Universal Finite Functorial Semi-norms

Abstract

Functorial semi-norms on singular homology measure the “size” of homology classes. A geometrically meaningful example is the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-semi-norm. However, the ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell ^1$$\\end{document}-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.