group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
geometric representation theory
representation, 2-representation, ∞-representation
Grothendieck group, lambda-ring, symmetric function, formal group
principal bundle, torsor, vector bundle, Atiyah Lie algebroid
Eilenberg-Moore category, algebra over an operad, actegory, crossed module
Be?linson-Bernstein localization?
The statement known as Segal’s conjecture (due to Graeme Segal in the 1970s, then proven by Carlsson 84) characterizes the stable cohomotopy groups of the classifying space of a finite group as the formal completion at the augmentation ideal (i.e. when regarded as a ring of functions: its restriction to the infinitesimal neighbourhood of the basepoint) of the ring of -equivariant stable cohomotopy groups of the point, the latter also being isomorphic to the Burnside ring of :
This statement is the direct analogue of the Atiyah-Segal completion theorem, which makes the analogous statement for the generalized cohomology not being (equivariant) stable cohomotopy but (equivariant) complex K-theory (with the role of the Burnside ring then being the representation ring of ).
A proof of the Sullivan conjecture follows with the Segal-Carlsson completion theorem
The statement was proven in
see also
Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, pdf)
Wolfgang Lück, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups (arXiv:math/0504051)
Review includes
Application to proof of the Sullivan conjecture is due to
See also
Last revised on September 10, 2018 at 12:14:28. See the history of this page for a list of all contributions to it.