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Adams operations compatible with the Chern character

Contents

Context

Algebraic topology

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Contents

Idea

The Adams operations ψ k\psi^k on complex topological K-theory are compatible with the Chern character map to rational cohomology in that the effect of ψ k\psi^k on the Chern character image in degree 2r2r is multiplication by k rk^r.

Statement

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}) \;\colon\; \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:

(1)H 2r(X;) ψ H k H 2r(X;) α 2k k rα 2k. \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 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) &\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.

References

The original statement:

Textbook accounts:

Review and exposition:

Last revised on January 7, 2021 at 08:42:04. See the history of this page for a list of all contributions to it.