Bounded cohomology, Codimension two submanifolds and Pontryagin-Thom constructions

Abstract

In this note we develop a novel approach for proving the non-vanishing of bounded cohomology. This utilizes a splitting argument whose simplest form is as follows: Let M denote an n-manifold of non-zero simplicial volume and N a codimension two submanifold of M, then one can conclude that the n-th bounded cohomology of the fundamental group of M \ N is non-zero. We then translate the existence of a complement with a given fundamental group into an easily accessible homology computation, which might be of independent interest.

arXiv commentThis paper represents a rework of the erroneous paper arXiv:2503.22511. It represents the attempts of the author to repair and further develop the approach of arXiv:2503.22511