Bounded acyclicity of the discrete embedding monoid Emb^δ(R^n)

Determine whether the discrete monoid Emb^δ(R^n) of self-embeddings of R^n is boundedly acyclic; concretely, show that H^k_b(Emb^δ(R^n)) = 0 for every k > 0.

Background

The paper shows that the maximal subgroup Homeo(R^n) of the monoid of self-embeddings Emb(R^n) is boundedly acyclic and discusses classical results (Kister, Segal) about the homotopy and homology of Emb(R^n) in topological and discrete settings. Extending bounded acyclicity from the group to the monoid is non-trivial and would further align discrete embedding monoids with their continuous counterparts.

This question targets the bounded cohomology of Emb^δ(R^n), asking whether it vanishes in positive degrees, which would parallel the authors’ main results for transformation groups and deepen the understanding of embedding monoids.