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Stable complexity and simplicial volume of manifolds

2012; Wiley; Volume: 5; Issue: 4 Linguagem: Inglês

10.1112/jtopol/jts026

1753-8424

Stefano Francaviglia, Roberto Frigerio, Bruno Martelli,

Homotopy and Cohomology in Algebraic Topology

Journal of TopologyVolume 5, Issue 4 p. 977-1010 Original articles Stable complexity and simplicial volume of manifolds Stefano Francaviglia, Corresponding Author Stefano Francaviglia [email protected] Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, ItalySearch for more papers by this authorRoberto Frigerio, Corresponding Author Roberto Frigerio [email protected] Dipartimento di Matematica 'L. Tonelli', Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, ItalySearch for more papers by this authorBruno Martelli, Corresponding Author Bruno Martelli [email protected] Dipartimento di Matematica 'L. Tonelli', Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, ItalySearch for more papers by this author Stefano Francaviglia, Corresponding Author Stefano Francaviglia [email protected] Dipartimento di Matematica, Università di Bologna, Piazza di Porta San Donato 5, 40126 Bologna, ItalySearch for more papers by this authorRoberto Frigerio, Corresponding Author Roberto Frigerio [email protected] Dipartimento di Matematica 'L. Tonelli', Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, ItalySearch for more papers by this authorBruno Martelli, Corresponding Author Bruno Martelli [email protected] Dipartimento di Matematica 'L. Tonelli', Università di Pisa, Largo Pontecorvo 5, 56127 Pisa, ItalySearch for more papers by this author First published: 23 December 2016 https://doi.org/10.1112/jtopol/jts026Citations: 15 2010 Mathematics Subject Classification 57M99, 57Q15 (primary), 53C23 (secondary). The authors are supported by the Italian FIRB project 'Geometry and topology of low-dimensional manifolds', RBFR10GHHH. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Abstract Let the Δ-complexity σ(M) of a closed manifold M be the minimal number of simplices in a triangulation of M. Such a quantity is clearly submultiplicative with respect to finite coverings, and by taking the infimum on all finite coverings of M normalized by the covering degree, we can promote σ to a multiplicative invariant, a characteristic number already considered by Milnor and Thurston, which we denote by σ∞(M) and call the stable Δ-complexity of M. We study here the relation between the stable Δ-complexity σ∞(M) of M and Gromov's simplicial volume ||M||. It is immediate to show that ||M|| ⩽ σ∞ (M) and it is natural to ask whether the two quantities coincide on aspherical manifolds with residually finite fundamental groups. We show that this is not always the case: there is a constant Cn < 1 such that ||M|| ⩽ Cnσ∞(M) for any hyperbolic manifold M of dimension n ⩾ 4. The question in dimension 3 is still open in general. We prove that σ∞(M) = ||M|| for any aspherical irreducible 3-manifold M whose JSJ decomposition consists of Seifert pieces and/or hyperbolic pieces commensurable with the figure-eight knot complement. The equality holds for all closed hyperbolic 3-manifolds if a particular three-dimensional version of the Ehrenpreis conjecture is true. Citing Literature Volume5, Issue4December 2012Pages 977-1010 RelatedInformation