Geometric cycles and bounded cohomology for a cocompact lattice in $SL_n(\mathbb R)$

TL;DR

This paper constructs a specific example of a locally symmetric manifold with a non-surjective bounded cohomology map in a certain degree, revealing new geometric and cohomological properties of such spaces.

Contribution

It demonstrates the existence of a closed locally symmetric manifold with a non-trivial homology class represented by a totally geodesic submanifold containing a circle factor, showing the non-surjectivity of the comparison map in a specific degree.

Findings

Existence of a closed locally symmetric manifold with specific geometric properties.

Identification of non-surjectivity of the comparison map in a particular degree.

Counterpart to previous surjectivity results by Lafont-Wang.

Abstract

We show there exists a closed locally symmetric manifold $M$ modeled on $SL_n(\mathbb R)/SO(n)$, and a non-trivial homology class in degree $dim(M)-rank(M)$ represented by a totally geodesic submanifold that contains a circle factor. As a result, the comparison map $c^k:H_b^k(M,\mathbb R)\rightarrow H^k(M,\mathbb R)$ is not surjective in degree $k=dim(M)-rank(M)$. This provides a counterpart to a result of Lafont-Wang which states that $c^k$ is always surjective in degree $k\geq dim(M)-rank(M)+2$.