Contents

Contents

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

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

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

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

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:

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

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

References

The original statement:

Textbook accounts:

Review and exposition:

Last revised on January 7, 2021 at 13:42:04. See the history of this page for a list of all contributions to it.