nLab
Adams operation

Contents

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

Given a complex line bundle LL over a space XX its kkth tensor power L kL^{\otimes k} is another line bundle for any kk \in \mathbb{N}. The line bundles define certain elements of topological K-theory group K(X)K(X), and there is a unique operation ψ k:K(X)K(X)\psi^k : K(X) \to K(X), the kkth Adams operation, such that:

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

  • ψ k:K(X)K(X)\psi^k : K(X) \to K(X) is a group homomorphism,

  • ψ k\psi^k is a natural transformation: any map f:XYf: X \to Y induces a map f *:K(Y)K(X)f^* : K(Y) \to K(X) on KK-theory, and ψ kf *=f *ψ k\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 XX be a compact topological space and write K(X)K(X) for its topological K-theory ring. For EE a vector bundle over XX, write [E]K(X)[E] \in K(X) for its class in K-theory. Given EE, write

λ t[E]k=0[ X kE]t kK(X)[[t]] \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)K(X) being the K-theory classes of the skew-symmetrized tensor product of vector bundles of EE with itself.

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

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

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

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

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

(explicit definition of Adams operation)

For EE a vector bundle over some topological space XX, write

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

for the bundle which over each connected component of XX is the trivial vector bundle of the same rank as EE over that component.

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

ψ t(E) k=0ψ k(E)t k ψ 0(E)tddtlogλ t(E)K(X)[[t]], \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 λ t\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:

ddtlogλ t(E) =1λ t(E)ddtλ t(E) =k=0(1λ t(E)) kddtλ t(E). \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 kkth Adams operation is the cohomology operation on topological K-theory

ψ k:K()K() \psi^k \;\colon\; K(-) \longrightarrow K(-)

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

Proposition

(basic properties and characterization of Adams operations)

The Adams operations

ψ k:K(X)K(X) \psi^k \;\colon\; K(X) \longrightarrow K(X)

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

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

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

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

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

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

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

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

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

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

    xK˜(S 2n)K(S 2n)ψ k(x)=k nxx \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.

e.g. Wirthmuller 12, section 11

Properties

Adams conjecture

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

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

References

General

In representation theory

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

Last revised on September 26, 2018 at 13:45:25. See the history of this page for a list of all contributions to it.