the structure of K-theory and its relation to other invariants
• Eigenspace decomposition: Adams operations can be used to decompose the K-theory of a space into eigenspaces corresponding to different eigenvalues
  • For a prime $p$, the $p$-adic completion of K-theory decomposes as $K(X)_p \cong \bigoplus_{i=1}^{p-1} K(X)_p^{(i)}$, where $K(X)_p^{(i)}$ is the eigenspace of $\psi^k$ with eigenvalue $k^i$ mod $p$
  • Eigenspace decomposition is a key ingredient in the proof of the Adams conjecture, which relates the Adams operations to the Bott periodicity in K-theory
• Relation to the gamma filtration: Adams operations can be used to define the gamma filtration on K-theory, which measures the complexity of vector bundles
  • Gamma filtration is defined as $\Gamma^n K(X) = \{ x \in K(X) : \psi^k(x) \equiv k^n x \mod \Gamma^{n+1} K(X) \text{ for all } k \in \mathbb{Z} \}$
  • Provides a connection between K-theory and the theory of lambda rings, which are rings equipped with a family of operations satisfying certain axioms
• Applications in algebraic geometry: Adams operations play a role in the study of algebraic cycles and the Grothendieck group of algebraic varieties
  • Grothendieck group $K_0(Var_k)$ of varieties over a field $k$ can be equipped with a lambda ring structure using the Adams operations
  • Relates the K-theory of varieties to their Chow groups and motivic cohomology
• Connections to representation theory: Adams operations are closely related to the representation theory of groups and the character theory of representations
  • For a compact Lie group $G$, the K-theory of the classifying space $BG$ is isomorphic to the completion of the representation ring $R(G)$ with respect to the augmentation ideal
  • Adams operations on $K(BG)$ correspond to the Adams operations on the representation ring, which can be computed using the characters of the representations

Advanced Topics and Open Problems

• Equivariant K-theory: Adams operations can be generalized to the equivariant setting, where a group action is considered on the space and the vector bundles
  • Equivariant Chern character maps the equivariant K-theory to the equivariant cohomology, taking into account the group action
  • Equivariant Adams operations satisfy similar properties to their non-equivariant counterparts, such as the composition rule and the relation to the equivariant lambda operations
• Higher algebraic K-theory: Adams operations can be extended to higher algebraic K-theory groups, such as $K_n(R)$ for a ring $R$ and $n > 0$
  • Involves the use of the relative Chern character and the boundary homomorphism in the long exact sequence of K-theory
  • Provides a deeper understanding of the structure of higher K-theory groups and their relation to other invariants, such as the cyclic homology of rings
• Motivic homotopy theory: Adams operations play a role in the study of motivic homotopy theory, which combines ideas from algebraic geometry and homotopy theory
  • Motivic Adams operations can be defined on the motivic K-theory of schemes, taking into account the motivic cohomology and the algebraic cycles
  • Relates the motivic K-theory to other motivic invariants, such as the motivic cohomology and the algebraic cobordism
• Open problems and conjectures:
  • Vandiver's conjecture: Relates the Adams operations on the K-theory of number fields to the class groups and the Bernoulli numbers
  • Quillen-Lichtenbaum conjecture: Proposes a relation between the algebraic K-theory of a scheme and its étale cohomology, involving the Adams operations and the Chern character
  • Bloch-Kato conjecture: Describes the structure of the motivic cohomology of a field in terms of the Milnor K-theory and the Galois cohomology, using the motivic Adams operations
• Further generalizations and applications: Adams operations continue to find new applications and generalizations in various areas of mathematics
  • Noncommutative geometry: Adams operations can be defined on the K-theory of C*-algebras and used to study the geometry of noncommutative spaces
  • Topological cyclic homology: Adams operations play a role in the study of topological cyclic homology, which is a refinement of algebraic K-theory that captures more arithmetic information
  • String theory and mathematical physics: Adams operations appear in the study of D-brane charges and the K-theory classification of Ramond-Ramond fields in string theory