Cohomology of classifying spaces of complex Lie groups and related discrete groups (Q799265)

The authors prove that the cohomology with \({\mathbb{Z}}/n\) coefficients of the classifying space BG of a complex reductive Lie group G can be computed as the cohomology of a locally finite group naturally associated to G. (In a forthcoming paper, ''Locally finite approximation of Lie groups I'', the authors generalize this theorem to any Lie group with finite component group.) Two consequences of this theorem: a) the identification originally due to M. Feshbach of \(H^*(BG,{\mathbb{Z}}/n)\) with the stable elements in the cohomology of the classifying space of the normalizer of a maximal torus of G; b) a sharpening of a theorem of J. Milnor asserting that \(H^*(BG,{\mathbb{Z}}/n)\) is a summand of the \({\mathbb{Z}}/n\) cohomology of G viewed as a discrete group. The so-called ''isomorphism conjecture'' asserts that this split injection is an isomorphism: The authors prove that this conjecture is equivalent to the conjecture that any \({\mathbb{Z}}/n\) cohomology class of G viewed as a discrete group can be detected on a finite subgroup of G. The authors avoid the use of the Becker-Gottlieb transfer throughout; their technique is to use algebraic geometry to relate G to its corresponding algebraic group in characteristic p.