algebraic topology – application of higher algebra and higher category theory to the study of (stable) homotopy theory
group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
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 $k^r$.
(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 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.