Vol. 259, No. 2, 2012

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Comparing seminorms on homology

Jean-François Lafont and Christophe Pittet

Vol. 259 (2012), No. 2, 373–385
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 Hn(X; ), with n3, the equality αman = α1 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; ), one has the inequality

δ3∥α∥man ≤ ∥α∥1 ≤ ∥α∥man.

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
Authors
Jean-François Lafont
Department of Mathematics
The Ohio State University
100 Math Tower
231 West 18th Avenue
Columbus, Ohio 43210-1174
United States
Christophe Pittet
Centre de Mathématiques et Informatique
Aix-Marseille Université
39 rue Frédéric Joliot-Curie
13453 Marseille Cedex 13
France