Complexity of PL manifolds

Martelli, Bruno

doi:10.2140/agt.2010.10.1107

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Complexity of PL manifolds

Bruno Martelli

Algebraic & Geometric Topology 10 (2010) 1107–1164

DOI: 10.2140/agt.2010.10.1107

Abstract

We extend Matveev’s complexity of $3$ –manifolds to PL compact manifolds of arbitrary dimension, and we study its properties. The complexity of a manifold is the minimum number of vertices in a simple spine. We study how this quantity changes under the most common topological operations (handle additions, finite coverings, drilling and surgery of spheres, products, connected sums) and its relations with some geometric invariants (Gromov norm, spherical volume, volume entropy, systolic constant).

Complexity distinguishes some homotopically equivalent manifolds and is positive on all closed aspherical manifolds (in particular, on manifolds with nonpositive sectional curvature). There are finitely many closed hyperbolic manifolds of any given complexity. On the other hand, there are many closed $4$ –manifolds of complexity zero (manifolds without $3$ –handles, doubles of $2$ –handlebodies, infinitely many exotic K3 surfaces, symplectic manifolds with arbitrary fundamental group).

Keywords

complexity, spine

Mathematical Subject Classification 2000

Primary: 57Q99

Secondary: 57M99

References

Publication

Received: 5 November 2009

Revised: 19 April 2010

Accepted: 21 April 2010

Published: 23 May 2010

Authors

Bruno Martelli
Dipartimento di Matematica “Tonelli” Università di Pisa Largo Pontecorvo 5 56127 Pisa Italy
http://www.dm.unipi.it/~martelli

Volume 10, issue 2 (2010)