Injectivity of ν^* below 2n for r > 1 (Hurder’s Problem 14.5)

Prove that for regularity r > 1, the map ν^*: H^*(BGL_n(R)_+; R) → H^*(BSΓ^r_n; R), induced by the classifying map for oriented normal bundles to codimension-n foliations, is injective in degrees strictly below 2n.

Background

The authors recall that ν is (n+1)-connected for all regularities, ensuring injectivity of ν* up to degree n+1, and that Tsuboi’s theorem implies isomorphism for r = 1 in all degrees. For r > 1, the stronger injectivity up to degree 2n is conjectural and appears as Problem 14.5 in the literature on foliation characteristic classes.

Establishing this injectivity would strengthen the link between classical characteristic classes of linear groups and those arising from Haefliger structures in higher regularity, impacting the understanding of secondary characteristic classes and boundedness questions.

References

For r=1, it is an isomorphism in all degrees by a remarkable theorem of Tsuboi and for r>1, it is conjecturally injective in degrees below 2n [Problem 14.5].

— The bounded cohomology of transformation groups of Euclidean spaces and discs (2405.20395 - Fournier-Facio et al., 30 May 2024) in Section “Unboundedness of characteristic classes” (paragraph before Theorem labeled “C^r unboundedness”)