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

2020/10/16 by Shi Wang, Wang, Shi
Mathematics · #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Geometry and complex manifolds #Homotopy and Cohomology in Algebraic Topology #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2010.08594

openalex publication_date 2020/10/16 · openalex created_date 2020/10/22 · openalex updated_date 2026/07/28

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)→ 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≥ dim(M)-rank(M)+2.

Related

• The Degree Theorem in higher rank
• Counting flat cycles in the homology of locally symmetric spaces
• Continuous cochains on Furstenberg boundaries and injectivity of the comparison map
• Special cycles in compact locally Hermitian symmetric spaces of type III associated with the Lie group SO₀(2,m)
• Cyclic Cohomology and Higher Rank Lattices