Universal finite functorial semi-norms

TL;DR

This paper proves the existence of universal finite functorial semi-norms on singular homology for spaces homotopy equivalent to finite CW-complexes, extending the understanding of size measures in homology.

Contribution

It establishes the existence of universal finite functorial semi-norms on singular homology, a significant advancement in the theory of homology size measures.

Findings

Universal finite functorial semi-norms exist on singular homology.

Arguments extend to more general settings of functorial semi-norms.

Provides a new perspective on measuring homology classes.

Abstract

Functorial semi-norms on singular homology measure the "size" of homology classes. A geometrically meaningful example is the $\ell^1$-semi-norm. However, the $\ell^1$-semi-norm is not universal in the sense that it does not vanish on as few classes as possible. We show that universal finite functorial semi-norms do exist on singular homology on the category of topological spaces that are homotopy equivalent to finite CW-complexes. Our arguments also apply to more general settings of functorial semi-norms.