Contents

# Contents

## Idea

Given a complex line bundle $L$ over a space $X$ its $k$th tensor power $L^{\otimes k}$ is another line bundle for any $k \in \mathbb{N}$. The line bundles define certain elements of complex topological K-theory group $K(X)$, and there is a unique cohomology operation

$\psi^k \colon K(X) \to K(X) \,,$

the $k$th Adams operation, such that:

• $\psi^k([L]) = [L^{\otimes k}]$ if $[L]$ is the K-theory class of any line bundle,

• $\psi^k \colon K(X) \to K(X)$ is a group homomorphism,

• $\psi^k$ is a natural transformation: any map $f: X \to Y$ induces a map $f^* : K(Y) \to K(X)$ on $K$-theory, and $\psi^k \circ f^* = f^* \circ \psi^k$.

More abstractly, Adams operations can be defined on any Lambda-ring. They are an example of power operations.

## Definition

The Adams operations have an explicit definition in terms of the Lambda-ring structure on topological K-theory, this we state as def. below. While explicit, this definition may look contrived on first sight. But it turns out that it satisfies a list of properties, of which two simple ones already uniquely characterize the Adams operations. This is proposition below.

###### Definition

(Lambda-ring structure on topological K-theory)

Let $X$ be a compact topological space and write $K(X)$ for its topological K-theory ring. For $E$ a vector bundle over $X$, write $[E] \in K(X)$ for its class in K-theory. Given $E$, write

$\lambda_t[E] \;\coloneqq\; \underoverset{k = 0}{\infty}{\sum} [\wedge^k_X E] t^k \;\;\in\;\; K(X)[ [t] ]$

for the formal power series with coefficients in the ring $K(X)$ being the K-theory classes of the skew-symmetrized tensor product of vector bundles of $E$ with itself.

Since the constant term of this power series is always the unit $[\wedge^0 E] = 1$, hence

$\lambda_t[E] \in 1 + (t) \cdot K(X)[ [t] ]$

there exists a multiplicative inverse formal power series $\lambda_t[E]^{-1}$.

Then given the class of a virtual vector bundle $[E] - [F] \in K(X)$, define more generally

$\lambda_t[[E- F]] \;\;\coloneqq\;\, \lambda_t[E] \cdot \lambda_t[F]^{-1} \;\;\in\;\; K(X)[ [t] ] \,.$
###### Definition

For $E$ a vector bundle over some topological space $X$, write

$\psi^0(E) \coloneqq rank(E)$

for the bundle which over each connected component of $X$ is the trivial vector bundle of the same rank as $E$ over that component.

Define a formal power series with coefficients in the K-theory ring $K(X)$ by

\begin{aligned} \psi_t(E) & \coloneqq \underoverset{\infty}{k = 0}{\sum} \psi^k(E) t^k \\ & \coloneqq \psi^0(E) - t \frac{d}{d t} log \lambda_{-t}(E) \;\;\in\;\; K(X)[ [t] ] \end{aligned} \,,

where $\lambda_t$ is the Lambda-ring operation from def. .

Here the derivative of the logarithm of formal power series stands for the usual expression in terms of the geometric series:

\begin{aligned} \frac{d}{d t} log \lambda_{-t}(E) & = \frac{1}{\lambda_{-t}(E)} \frac{d}{d t} \lambda_{-t}(E) \\ & = \underoverset{\infty}{k = 0}{\sum} \left( 1 - \lambda_{-t}(E) \right)^k \cdot \frac{d}{d t} \lambda_{-t}(E) \end{aligned} \,.

The $k$th Adams operation is the cohomology operation on topological K-theory

$\psi^k \;\colon\; K(-) \longrightarrow K(-)$

which is the coefficient of $t^k$ in $\psi_t$.

###### Proposition

(basic properties and characterization of Adams operations)

$\psi^k \;\colon\; K(X) \longrightarrow K(X)$

have the following properties, for all elements $x,y \in K(X)$ and $k, l \in \mathbb{N}$ and $p \; \text{prime}$:

1. $\psi^k(x + y) = \psi^k(x) + \psi^k(y)$

($\psi^k$ is a natural abelian group homomorphism)

2. $x \,\text{a line} \;\;\Rightarrow\;\; \psi^k(x) = x^k$

(applied to a class $x \coloneqq [L]$ represented by a line bundle $L$, $\psi^k$ is the $k$th tensor power)

3. $\psi^k(x \cdot y) = \psi^k(x) \cdot \psi^k(y)$

($\psi^k$ is in fact a natural ring homomorphism)

4. $\psi^k(\psi^l(x)) = \psi^{k l}(x)$

5. $\psi^p(x) = x^p \, \text{mod}\, p$

6. if $x \in \tilde K(S^{2n})$ (reduced cohomology) then

$x \in \tilde K(S^{2n}) \hookrightarrow K(S^{2n}) \;\;\Rightarrow\;\;\psi^k(x) = k^n \cdot x$.

Moreover, the first two of these already uniquely characterize the Adams operations.

## Properties

### Basic properties

###### Proposition

(Adams operations on complex topological K-theory of n-spheres)

For $n \in \mathbb{N}$, the Adams operations on the reduced K-theory of the 2n-sphere are given by:

$\array{ \widetilde K \big( S^{2n} \big) & \overset{ \;\;\; \psi^k\;\;\; }{\longrightarrow} & \widetilde K \big( S^{2n} \big) \\ V &\mapsto& k^n \cdot V }$

### Compatibility with complexification

The Adams operations are compatible with the complexification map $(-) \otimes_{\mathbb{R}} \mathbb{C}$ from real vector bundles to complex vector bundles, hence from KO-cohomology to KU-cohomology, in that the following diagram commutes, for all $k$:

$\array{ KO(X) &\overset{ \psi^k }{\longrightarrow}& KO(X) \\ {}^{ \mathllap{ (-) \otimes_{\mathbb{R}} \mathbb{C} } } \big\downarrow && \big\downarrow {}^{ \mathrlap{ (-) \otimes_{\mathbb{R}} \mathbb{C} } } \\ KO(X) &\overset{ \psi^k }{\longrightarrow}& KO(X) }$

### Compatibility with the Chern character

The Adams operation are compatible with the Chern character map in the following way:

###### Definition

For $X$ a topological space, with rational cohomology in even degrees denoted

$H^{ev}(X;\, \mathbb{Q}) \;\coloneqq\; \underset{r \in \mathbb{N}}{\prod} H^{2 r}(X;\, \mathbb{Q})$

$\psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)$

for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$

$\array{ H^{2r}(X;\mathbb{Q}) &\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}& H^{2r}(X;\mathbb{Q}) \\ \alpha_{2k} &\mapsto& k^{r} \cdot \alpha_{2r} \,. }$
###### Proposition

(Adams operations compatible with the Chern character)

For $X$ a topological space with a finite CW-complex-mathematical structure, the Chern character $ch$ on the complex topological K-theory of $X$ intertwines the Adams operations $\psi^n$ on K-theory with the Adams-like operations $\psi^n_H$ on rational cohomology from Def. , for $k \geq 1$, in that the following diagram commutes:

$\array{ K(X) &\overset{\;\;\;ch\;\;\;}{\longrightarrow}& H^{ev}(X;\,\mathbb{Q}) \\ {}^{ \mathllap{ \psi^k } } \big\downarrow && \big\downarrow {}^{ \mathrlap{ \psi^k_H } } \\ K(X) &\overset{\;\;\;ch\;\;\;}{\longrightarrow}& H^{ev}(X;\,\mathbb{Q}) \,, }$
###### Proof idea

Use the exponentional-formula for the Chern character with the splitting principle.

The Adams conjecture (a theorem) says that for all $k \in \mathbb{N}$ and $V \in K(X)$ there is $n \in \mathbb{N}$ such that the spherical fibration assigned to the K-theory class $k^n (\psi^k(V)-V)$ under the J-homomorphism is trivial, hence that

$J \left( k^n \left( \psi^k(V) - V \right) \right) = 0 \,.$

### General

The original article:

Review:

### In representation theory

Adams operations on the representation ring (the equivariant K-theory of the point) are discussed in

### In knot theory

Adams operations on Jacobi diagrams modulo STU relations (Adams operation on Jacobi diagrams) are discussed in