Contents

cohomology

# 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 topological K-theory group $K(X)$, and there is a unique operation $\psi^k : K(X) \to K(X)$, the $k$th Adams operation, such that:

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

• $\psi^k : K(X) \to K(X)$ 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

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

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

### 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