nLab
Adams operation

Contents

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 complex topological K-theory group K(X)K(X), and there is a unique cohomology operation

ψ k:K(X)K(X), \psi^k \colon 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 K-theory class of any line bundle,

  • ψ k:K(X)K(X)\psi^k \colon 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

Basic properties

Proposition

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

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

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

(e.g. Wirthmüller, Prop. on p. 45 (47 of 67))

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 kk:

KO(X) ψ k KO(X) () () KO(X) ψ k KO(X) \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) }

(Adams 62, Thm. 5.1. (iv), Karoubi 78, Prop. IV.7.25)

Compatibility with the Chern character

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

Definition

(Adams-like operations on rational cohomology)

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

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

define graded linear maps

ψ H k:H ev(X)H ev(X) \psi^k_{H} \;\colon\; H^{ev}(X) \longrightarrow H^{ev}(X)

for kk \in \mathbb{N} by taking their restriction to degree 2r2r to act by multiplication with k rk^r

H 2r(X;) ψ H k H 2r(X;) α 2k k rα 2r. \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 XX a topological space with a finite CW-complex-mathematical structure, the Chern character chch on the complex topological K-theory of XX intertwines the Adams operations ψ n\psi^n on K-theory with the Adams-like operations ψ H n\psi^n_H on rational cohomology from Def. , for k1k \geq 1, in that the following diagram commutes:

K(X) ch H ev(X;) ψ k ψ H k K(X) ch H ev(X;), \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}) \,, }

(Adams 62, Thm. 5.1. (vi), review in Karoubi 78, Chapter V, Theorem 3.27, Maakestad 06, Thm. 4.9)

Proof idea

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


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

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

Last revised on January 19, 2021 at 01:38:11. See the history of this page for a list of all contributions to it.