A geometric criterion for the boundedness of characteristic classes

Abstract

previously a preprint with title "On the bounded cohomology of Lie groups".International audienceWe show that each integral Borel cohomology class of a connected Lie group G can be represented by a Borel bounded cocycle if and only if the radical of G is linear. This leads to a generalization of Gromov's boundedness theorem on characteristic classes of flat bundles

Flat Bundles With Complex Analytic Holonomy

“…Let Q → Q be the universal cover. The pull-back of G → Q overQ yields a short exact sequence of Lie groups (2) R →Ḡ →Q which is split becauseQ is simply connected (see Lemma 14 of [2]). The natural map p :Ḡ → G is a surjective homomorphism of connected Lie groups with finite kernel K isomorphic to π 1 (Q).…”

“…The first equality (as well as the fourth one) is true because the embedding of a connected solvable Lie group in its universal complexification is a homotopy equivalence, the second equality (as well as the last one) is a general fact (see [2,Lemma 6]) about closed normal subgroups in Lie groups, the third equality follows from the fact that [ R, R] + ∩ R ⊂ [ R, R] which is deduced from the corresponding inclusion between Lie algebras. The fifth equality is true because we have already proved the lemma for simply connected solvable Lie groups hence [ R, R] + = [( R) + , ( R) + ].…”

Bounded characteristic classes and flat bundles

“…There are examples of classes x δ in the image of Λ, which are bounded, but which do not admit a bounded Borel cocycle representative. However, this can only happen, if the derived subgroup [R, R] of the radical of G is not closed in G (see [7] for such examples).…”

“…The image of the natural map H * (BG, R) → H * (BG δ , R) is generated as a ring by bounded classes together with the image in degree two [7,Lemma 51,Theorem 54].…”

“…Previously known sufficient conditions where: G is linear algebraic (Gromov [11,Theorem p. 23], resp. Bucher [5]), and the weaker condition that the radical R of G is linear [7,Theorem 54]; the radical of G is the largest connected, normal, solvable subgroup of G. It is well-known that R is linear if and only if the closure of its derived subgroup [R, R] is simply connected. Hence linearity of R is stronger than simple connectedness of [R, R] because…”