Relative measure homology and continuous bounded cohomology of topological pairs

Roberto Frigerio and Cristina Pagliantini

Vol. 257 (2012), No. 1, 91–130

DOI: 10.2140/pjm.2012.257.91

Abstract

Measure homology was introduced by Thurston in his notes about the geometry and topology of 3-manifolds, where it was exploited in the computation of the simplicial volume of hyperbolic manifolds. Zastrow and Hansen independently proved that there exists a canonical isomorphism between measure homology and singular homology (on the category of CW-complexes), and it was then shown by Löh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L¹-seminorm on singular homology and the total variation seminorm on measure homology. Löh’s result plays a fundamental rôle in the use of measure homology as a tool for computing the simplicial volume of Riemannian manifolds.

This paper deals with an extension of Löh’s result to the relative case. We prove that relative singular homology and relative measure homology are isometrically isomorphic for a wide class of topological pairs. Our results can be applied for instance in computing the simplicial volume of Riemannian manifolds with boundary.

Our arguments are based on new results about continuous (bounded) cohomology of topological pairs, which are probably of independent interest.

Keywords

simplicial volume, singular homology, bounded cohomology of groups, CAT(0) spaces

Mathematical Subject Classification 2010

Primary: 55N10, 55N35

Secondary: 20J06, 55U15, 57N65

Milestones

Received: 8 June 2011

Revised: 9 February 2012

Accepted: 15 May 2012

Published: 19 June 2012