nLab Adams operations compatible with the Chern character

Idea

The Adams operations $\psi^k$ on complex topological K-theory are compatible with the Chern character map to rational cohomology in that the effect of $\psi^k$ on the Chern character image in degree $2r$ is multiplication by $k^r$.

Statement

Definition

(Adams-like operations on rational cohomology)

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

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

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 $k^r$:

(1)$$\array{ H^{2r}(X;\mathbb{Q}) &\overset{\;\;\;\psi^k_H\;\;\;}{\longrightarrow}& H^{2r}(X;\mathbb{Q}) \\ \alpha_{2k} &\mapsto& k^{r} \cdot \alpha_{2k} \,. }$$

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:

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

(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.

Related entries

• e-invariant

References

The original statement:

• John Adams, Theorem 5.1 (vi) Vector fields on spheres, Bull. Amer. Math. Soc. Volume 68, Number 1 (1962), 39-41 (euclid:bams/1183524456, pdf)

Textbook accounts:

• Max Karoubi, Theorem V3.27 in: K-Theory – An introduction, Grundlehren der mathematischen Wissenschaften 226, Springer 1978 (pdf, doi:10.1007/978-3-540-79890-3)

Review and exposition:

• 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)