group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
symmetric monoidal (∞,1)-category of spectra
By the general discussion at equivariant K-theory, given a suitable topological group with an action on a topological space there is a canonical map
from the equivariant K-theory of to the ordinary topological K-theory of the homotopy quotient (Borel construction).
While this map is never an isomorphism unless is the trivial group, the Atiyah-Segal completion theorem says that this map exhibits as the formal completion of the cohomology ring at the augmentation ideal of the representation ring of (hence, regarded as a ring of functions, the restriction to an infinitesimal neighbourhood of the base point).
See also at formal completion – Examples – Atiyah-Segal theorem.
The analog stable for stable cohomotopy is the Segal-Carlsson completion theorem:
In the case where (i.e. a point), we have that and , thus we conclude that
where is the augmentation ideal of the representation ring of .
(complex topological K-theory of )
The complex topological K-theory of the classifying space of the circle group is the power series ring:
where is any complex orientation of KU (see this Prop).
On the other hand, the representation ring of the circle group is (see this Example)
Here is the class of the 1-dimensional irrep , the augmentation ideal is clearly generated by . The corresponding completion is again
where the first step is this example and the third step observes that is invertible in the -power series ring (by this Prop.).
(e.g. Buchholtz 08, Sec. 8.2, also Math.SE:a/3282578)
compare also Greenlees 1994, p. 74
As explained in Equivariant stable homotopy theory, there is a related characterization of the K-homology of as a localization.
The original articles:
Michael Atiyah, Characters and cohomology of finite groups, Publications Mathématiques de l’IHÉS, Volume 9 (1961) , p. 23-64 (numdam:PMIHES_1961__9__23_0)
Michael Atiyah, Friedrich Hirzebruch, Vector bundle and homogeneous spaces, Proc. Sympos. Pure Math., Vol. III, American Mathematical Society, Providence, R.I., 1961, 3, 7–38 (pdf)
Michael Atiyah, Graeme Segal, Equivariant -theory and completion, J. Differential Geom. Volume 3, Number 1-2 (1969), 1-18. (euclid:jdg/1214428815)
Review and survey:
Ulrik Buchholtz, The Atiyah-Segal completion theorem, Master Thesis 2008 (pdf, pdf)
Wikipedia, Atiyah-Segal completion theorem
Last revised on June 7, 2024 at 15:45:48. See the history of this page for a list of all contributions to it.