Universal finite functorial semi-norms

Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the ${\mathrm{\ell }}^1$-semi-norm. However, the ${\mathrm{\ell }}^1$-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.