nLab Adams operation

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

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

for the formal power series whose coefficients in the ring $K(X)$ being the K-theory classes of the exterior powers $\Lambda^k X$, i.e. the skew-symmetrized tensor powers of $E$.

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

(explicit definition of Adams operation)

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

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

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:

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

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)

The 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.

For a proof see e.g. Wirthmuller 12, section 11. For item 1 recall that for any vector bundles $E$ and $F$ we have

$$\Lambda^n (E \oplus F) \cong \bigoplus_{k = 0}^n \Lambda^k E \otimes \Lambda^{n-k} F$$

hence a calculation gives

$$\lambda_t(x + y) = \lambda_t(x) \cdot \lambda_t(y)$$

and thus

$$\log \lambda_t(x + y) = \lambda_t(x) + \lambda_t(y).$$

That is, $\log \lambda_t(x)$ is additive as a function of $x$. Since

$$\sum_{k=0}^\infty t^k \psi^k(x) = \mathrm{rank}(x) - t \frac{d}{d t} \log \lambda_{-t}(x)$$

and both terms are right are additive as a function of $x$, we conclude

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

For item 2 we can do the following calculation. Suppose that $x$ is the class of a line bundle $L$, so that $\Lambda^2 L = 0$ hence $\lambda_{-t}(x) = 1 - t x$. Then

$$\begin{array}{ccl} \sum_{k=0}^\infty t^k \psi^k(x) &=& 1 - t \frac{d}{d t} \log \lambda_{-t}(x) \\ &=& 1 - t \frac{d}{d t} \log(1 - t x) \\ &=& 1 + \frac{t x}{1 - t x} \\ &=& \frac{1}{1 - t x} \\ &=& \sum_{k=0}^\infty t^k x^k \end{array}$$

so $\psi_k(x) = x^k$.

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:

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

(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 $k$:

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

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

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

define graded linear maps

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

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:

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

(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 $k \in \mathbb{N}$ 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

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

Related concepts

• Lambda-ring

• Steenrod square

• Steenrod operation

• power operation

• Adams operation on Jacobi diagrams

References

General

The original article:

• John Adams, Section 5 of: Vector fields on spheres, Bull. Amer. Math. Soc. Volume 68, Number 1 (1962), 39-41 (euclid:bams/1183524456, pdf)

See also:

• Victor M. Buchstaber, Sergei P. Novikov, Formal groups, power systems and Adams operators, Math. USSR-Sb.13 (1971) 80-116 [doi:10.1070/SM1971v013n01ABEH001030]

Review:

• Max Karoubi, Section IV.7 in in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007/978-3-540-79890-3)

• Marcelo Aguilar, Samuel Gitler, Carlos Prieto, Section 10 of: Algebraic topology from a homotopical viewpoint, Springer (2002)

• Helge Maakestad, Notes on the Chern-character, Journal of Generalized Lie Theory and Applications, 2017, 11:1 (arXiv:math/0612060, doi:10.4172/1736-4337.100025)

• Michael Hopkins (notes by Akhil Mathew), Lecture 10 in: Spectra and stable homotopy theory, 2012 (pdf, pdf)

• Klaus Wirthmüller, Section 11 of: Vector Bundles and K-theory, 2012 (pdf)

• Allen Hatcher, section 2.3 of Vector bundles and K-theory (web)

• Wikipedia, Adams operation

• Jacob Lurie, remark 2 in: Chromatic Homotopy Theory, Lecture series 2010, Lecture 35 The image of $J$ (pdf)

In representation theory

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

• Tammo tom Dieck, section 3.5 of Transformation Groups and Representation Theory, Lecture Notes in Mathematics 766 Springer 1979

• Robert Boltje, A characterization of Adams operations on representation rings, 2001 (pdf)

• Dai Tamaki, Akira Kono, Section 4.3 in: Generalized Cohomology, Translations of Mathematical Monographs, American Mathematical Society, 2006 (ISBN: 978-0-8218-3514-2)

• Tammo tom Dieck, section 6.2 of Representation theory, 2009 (pdf)

• Michael Boardman, Adams operations on Group representations, 2007 (pdf)

• Ehud Meir, Markus Szymik, Adams operations and symmetries of representation categories (arXiv:1704.03389)

In knot theory

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

• Dror Bar-Natan, Theorem 7, Def. 3.11 and Section 6.3.4 of: On the Vassiliev knot invariants, Topology Volume 34, Issue 2, April 1995, Pages 423-472 (doi:10.1016/0040-9383(95)93237-2, pdf)