Commutative algebra for cohomology rings of classifying spaces of compact Lie groups

Self-maps of Classifying Spaces of Compact Simple Lie Groups

We describe here the set [BG, BG] of homotopy classes of self-maps of the classifying space BG , for any compact connected simple Lie group G. In particular, we show that two maps ƒ , f : BG —• BG are homotopic if and only if they are homotopic after restricting to the maximal torus of G ; or equivalently if and only if they induce the same homomorphism in rational cohomology. In addition, we i...

Homotopy Theory of Classifying Spaces of Compact Lie Groups

The basic problem of homotopy theory is to classify spaces and maps between spaces, up to homotopy, by means of invariants like cohomology. In the last decade some striking progress has been made with this problem when the spaces involved are classifying spaces of compact Lie groups. For example, it has been shown, for G connected and simple, that if two self maps of BG agree in rational cohomo...

Vector Bundles over Classifying Spaces of Compact Lie Groups

The completion theorem of Atiyah and Segal [AS] says that the complex K-theory group K(BG) of the classifying space of any compact Lie group G is isomorphic to R(G)̂ : the representation ring completed with respect to its augmentation ideal. However, the group K(BG) = [BG,Z × BU ] does not directly contain information about vector bundles over the infinite dimensional complex BG itself. The purp...

Classifying spaces of compact Lie groups that are p–compact for all prime numbers

A p–compact group (see Dwyer–Wilkerson [8]) is a loop space X such that X is Fp –finite and that its classifying space BX is Fp –complete (see Andersen–Grodal– Møller–Viruel [2] and Dwyer–Wilkerson [11]). We recall that the p–completion of a compact Lie group G is a p–compact group if π0(G) is a p–group. Next, if C(ρ) denotes the centralizer of a group homomorphism ρ from a p–toral group to a c...

Lie Algebra Cohomology and the Representations Ofsemisimple Lie Groups