The ${\ell }^{\infty }$-semi-norm on uniformly finite homology

Abstract

Uniformly finite homology is a coarse homology theory, defined via chains that satisfy a uniform boundedness condition. By construction, uniformly finite homology carries a canonical ${\ell }^{\infty }$-semi-norm. We show that, for uniformly discrete spaces of bounded geometry, this semi-norm on uniformly finite homology in degree 0 with integral coefficients allows for a new formulation of Whyte's rigidity result. In contrast, we prove that this semi-norm is trivial on uniformly finite homology in higher degrees with real coefficients.

16 pages; v2: small changes in the introduction

References in corpus (1)

• On macroscopic dimension of rationally essential manifolds