Comparing seminorms on homology

Jean-François Lafont and Christophe Pittet

Vol. 259 (2012), No. 2, 373–385

DOI: 10.2140/pjm.2012.259.373

Abstract

We compare the l¹-seminorm ∥⋅∥₁ and the manifold seminorm ∥⋅∥_man on n-dimensional integral homology classes. Crowley and Löh showed that for any topological space X and any α ∈ H_n(X; ℤ), with n≠3, the equality ∥α∥_man = ∥α∥₁ holds. We compute the simplicial volume of the 3-dimensional Tomei manifold and apply Gaifullin’s desingularization to establish the existence of a constant δ₃ ≈ 0.0115416, with the property that for any X and any α ∈ H₃(X; ℤ), one has the inequality

Keywords

l¹-norm, simplicial volume, singular homology, manifold norm, Steenrod’s realization problem, Thurston norm, Tomei manifold

Mathematical Subject Classification 2010

Primary: 53C23

Secondary: 57M50

Milestones

Received: 28 March 2012

Revised: 2 August 2012

Accepted: 21 August 2012

Published: 3 October 2012