Fundamental classes of negatively curved manifolds cannot be represented by products of manifolds

TL;DR

This paper proves that fundamental classes of negatively curved manifolds cannot be represented as products of manifolds, revealing limitations in their homological representations and impacting the understanding of functorial semi-norms.

Contribution

It establishes a fundamental restriction on representing negatively curved manifold classes as products, advancing the understanding of their homological properties.

Findings

Fundamental classes of negatively curved manifolds cannot be expressed as products.

Highlights limitations of functorial semi-norms on singular homology.

Provides insight into the structure of homology classes of negatively curved manifolds.

Abstract

Not every singular homology class is the push-forward of the fundamental class of some manifold. In the same spirit, one can study the following problem: Which singular homology classes are the push-forward of the fundamental class of a given type of manifolds? In the present article, we show that the fundamental classes of negatively curved manifolds cannot be represented by a non-trivial product of manifolds. This observation sheds some light on the functorial semi-norm on singular homology given by products of compact surfaces.