Surjectivity of the Comparison Map in Bounded Cohomology for Hermitian Lie Groups

Abstract

We prove surjectivity of the comparison map from continuous bounded cohomology to continuous cohomology for Hermitian Lie groups with finite center. For general semisimple Lie groups with finite center, the same argument shows that the image of the comparison map contains all the even generators. Our proof uses a Hirzebruch type proportionality principle in combination with Gromov's results on boundedness of primary characteristic classes and classical results of Cartan and Borel on the cohomology of compact homogeneous spaces. orem 1 we learned from Michelle Bucher-Karlsson that a similar theorem should be true for certain even-degree classes in arbitrary semisimple Lie groups. We are indepted to her for this suggestion, which led us to discover Theorem 2. The second author would like to thank the Department of Mathematics at Rutgers University for their hospitality and excellent working conditions. Outline of the proofIn this section we outline the proofs of Theorem 1 and Theorem 2.Let G be an arbitrary semisimple Lie group with finite center and without compact factors. The main technical result of this article is Proposition 2.2 below,

On splitting rank of non-compact-type symmetric spaces and bounded cohomology

“…This conjecture remains open in the specific case of SL(n, R). Prior work includes Hartnick and Ott [14], which confirmed the conjecture for Lie groups of Hermitian type (as well as some other cases). Domic and Toledo gave explicit bounds in degree two [8], and this was later generalized by Clerc and Ørsted in [5].…”

“…The codimension takes its second minimal value 30 when Y ′ = SL(6, C)/SU (6), hence the gap is 10. If r = 7, then X = SO(14, C)/SO (14) and Y = SO(12, C)/SO (12). The codimension takes its second minimal value 42 when Y ′ is either H 3 × SO(10, C)/SO (10) or SL(7, C)/SU (7), hence the gap is 18.…”

Higher genera for proper actions of Lie groups

“…In fact, the conjecture in [9] is that for semisimple Lie groups all these cocycles are bounded. For recent work on this conjecture, see [13,21]. In this last reference, different simplices are used, given by the barycentric subdivision of the geodesic ones, to prove boundedness of the top dimensional cocycle for general connected semisimple Lie groups.…”

“…We remark that item (2) above has in independent interest, and should be compared with the literature on bounded cohomology of Lie groups, c.f. [13,21] The geometric applications stated in Theorem 1.5 are then a direct consequence of the G homotopy invariance of the signature index class, established by Fukumoto in [10] and, for the higher A-genera, of the vanishing of the index class Ind C * r (G) (ð) ∈ K * (C * r (G)) of the spin Dirac operator ð of a G-spin G-proper manifold endowed with a G-metric of positive scalar curvature, established by Guo, Mathai and Wang in [11]. In the odd dimensional case we argue by suspension.…”

On the bounded cohomology of Lie groups

“…) is conjectured to be onto for semisimple Lie groups with finite center (see [12,Conjecture 18.1,page 56]). The conjecture has been proved for Hermitian Lie groups with finite center (see [28]).…”