# The zero-norm subspace of bounded cohomology

### T. Soma

Tokyo Denki University, Japan

## Abstract

Let $\Sigma$ be a closed, orientable surface of genus > 1. In this paper, non-trivial elements $\alpha$ of the third bounded cohomology $H_b^3(\Sigma;\bf {R})$ with $\Vert \alpha \Vert = 0$ are given constructively by using both a hyperbolic metric and a singular euclidean metric on $\Sigma \times \bf {R}$ . Furthermore, it is shown that the dimension of the subspace $N^3(\Sigma)$ of $H_b^3(\Sigma;\bf {R})$ consisting of zero-norm elements is the cardinality of the continuum.

## Cite this article

T. Soma, The zero-norm subspace of bounded cohomology. Comment. Math. Helv. 72 (1997), no. 4, pp. 582–592

DOI 10.1007/S000140050035