algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
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
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.
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.
(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
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
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
(explicit definition of Adams operation)
For $E$ a vector bundle over some topological space $X$, write
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
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:
The $k$th Adams operation is the cohomology operation on topological K-theory
which is the coefficient of $t^k$ in $\psi_t$.
(basic properties and characterization of Adams operations)
The Adams operations
have the following properties, for all elements $x,y \in K(X)$ and $k, l \in \mathbb{N}$ and $p \; \text{prime}$:
$\psi^k(x + y) = \psi^k(x) + \psi^k(y)$
($\psi^k$ is a natural abelian group homomorphism)
$x \,\text{a line} \;\;\Rightarrow\;\; \psi^k(x) = x^k$
(applied to a class $x \coloneqq [L]$ represented by a line bundle $L$, $\psi^k$ is the $k$th tensor power)
$\psi^k(x \cdot y) = \psi^k(x) \cdot \psi^k(y)$
($\psi^k$ is in fact a natural ring homomorphism)
$\psi^k(\psi^l(x)) = \psi^{k l}(x)$
$\psi^p(x) = x^p \, \text{mod}\, p$
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.
e.g. Wirthmuller 12, section 11
(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:
(e.g. Wirthmüller, Prop. on p. 45 (47 of 67))
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 $k$:
(Adams 62, Thm. 5.1. (iv), Karoubi 78, Prop. IV.7.25)
The Adams operation are compatible with the Chern character map in the following way:
(Adams-like operations on rational cohomology)
For $X$ a topological space, with rational cohomology in even degrees denoted
define graded linear maps
for $k \in \mathbb{N}$ by taking their restriction to degree $2r$ to act by multiplication with $k^r$
(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:
(Adams 62, Thm. 5.1. (vi), review in Karoubi 78, Chapter V, Theorem 3.27, Maakestad 06, Thm. 4.9)
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
The original article:
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)
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)
Adams operations on Jacobi diagrams modulo STU relations (Adams operation on Jacobi diagrams) are discussed in
Last revised on January 19, 2021 at 06:38:11. See the history of this page for a list of all contributions to it.