Relative measure homology and continuous bounded cohomology of topological pairs

TL;DR

This paper extends Loeh's isometric isomorphism between measure and singular homology to the relative case for topological pairs, aiding in computing simplicial volumes of manifolds with boundary.

Contribution

It establishes a new isometric isomorphism between relative measure and singular homology for a broad class of topological pairs, expanding the tools for geometric topology.

Findings

Proves relative measure homology and singular homology are isometrically isomorphic.

Provides methods to compute simplicial volume of manifolds with boundary.

Introduces new results on continuous (bounded) cohomology of topological pairs.

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 Loeh that, in the absolute case, such isomorphism is in fact an isometry with respect to the L^1-seminorm on singular homology and the total variation seminorm on measure homology. Loeh's result plays a fundamental role in the use of measure homology as a tool for computing the simplicial volume of Riemannian manifolds. This paper deals with an extension of Loeh'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.